Classes of Finite Groups

Classes of Finite Groups Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and C...

Author: Adolfo Ballester-Bolinches | Luis M. Ezquerro

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Classes of Finite Groups

Mathematics and Its Applications

Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

Volume 584

Classes of Finite Groups by

Adolfo Ballester-Bolinches Universitat de València, València, Spain and

Luis M. Ezquerro Universidad Pública de Navarra, Pamplona, Spain

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 ISBN-13 ISBN-10 ISBN-13

1-4020-4718-5 (HB) 978-1-4020-4718-3 (HB) 1-4020-4719-3 (e-book) 978-1-4020-4719-0 (e-book)

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com

Printed on acid-free paper

All Rights Reserved © 2006 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in the Netherlands.

For the ones we love: Fran, Isabel, Eneko

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1

Maximal subgroups and chief factors . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Primitive groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 A generalisation of the Jordan-H¨ older theorem . . . . . . . . . . . . . . 40 1.3 Crowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 1.4 Systems of maximal subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2

Classes of groups and their properties . . . . . . . . . . . . . . . . . . . . . 87 2.1 Classes of groups and closure operators . . . . . . . . . . . . . . . . . . . . . 87 2.2 Formations: Basic properties and results . . . . . . . . . . . . . . . . . . . . 91 2.3 Schunck classes and projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 2.4 Fitting classes, Fitting sets, and injectors . . . . . . . . . . . . . . . . . . . 109 2.5 Fitting formations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

3

X-local formations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.1 X-local formations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.2 A generalisation of Gasch¨ utz-Lubeseder-Schmid-Baer theorem . . 144 3.3 Products of X-local formations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 3.4 ω-local formations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

4

Normalisers and prefrattini subgroups . . . . . . . . . . . . . . . . . . . . . 169 4.1 H-normalisers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.2 Normalisers of groups with soluble residual . . . . . . . . . . . . . . . . . 179 4.3 Subgroups of prefrattini type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

5

Subgroups of soluble type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 5.1 Subgroup functors and subgroups of soluble type: elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 5.2 Existence criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 5.3 Projectors of soluble type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

vii

viii

Contents

6

F-subnormality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 6.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 6.2 F-subnormal closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 6.3 Lattice formations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 6.4 F-subnormal subgroups and F-critical groups . . . . . . . . . . . . . . . . 265 6.5 Wielandt operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

7

Fitting classes and injectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 7.1 A non-injective Fitting class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 7.2 Injective Fitting classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 7.3 Supersoluble Fitting classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 7.4 Fitting sets, Fitting sets pairs, and outer Fitting sets pairs . . . . 339

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 Index of authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

Preface

. . . [El caballero andante] ha de saber las matem´ aticas, porque a cada paso se le ofrecer´ a tener necesidad dellas; . . . Miguel de Cervantes Saavedra Segunda parte del ingenioso caballero don Quijote de la Mancha, chapter 18

In the sixties and seventies of the last century, in parallel to the tremendous effort to classify the simple groups, a large number a papers created a beautiful and comprehensive view of finite soluble groups. In 1980, when the classification was almost completed, Helmut Wielandt proposed giving priority after the classification to the extension of these brilliant results of the theory of finite soluble groups to the more ambitious universe of all finite groups. Almost at the same time Klaus Doerk and Trevor Hawkes started to write a volume gathering, ordering, and systematising the rich stuff of soluble groups. This encyclopedic work took more than ten years to accomplish. The publication of Finite soluble groups (De Gruyter, 1992) is a crucial milestone in the history of the development of the theory of classes of finite soluble groups. In fact lots of separate pieces of the manuscript, generously distributed by the authors to all interested specialists, had a strong influence on the research of the area even before the publication of the volume. In the last decade, the Doerk-Hawkes’ book has been one of the most powerful tools for undertaking Wielandt’s task. The consequence is an impressive flourishing of ideas, methods and results illuminating the structure of finite groups. Furthermore, this process has produced a new arithmetic-free approach to understand some aspects of the soluble case. We believe that there is already a lot of work published in this area and consequently there is a need for a detailed account of the theory of classes of

ix

x

Preface

groups in the general finite universe. The present book represents an attempt to meet this need. Our main objective in this book is to present the latest achievements and investigations continuing the Doerk-Hawkes book to enlarge and adapt the methods of the soluble case to classes of finite non-necessarily soluble, according to Wielandt’s proposal. The contents of the book are organised in seven chapters. Chapter 1 begins with primitive groups and crowns. These concepts are central to our approach. It continues with the study of solid sets and systems of maximal subgroups. They are, together with the generalised Jordan-H¨ older theorem, the ingredients combined to introduce the prefrattini subgroups in Chapter 4. Chapter 2 contains definitions, and elementary and basic results on classes of groups. Chapter 3 deals with partially saturated formations. A unified extension of the theorems of Gasch¨ utz-Lubeseder-Schmid and Baer on the local character of the saturated and solubly saturated formations is presented there. Normalisers associated with Schunck classes H of the form EΦ F for some formation F and prefrattini subgroups associated to arbitrary Schunck classes are studied in Chapter 4, whereas Chapter 5 is devoted to presenting an alternative approach to a theory of projectors and covering subgroups in arbitrary finite groups resembling the corresponding theory in finite soluble groups. It is based on Salomon’s Dissertation Strukturerhaltende Untergruppen, Schunckklassen und extreme Klassen endlicher Gruppen, Johannes Gutenberg-Universität, Mainz, 1987. Subnormal subgroups associated to formations are the main theme of Chapter 6. This concept was introduced by Hawkes in 1969 in the soluble universe and it turns out to be very useful in the study of the structure of finite groups. The last chapter contains some of the recent developments of the theory of Fitting classes, focusing our attention on injective Fitting classes and supersoluble Fitting classes. In particular, a detailed account of Salomon’s unpublished example of a non-injective Fitting class is included. To end this preface, we would like to pay a tribute to the figure of Professor Klaus Doerk, recently deceased. Without Doerk and his research team’s collaboration, this book would have never ever come to be.

Acknowledgements We would like to conclude by expressing our deepest gratitude to Ram´ on Esteban-Romero for his patient work with our manuscripts. His knowledge on this book’s issues as well as his master skills on the use of TEX made him the best helper for the meticulous task of editing this book. We would also like to thank Homer Bechtell, John Cossey, Arny Feldman, Mar´ıa Jesu´ s Iranzo, Paz Jimé nez-Seral, Carmen Lacasa-Esteban, Julio Lafuente, Inmaculada Lizasoain, Mar´ıa del Carmen Pedraza-Aguilera, Tatiana Pedraza and Francisco Pérez-Monasor for their valious collaborations, as well as to the Ministerio de Educaci´ on y Ciencia (Spanish Government)

Preface

xi

and FEDER (European Union) for their financial support via the grants MTM2004-08219-C02-01 and MTM2004-08219-C02-02. To conclude, we must thank Springer for converting this project into a reality and for their continuous patience and help while writing this book.

Torres-Torres, Pamplona, January, 2006

A. Ballester-Bolinches Luis M. Ezquerro

1 Maximal subgroups and chief factors

1.1 Primitive groups This book, devoted to classes of finite groups, begins with the study of a class, the class of primitive groups, with no hereditary properties, the usual requirement for a class of groups, but whose importance is overwhelming to understand the remainder. We shall present the classification of primitive groups made by R. Baer and the refinement of this classification known as the O’NanScott Theorem. The book of H. Kurzweil and B. Stellmacher [KS04], recently appeared, presents an elegant proof of this theorem. Our approach includes the results of F. Gross and L. G. Kov´ acs on induced extensions ([GK84]) which are essential in some parts of this book. We will assume our reader to be familiar with the basic concepts of permutation representations: G-sets, orbits, faithful representation, stabilisers, transitivity, the Orbit-Stabiliser Theorem, . . . (see [DH92, A, 5]). In particular we recall that the stabilisers of the elements of a transitive G-set are conjugate subgroups of G and any transitive G-set Ω is isomorphic to the G-set of right cosets of the stabiliser of an element of Ω in G. Definition 1.1.1. Let G be a group and Ω a transitive G-set. A subset Φ ⊆ Ω is said to be a block if, for every g ∈ G, we have that Φg = Φ or Φg ∩ Φ = ∅. Given a G-set Ω, trivial examples of blocks are ∅, Ω and any subset with a single element {ω}, for any ω ∈ Ω. In fact, these are called trivial blocks. Proposition 1.1.2. Let G be a group which acts transitively on a set Ω and ω ∈ Ω. There exists a bijection {block Φ of Ω : ω ∈ Φ} −→ {H ≤ G : Gω ≤ H} which preserves the containments. 1

2


Proof. Given a block Φ in Ω such that ω ∈ Φ, then GΦ = {g ∈ G : Φg = Φ} is a subgroup of G and the stabiliser Gω is a subgroup of GΦ . Conversely, if H is a subgroup of G containing Gω , then the set Φ = {ω h : h ∈ H} is a block and ω ∈ Φ. These are the mutually inverse bijections required. The following result is well-known and its proof appears, for instance, in Huppert’s book [Hup67, II, 1.2]. Theorem 1.1.3. Let G be a group which acts transitively on a set Ω and assume that Φ is a non-trivial block of the action of G on Ω. Set H = {g ∈ G : Φg = Φ}. Then H is a subgroup of G. Let T be a right transversal of H in G. Then 1. {Φt : t ∈ T } is a partition of Ω. 2. We have that |Ω| = |T ||Φ|. In particular |Φ| divides |Ω|. 3. The subgroup H acts transitively on Φ. Notation 1.1.4. If H is a subgroup of a group G, the core of H in G is the subgroup Hg. CoreG (H) = g∈G

Along this chapter, in order to make the notation more compact, the core of a subgroup H in a group G will often be denoted by HG instead of CoreG (H). Theorem 1.1.5. Let G be a group. The following conditions are equivalent: 1. G possesses a faithful transitive permutation representation with no nontrivial blocks; 2. there exists a core-free maximal subgroup of G. Proof. 1 implies 2. Suppose that there exists a transitive G-set Ω with no non-trivial blocks and consider any ω ∈ Ω. The action of G on Ω is equivalent to the action of G on the set of right cosets of Gω in G. The kernel of this action is CoreG (Gω ) and, by hypothesis, is trivial. By Proposition 1.1.2, if H is a subgroup containing Gω , there exists a block Φ = {ω h : h ∈ H} of Ω such that ω ∈ Φ and H = GΦ = {g ∈ G : Φg = Φ}. Since G has no non-trivial blocks, either Φ = {ω} or Φ = Ω. If Φ = {ω}, then Gω = H and if Φ = Ω, then H = GΩ = G. Hence the stabiliser Gω is a core-free maximal subgroup of G. 2 implies 1. If U is a core-free maximal subgroup of G, then the action of G on the set of right cosets of U in G is faithful and transitive. By maximality of U , this action has no non-trivial blocks by Proposition 1.1.2. Definitions 1.1.6. A a faithful transitive permutation representation of a group is said to be primitive if it does not have non-trivial blocks. A primitive group is a group which possesses a primitive permutation representation. Equivalently, a group is primitive if it possesses a core-free maximal subgroup.

1.1 Primitive groups

3

A primitive pair is a pair (G, U ), where G is a primitive group and U a core-free maximal subgroup of G, Each conjugacy class of core-free maximal subgroups affords a faithful transitive and primitive permutation representation of the group. Thus, in general, it is more precise to speak of primitive pairs. Consider, for instance, the alternating group of degree 5, G = Alt(5). There exist three conjugacy classes of maximal subgroups, namely the normalisers of each type of Sylow subgroup. Obviously all of them are core-free. This gives three non-equivalent primitive representations of degrees 5 (for the normalisers of the Sylow 2subgroups), 10 (for the normalisers of the Sylow 3-subgroups) and 6 (for the normalisers of the Sylow 5-subgroups). The remarkable result that follows, due to R. Baer, classifies all primitive groups (a property defined in terms of maximal subgroups) according to the structure of the socle, i.e. the product of all minimal normal subgroups. Theorem 1.1.7 ([Bae57]). 1. A group G is primitive if and only if there exists a subgroup M of G such that G = M N for all minimal normal subgroups N of G. 2. Let G be a primitive group. Assume that U is a core-free maximal subgroup of G and that N is a non-trivial normal subgroup of G. Write C = CG (N ). Then C ∩ U = 1. Moreover, either C = 1 or C is a minimal normal subgroup of G. 3. If G is a primitive group and U is a core-free maximal subgroup of G, then exactly one of the following statements holds: a) Soc(G) = S is a self-centralising abelian minimal normal subgroup of G which is complemented by U : G = U S and U ∩ S = 1. b) Soc(G) = S is a non-abelian minimal normal subgroup of G which is supplemented by U : G = U S. In this case CG (S) = 1. c) Soc(G) = A × B, where A and B are the two unique minimal normal subgroups of G and both are complemented by U : G = AU = BU and A ∩ U = B ∩ U = A ∩ B = 1. In this case A = CG (B), B = CG (A), and A, B and AB ∩ U are non-abelian isomorphic groups. Proof. 1. If G is a primitive group, and U is a core-free maximal subgroup of G, then it is clear that G = U N for every minimal normal subgroup N of G. Conversely, if there exists a subgroup M of G, such that G = M N for every minimal normal subgroup N of G and U is a maximal subgroup of G such that M ≤ U , then U cannot contain any minimal normal subgroup of G, and therefore U is a core-free maximal subgroup of G. 2. Since U is core-free in G, we have that G = U N . Since N is normal, then C is normal in G and then C ∩ U is normal in U . Since C ∩ U centralises N , then C ∩ U is in fact normal in G. Therefore C ∩ U = 1. If C = 1, consider a minimal normal subgroup X of G such that X ≤ C. Since X is not contained in U, then G = XU. Then C = C∩XU = X(C∩U) = X.

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3. Let us assume that N1 , N2 , and N3 are three pairwise distinct minimal normal subgroups. Since N1 ∩ N2 = N1 ∩ N3 = N2 ∩ N3 = 1, we have that N2 ×N3 ≤ CG (N1 ). But then CG (N1 ) is not a minimal normal subgroup of G, and this contradicts 2. Hence, in a primitive group there exist at most two distinct minimal normal subgroups. Suppose that N is a non-trivial abelian normal subgroup of G. Then N ≤ CG (N ). Since by 2, CG (N ) is a minimal normal subgroup of G, we have that N is self-centralising. Thus, in a primitive group G there exists at most one abelian minimal normal subgroup N of G. Moreover, G = N U and N is self-centralising. Then N ∩ U = CG (N ) ∩ U = 1. If there exists a unique minimal non-abelian normal subgroup N , then G = N U and CG (N ) = 1. If there exist two minimal normal subgroups A and B, then A ∩ B = 1 and then B ≤ CG (A) and A ≤ CG (B). Since CG (A) and CG (B) are minimal normal subgroups, we have that B = CG (A) and A = CG (B). Now A ∩ U = CG (B) ∩ U = 1 and B ∩ U = CG (A) ∩ U = 1. Hence G = AU = BU . Since A = CG (B), it follows that B is non-abelian. Analogously we have that A is non-abelian. By the Dedekind law [DH92, I, 1.3], we have A(AB ∩ U ) = AB = B(AB ∩ U ). Hence A ∼ = A/(A ∩ B) ∼ = AB/B ∼ = B(AB ∩ U )/B = AB ∩ U . Analogously ∼ B = AB ∩ U . Baer’s theorem enables us to classify the primitive groups as three different types. Definition 1.1.8. A primitive group G is said to be 1. a primitive group of type 1 if G has an abelian minimal normal subgroup, 2. a primitive group of type 2 if G has a unique non-abelian minimal normal subgroup, 3. a primitive group of type 3 if G has two distinct non-abelian minimal normal subgroups. We say that G is a monolithic primitive group if G is a primitive group of type 1 or 2. Definition 1.1.9. Let U be a maximal subgroup of a group G. Then U/UG is a core-free maximal subgroup of the quotient group G/UG . Then U is said to be 1. a maximal subgroup of type 1 if G/UG is a primitive group of type 1, 2. a maximal subgroup of type 2 if G/UG is a primitive group of type 2, 3. a maximal subgroup of type 3 if G/UG is a primitive group of type 3. We say that U is a monolithic maximal subgroup if G/UG is a monolithic primitive group.


5

Obviously all primitive soluble groups are of type 1. For these groups, there exists a well-known description called Galois’ theorem. The proof appears in Huppert’s book [Hup67, II, 3.2 and 3.3]. Theorem 1.1.10. 1. (Galois) If G is a soluble primitive group, then all corefree maximal subgroups are conjugate. 2. If N is a self-centralising minimal normal subgroup of a soluble group G, then G is primitive, N is complemented in G, and all complements are conjugate. Remarks 1.1.11. 1. The statement of Theorem 1.1.10 (1) is also valid if G is p-soluble for all primes dividing the order of Soc(G). 2. If G is a primitive group of type 1, then its minimal normal subgroup N is an elementary abelian p-subgroup for some prime p. Hence, N is a vector space over the field GF(p). Put dim N = n, i.e. |N | = pn . If M is a core-free subgroup of G, then M is isomorphic to a subgroup of Aut(N ) = GL(n, p). Therefore G can be embedded in the affine group AGL(n, p) = [Cpn ] GL(n, p) in such a way that N is the translation group and G∩GL(n, p) acts irreducibly on N . Thus, clearly, primitive groups of type 1 are not always soluble. 3. In his book B. Huppert shows that the affine group AGL(3, 2) = [C2 × C2 × C2 ] GL(3, 2) is an example of a primitive group of type 1 with nonconjugate core-free maximal subgroups (see [Hup67, page 161]). 4. Let G be a primitive group of type 2. If N is the minimal normal subgroup of G, then N is a direct product of copies of some non-abelian simple group and, in particular, the order of N has more than two prime divisors. If p is a prime dividing the order of N and P ∈ Sylp (N ), then G = NG (P )N by the Frattini argument. Since P is a proper subgroup of N , then NG (P ) is a proper subgroup of G. If U is a maximal subgroup of G such that NG (P ) ≤ U , then necessarily U is core-free. Observe that if P0 ∈ Sylp (G) such that P ≤ P0 , then P = P0 ∩ N is normal in P0 and so P0 ≤ U . In other words, U has p -index in G. This argument can be done for each prime dividing |N |. Hence, the set of all core-free maximal subgroups of a primitive group of type 2 is not a conjugacy class. 5. In non-soluble groups, part 2 of Theorem 1.1.10 does not hold in general. Let G be a non-abelian simple group, p a prime dividing |G| and P ∈ Sylp (G). Suppose that P is cyclic. Let GΦ,p be the maximal Frattini extension of G with p-elementary abelian kernel A = Ap (G) (see [DH 92; Appendix β] for details of this construction). Write J = J(KG) for the Jacobson radical of the group algebra KG of G, over the field K = GF(p). Then the section N = A/AJ is irreducible and CG (N ) = Op ,p (G) = 1. Consequently GΦ,p /AJ is a group with a unique minimal normal subgroup, isomorphic to N , self-centralising and non-supplemented. In primitive groups of type 1 or 3, the core-free maximal subgroups complement each minimal subgroup. This characterises these types of primitive groups. In case of primitive groups of type 2 we will see later that the minimal

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normal subgroup could be complemented by some core-free maximal subgroup in some cases; but even then, there are always core-free maximal subgroups supplementing and not complementing the socle. Proposition 1.1.12 ([Laf84a]). For a group G, the following are pairwise equivalent: 1. G is a primitive group of type 1 or 3; 2. there exists a minimal normal subgroup N of G complemented by a subgroup M which also complements CG (N ); 3. there exists a minimal normal subgroup N of G such that G is isomorphic to the semidirect product X = [N ] G/ CG (N ) . Proof. Clearly 1 implies 2. For 2 implies 1 observe that, since N ∩ MG = 1, then MG ≤ CG (N ). But, since also MG ∩ CG (N ) = 1, we have that MG = 1. Suppose that S is a proper subgroup of G such that M ≤ S. Then the subgroup S ∩ N is normal in S and is centralised by CG (N ). Hence S ∩ N is normal in S CG (N ) = G. By minimality of N , we have that S ∩ N = 1 and then S = M . Then M is a core-free maximal subgroup of G and the group G is primitive. Observe that the minimal normal subgroup of a primitive group of type 2 has trivial centraliser. ∼ 2 implies 3. Observe that G = NM , with N ∩M = 1, and M = G/ C G (N ). α The map α : G −→ [N ] G/ CG (N ) given by (nm) = n, m CG (N ) is the desired isomorphism. 3 implies 2. Write C = CG (N ). Assume that there exists an isomorphism α : [N ](G/C) −→ G α and consider the following subgroups N ∗ = {(n, C) : n ∈ N } , M ∗ = α α {(1, gC) : g ∈ G} , and C ∗ = {(n, gC) : ng ∈ C} . For each n ∈ N , −1 α ∗ the element (n , nC) is a non-trivial element of C . Hence C ∗ = 1. It is an easy calculation to show that N ∗ is a minimal normal subgroup of G, C ∗ = CG (N ∗ ) and M ∗ complements N ∗ and C ∗ . Corollary 1.1.13. The following conditions for a group G are equivalent: 1. G is a primitive group of type 3. 2. The group G possesses two distinct minimal normal subgroups N1, N2, such that a) N1 and N2 have a common complement in G; b) the quotient groups G/Ni , for i = 1, 2, are primitive groups of type 2. Proof. 1 implies 2. By Theorem 1, if G is a primitive group of type 3, then G possesses two distinct minimal normal subgroups N1 , N2 which have a common complement M in G. Observe that M ∼ = G/N1 and N2 N1 /N1 is a minimal normal subgroup of G/N1 . If gN1 ∈ CG/N1 (N2 N1 /N1 ), then [n, g] ∈ N1 , for all n ∈ N2 . But then [n, g] ∈ N1 ∩N2 = 1, and therefore g ∈ CG (N2 ) = N1 . Hence


7

CG/N1 (N2 N1 /N1 ) = 1. Consequently G/N1 is a primitive group of type 2 and therefore so are M and G/N2 . 2 implies 1. Let M be a common complement of N1 and N2 . Then G/Ni ∼ = M is a primitive group of type 2 such that Soc(G/Ni ) = N1 N2 /Ni and CG (N1 N2 /Ni ) = Ni . Therefore CG (N2 ) = N1 and CG (N1 ) = N2 . By Proposition 1.1.12, this means that G is a primitive group of type 3. Proposition 1.1.14 ([Laf84a]). For a group G, the following statements are pairwise equivalent. 1. G is a primitive group of type 2. 2. G possesses a minimal normal subgroup N such that CG (N ) = 1. 3. There exists a primitive group X of type 3 such that G ∼ = X/A for a minimal normal subgroup A of X. Proof. 3 implies 2 is Corollary 1.1.13 and 2 implies 1 is the characterisation of primitive groups of type 2 in Theorem 1. Thus it only remains to prove that 1 implies 3. If G is a primitive group of type 2 and N is the unique minimal normal subgroup of G, then N is non-abelian and CG (N ) = 1. By Proposition 1.1.12, the semidirect product X = [N ]G is a primitive group of type 3. Clearly if A = {(n, 1) : n ∈ N }, then X/A ∼ = G. Consequently, if M is a core-free maximal subgroup of a primitive group G of type 3, then M is a primitive group of type 2 and Soc(M ) is isomorphic to a minimal normal subgroup of G. According to Baer’s Theorem, the socle of a primitive group of type 2 is a non-abelian minimal normal subgroup and therefore is a direct product of copies of a non-abelian simple group (see [Hup67, I, 9.12]). Obviously, the simplest examples of primitive groups of type 2 are the non-abelian simple groups. Observe that if S is a non-abelian simple group, then Z(S) = 1 and we can identify S and the group of inner automorphisms Inn(S) and write S ≤ Aut(S). Since CAut(S) (S) = 1, any group G such that S ≤ G ≤ Aut(S) is a primitive group of type 2 such that Soc(G) is a non-abelian simple group. Conversely, if G is a primitive group of type 2 and S = Soc(G) is a simple group, then, since CG (S) = 1, we can embed G in Aut(S). Definition 1.1.15. An almost simple group G is a subgroup of Aut(S) for some simple group S, such that S ≤ G. If G is an almost simple group and S ≤ G ≤ Aut(S), for a non-abelian simple group S, then CG (S) = 1. Hence G possesses a unique minimal normal subgroup S and every maximal subgroup U of G such that S ≤ U is core-free in G. Proposition 1.1.16. Suppose that S is a non-abelian simple group and let G be an almost simple group such that S ≤ G ≤ Aut(S). If U is a core-free maximal subgroup of G, then U ∩ S = 1.

8


Proof. Recall Schreier’s conjecture ([KS04, page 151]) which states that the group of outer automorphisms Out(S) = Aut(S)/ Inn(S) of a non-abelian simple group S is always soluble. The classification of simple groups has allowed us to check that this conjecture is true. Suppose that U ∩ S = 1. We know that U ∼ = U S/S ≤ Aut(S)/ Inn(S) and, by Schreier’s conjecture ([KS04, page 151]) we deduce that U is soluble. Let Q be a minimal normal subgroup of U . Then Q is an elementary abelian q-group for some prime q. Observe that CG (Q) is normalised by U . Therefore CS (Q) is normalised by U and then U CS (Q) is a subgroup of G. Since U is maximal in G and CG (S) = 1, then CS (Q) = 1. The q-group Q acts fixed-point-freely on S and then S is a q -group. By the Odd Order Theorem ([FT63]), we have that q = 2. Now Q acts by conjugation on the elements of the set Syl2 (S) and by the Orbit-Stabiliser Theorem ([DH92, A, 5.2]) we deduce that Q normalises −1 some P ∈ Syl2 (S). If P and P x , for x ∈ S, are two Sylow 2-subgroups of S which are normalised by Q, then Q, Qx ∈ Sylq NQS (P ) and there exists an element g ∈ NQS (P ), such that Qg = Qx . Write g = yz, with y ∈ Q and z ∈ S. Then Qx = Qz with z ∈ NS (P ). Hence [Q, xz −1 ] ≤ Q ∩ S = 1 and xz −1 ∈ CS (Q) = 1. Therefore x = z ∈ NS (P ) and we conclude that Q normalises exactly one Sylow 2-subgroup P of S. Hence NG (Q) ≤ NG (P ). But U = NG (Q), by maximality of U . The subgroup U P is a proper subgroup of G which contains properly the maximal subgroup U . This is a contradiction. Hence U ∩ S = 1. For our purposes, it will be necessary to embed the primitive group G in a larger group. Suppose that Soc(G) = S1 × · · · × Sn , where the Si are copies ∼ n of a non-abelian simple group S, i.e. Soc(G) = S , the direct product ofn n copies of S. Since CG Soc(G) = 1, the group G can be embedded in Aut(S ). The automorphism group of a direct product of copies of a non-abelian simple group has a well-known structure: it is a wreath product. Thus, the study of some relevant types of subgroups of groups which are wreath products and the analysis of some special types of subgroups of a direct product of isomorphic non-abelian simple groups will be essential. Definition 1.1.17. Let X and H be two groups and suppose that H has a permutation representation ϕ on a finite set I = {1, . . . , n} of n elements. The wreath product X ϕ H (or simply X H if the action is well-known) is the semidirect product [X ]H, where X is the direct product of n copies of X: X = X1 × · · · × Xn , with Xi = X for all i ∈ I, and the action is (x1 , . . . , xn )h = (x1(h−1 )ϕ , . . . , xn(h−1 )ϕ ) for h ∈ H and xi ∈ X, for all i ∈ I. The subgroup X is called the base group of X H. Remarks 1.1.18. Consider a wreath product G = X ϕ H. 1. If ϕ is faithful, then CG (X ) ≤ X .

(1.1)


9

2. For any g ∈ G, then g = xh, with x ∈ X and h ∈ H. For each i = 1, . . . , n, we have that Xig = Xih = Xihϕ . 3. Thus, the group G acts on I by the following rule: if i ∈ I, for any ϕ ϕ g = xh ∈ G, with x ∈ X and h ∈ H, then ig = ih . In particular ih = ih , if h ∈ H. 4. If S ⊆ I, then write Xj πS : X −→ j∈S

for the projection of X onto we have that

j∈S

Xj . Then for any y ∈ X and any g ∈ G,

(y g )πS g = (y πS )g . Proposition 1.1.19. Let S be a non-abelian simple group and write S n = S1 × · · · × Sn for the direct product of n copies S1 , . . . , Sn of S, for some positive integer n. Then the minimal normal subgroups of S n are exactly the Si , for any i = 1, . . . , n, Proof. Let N be a minimal normal subgroup of S n . Suppose that N ∩ Si = 1 for all i = 1, . . . , n. Then N centralises all Si and hence N ≤ Z(S n ) = 1. This is a contradiction. Therefore N ∩ Si = N for some index i. Then N = Si . Proposition 1.1.20. Let S be a non-abelian simple group and write S n = S1 × · · · × Sn for the direct product of n copies S1 , . . . , Sn of S, for some positive integer n. Then Aut(S n ) ∼ = Aut(S) Sym(n), where Sym(n) is the symmetric group of degree n. Proof. If σ is a permutation in Sym(n), the map ασ defined by (x1 , . . . , xn )ασ = (x1σ−1 , . . . , xnσ−1 ) is an element of Aut (S n ) associated with σ. Now H = {ασ ∈ Aut (S n ): σ ∈ Sym(n)} is a subgroup of Aut(S n ) and σ −→ ασ defines an isomorphism between Sym(n) and H. By Proposition 1.1.19, the minimal normal subgroups of the direct product S1 × · · · × Sn are exactly the S1 , . . . , Sn . Therefore, if γ ∈ Aut(S n ), then there exists a σ ∈ Sym(n) such that Siγ = Siσ = Siασ , for all i = 1, . . . , n. Let D be the subgroup of all elements β in Aut(S n ) such that Siβ = Si for all i. The maps β1 , . . . , βn defined by (x1 , . . . , xn )β = (xβ1 1 , . . . , xβnn ) are automorphisms of S and the map β → (β1 , . . . , βn ) defines an isomorphism between D and Aut(S)n . Moreover, by Proposition 1.1.19 again, if β ∈ D and γ ∈ Aut(S n ), then (Siγ )β = Siγ . This means that D is a normal subgroup of Aut(S n ). Observe that ασ ∈ D if and only if σ = 1, or, in other words, D ∩ H = 1. Moreover for all γ ∈ Aut(S n ), we have that γασ−1 ∈ D. Therefore Aut(S n ) = [D]H. This allows us to define a bijective map between Aut(S n ) and Aut(S) Sym(n) which is an isomorphism.

10


F. Gross and L. G. Kov´ acs published in [GK84] a construction of groups, the so-called induced extensions, which is crucial to understand the structure of a, non-necessarily finite, group that possesses a normal subgroup which is a direct product of copies of a group. It is clear that primitive groups of type 2 are examples of this situation. We present in the sequel an adaptation of this construction to finite groups. Proposition 1.1.21. Consider the following diagram of groups and group homomorphisms: (1.2) Z g

X

f

/Y

where g is a monomorphism. Let G be the following subset of X: G = {x ∈ X : xf = z g for some z ∈ Z}, and the following mapping h : G −→ Z

xh = xf g

−1

for every x ∈ G.

Then G is a subgroup of X and h is a well-defined group homomorphism such that the following diagram of groups and group homomorphisms is commutative: h / Z G g

ι

X

f

/Y

(where ι is the canonical inclusion of G in X). Moreover Ker(hι ) = Ker(f ). Further, if (G0 , ι0 , h0 ) is a triple, with G0 a group, ι0 : G0 −→ X a monomorphism and h0 : G0 −→ Z is a group homomorphism, such that the diagram G0

h0

/Z

f

/Y

g

ι0

X

is commutative, then there exists a monomorphism Φ : G0 −→ G, such that Φ ι Φh = h0 , Φι = ι0 and Ker(h0 ) ≤ Ker(h) = Ker(f ). Proof. It is an easy exercise to prove that G is a subgroup of X and, since g is a monomorphism, the mapping h is a well-defined group homomorphism. It is not difficult to see that Ker(h)ι = Ker(f ). For the second statement, let x ∈ G0 and observe that xh0 is an element of Z such that (xh0 )g = (xι0 )f , and then xι0 ∈ G and (xι0 )h = xh0 . Write Φ : G0 −→ G such that xΦ = xι0 .


11

Definition 1.1.22. The triple (G, ι, h) introduced in Proposition 1.1.21 is said to be the pull-back of the diagram (1.2). Proposition 1.1.23. Consider the following extension of groups: 1

/X

/K

/1

/Y

f

and a monomorphism g : Z −→ Y . Consider the triple (G, ι, h), the pull-back of the diagram (1.2). 1. There exists an extension Eg : 1

/K

/G

h

/1

/Z

such that the following diagram of groups and group homomorphisms is commutative: Eg : 1

/K

E: 1

/K

/K

id

h

/Z

/X

f

/Y

/ G0

h0

/Z

/G

/1

g

ι

/1

2. Moreover, if E0 : 1

/1

is another extension such that the diagram E0 : 1

/K

E: 1

/K

id

/ G0

h0

/Z

f

/Y

g

ι0

/X

/1 /1

is commutative, there exists a group isomorphism Φ : G0 −→ G such that Φh = h0 , Φι = ι0 and Φ|K = idK . Proof. The proof of 1 is a direct exercise. To see 2, first notice that, by the Short Five Lemma ([Hun80, IV, 1.17]), the homomorphism ι0 is a monomorphism. By Proposition 1.1.21, there exists a group monomorphism Φ : G0 −→ G such that Φh = h0 , Φι = ι0 and Φ|K = idK . Furthermore, since |G| = |Z|/|K| = |G0 |, we have that Φ is an isomorphism. Definition 1.1.24. The extension Eg is said to be the pull-back extension of the extension E and the monomorphism g. Hypotheses 1.1.25. Let B be a group. Assume that C a subgroup of a group B such that |B : C| = n and let T = {t1 = 1, . . . , tn } be a right transversal

12


of C in B. Then B, acting by right multiplication on the set of right cosets of C in B, induces a transitive action ρ : B −→ Sym(n) on the set of indices I = {1, . . . , n} in the following way. For each i ∈ I and each h ∈ B, the element ti h belongs to some coset Ctj , i.e. ti h = ci,h tj , for some ci,h ∈ C. ρ Then ih = j. Write P = B ρ ≤ Sym(n). Let α : A −→ B be a group homomorphism and write C = Aα and S = ¯ : A ρ P −→ Ker(α). Write W = Aρ P . Thereexists an induced epimorphism α α ¯ α C ρ P defined by (a1 , . . . , an )x = (aα , . . . , a )x, for a , . . . , an ∈ A and 1 n 1 x ∈ P . Write M = Ker(¯ α). Observe that (a1 , . . . , an )x ∈ M if and only ¯ = if aα j = 1, for all j ∈ I and x = 1. This is to say that M = Ker(α) Ker(α) × . . . × Ker(α) = S1 × . . . × Sn . We have the exact sequence: E: 1

/ A ρ P

/M

α ¯

/1

/ C ρ P

Lemma 1.1.26. Assume the hypotheses and notation of Hypotheses 1.1.25. 1. The mapping λ = λT : B −→ C ρ P such that hλ = (c1,h , . . . , cn,h )hρ , for any h ∈ B, is a group monomorphism. 2. Consider the pull-back exact sequence Eλ: Eλ : 1

/M

E: 1

/M

/G

id

/ A ρ P

σ

α ¯

/1

/B

λ

/ C ρ P

/1

Then, the isomorphism class of the group G is independent from the choice of transversal of C in B. Proof.

1. Let h, h ∈ B. Observe that ci,hh ti(hh )ρ = ti hh = ci,h tihρ h = ci,h cihρ ,h ti(hh )ρ .

Hence, by (1.1) in Definition 1.1.17, we have that hλ hλ = (c1,h , . . . , cn,h )hρ (c1,h , . . . , cn,h )h

ρ

ρ −1

= (c1,h , . . . , cn,h )(c1,h , . . . , cn,h )(h ) (hh )ρ = (c1,h , . . . , cn,h )(c1hρ ,h , . . . , cnhρ ,h )(hh )ρ = (c1,hh , . . . , cn,hh )(hh )ρ = (hh )λ and λ is a group homomorphism. ρ Suppose that hλ = hλ . Then (c1,h , . . . , cn,h )hρ = (c1,h , . . . , cn,h )h and n therefore, since C ρ Pn = [C ]Pn is a semidirect product, we have that cj,h = cj,h = cj ,

j ∈ I;

hρ = hρ = τ.

Therefore, for any index j ∈ I, we have that tj h = cj tj τ = tj h and then h = t−1 j cj tj τ = h . Hence λ is a group monomorphism.


13

2. Let T = {t1 , . . . , tn } be some other right transversal of C in B such that Cti = Cti , for each i ∈ I: there exist elements b1 , . . . , bn ∈ C such that ti = bi ti , for i = 1, . . . , n. For each i ∈ I and each h ∈ B, the element ti h ρ belongs to the coset Ctj = Ctj , for ih = j, and ti h = ci,h tj , for some ci,h ∈ C. Then ti h = bi ti h = bi ci,h tj = ci,h tj = ci,h bj tj

and ci,h = b−1 i ci,h bj

and it appears the element (b1 , . . . , bn ) ∈ C associated with T . Then, for λ = λT , we have that hλ = (c1,h , . . . , cn,h )hρ = (b1 , . . . , bn )−1 (c1,h , . . . , cn,h )(b1hρ , . . . , bnhρ ) hρ −1 ρ = (b1 , . . . , bn )−1 (c1,h , . . . , cn,h )(b1 , . . . , bn )(h ) hρ = (b1 , . . . , bn )−1 (c1,h , . . . , cn,h )hρ (b1 , . . . , bn ) (b1 ,...,bn ) = (c1,h , . . . , cn,h )hρ = (hλ )(b1 ,...,bn ) , (b1 ,...,bn ) = Im(λ). For each i ∈ I, let ai be an for any h ∈ B, and then Im(λ ) element of A such that aiα = bi . This is to say that (a1 , . . . , an )α¯ = (b1 , . . . , bn ). If x ∈ G, then

(x(a1 ,...,an ) )α¯ = (xα¯ )(b1 ,...,bn ) = (hλ )(b1 ,...,bn ) = hλ

and then x(a1 ,...,an ) ∈ G∗ = {w ∈ W : wα¯ = hλ for some h ∈ B}, which is the pull-back defined with the monomorphism λ : Eλ : 1

/M id

E: 1

/M

/ G∗

/ A ρ P

σ

α ¯

/B

/1

λ

/ C ρ P

/1

Thus, G∗ = Ga for some a ∈ A associated with the transversals T and T , i.e. the pull-back groups constructed from two different transversals are conjugate in W . In other words, the isomorphism class of the group G is independent from the choice of transversal.

Definition 1.1.27 ([GK84]). In the above situation and with that notation, we will say that Eλ is the induced extension defined by α : A −→ B. Recall that G is a subgroup of W = A ρ P defined by: G = {x ∈ W : xα¯ = hλ , for some h ∈ B} and σ is defined by σ = α ¯ |G λ−1 .

14


Proposition 1.1.28. With the notation introduced above, we have the following. 1. NG (A1 ) = NG (S1 ) = NG (S2 × · · · × Sn ) = N = {x ∈ W : xα¯ = hλ , for some h ∈ C}. 2. N/(S2 × · · · × Sn ) ∼ = A. Moreover, the image of M/(S2 × · · · × Sn ) under this isomorphism is S = Ker(α). 3. In particular N σ = C and |G : N | = |B : C| = n. Thus, if ρ : G −→ Sym(n) is the action of G on the right cosets of N in G by multiplication, then ρ = σρ. 4. The set {S1 , . . . , Sn } is the conjugacy class of the subgroup S1 in G. Proof.

1. We can consider the subgroup N = {w ∈ W : wα¯ = hλ , for some h ∈ C}.

Observe that if (a1 , . . . , an )x ∈ N , for ai ∈ A and x ∈ P , then there exists h ∈ C, such that α hλ = (c1,h , . . . , cn,h )hρ = (aα 1 , . . . , an )x.

Since h ∈ C, it is clear that c1,h = h and hρ belongs to the stabiliser P1 of 1. In other words N ≤ A1 × (A2 × · · · × An )P1 = NW (A1 ) = NW (S1 ) = NW (S2 × · · · × Sn ) and hence N ≤ NG (A1 ). Conversely, if (a1 , . . . , an )x ∈ NG (A1 ), then x ∈ P1 ρ hρ and there exists h ∈ B such that aα = 1. i = ci,h and x = h ∈ P1 , i.e. 1 Hence h = t1 h = c1,h t1 = c1,h = aα 1 ∈ C. Then NG (A1 ) ≤ N . Hence N = NG (A1 ) = NG (S1 ). 2. Consider the projection e1 : A1 × (A2 × · · · × An )P1 = NW (A1 ) −→ A on the first component. Obviously, Ker(e1 ) = (A2 × · · · × An )P1 . Let e be the restriction to N of the projection e1 : e = e1 |N : N −→ A. Observe that if x ∈ N , then xα¯ = cλ for some c ∈ C. We can characterise this c = xσ in the following way. Assume that x = (a1 , . . . , an )y. Then xα¯ = α λ ρ α eα (aα 1 , . . . , an )y = c = (c, c2,c , . . . , cn,c )c . Hence c = a1 = x . We have that Ker(e) = Ker(e1 ) ∩ N . If x ∈ Ker(e), then xα¯ = (xeα )λ = 1. Thus x ∈ Ker(¯ α) = M and then Ker(e) ≤ M . Therefore Ker(e) = Ker(e1 ) ∩ M = (A2 × · · · × An )P1 ∩ M = S2 × · · · × Sn . For any a ∈ A, consider the element c = aα ∈ C. Then cρ ∈ P1 and ρ ∈ C, where j = ic , for i = 2, . . . , n. Since C = Aα , there exist ci,c = ti ct−1 j elements a2 , . . . , an in A such that aα j = cj,c , for j = 2, . . . , n. The element x = α ρ ρ (a, a2 , . . . , an )cρ ∈ N , since xα¯ = (aα , aα 2 , . . . , an )c = (c, c2,c , . . . , cn,c )c = λ e c . Now x = a, and then e is an epimorphism. Hence N/ Ker(e) = N/(S2 × · · · × Sn ) ∼ = A. Finally observe that M e ∼ = S. Since = M/ Ker(e|M ) = M/(S2 × · · · × Sn ) ∼ M e ≤ S = Ker(α) and these two subgroups have the same order, equality holds.


15

3. Choose a right transversal of N in G, {g1 = 1, . . . , gn } such that giσ = ti . Then for each g ∈ G, we have that gi g = xi,g gigρ , for some xi,g ∈ N . Then ci,gσ tigσρ = ti g σ = giσ g σ = xσi,g g σgρ = xσi,g tigρ i

σρ

ρ

and then ig = ig , for every i ∈ I. Therefore g σρ = g ρ for each g ∈ G, and then σρ = ρ . 4. Observe that for each i ∈ I, the permutation tρi moves 1 to i. Therefore, having in mind (1.1) of Definition 1.1.17, we see that S1gi = Si , and then {S1 , . . . , Sn } is the conjugacy class of the subgroup S1 in G. We prove next that in fact the structure of the group G analysed in Proposition 1.1.28 characterises the induced extensions. Theorem 1.1.29. Let G be a group. Suppose that we have in G the following situation: there exist a normal subgroup M of G and a normal subgroup S of M such that {S1 , . . . , Sn } is the set of all conjugate subgroups of S in G and M = S1 × · · · × Sn . Write N = NG (S1 ) and K = S2 × · · · × Sn . Let α : N/K −→ G/M be defined by (Kx)α = M x. Then G is the induced extension defined by α. Proof. Let σ : G −→ G/M and e : N −→ N/K be the natural epimorphisms. If T = {t1 = 1, . . . , tn } is a right transversal of N in G, then T σ is a right transversal of N/M in G/M . Consider ρ : G/M −→ Sym(n) the permutation representation of G/M on the right cosets of N/M in G/M . Then ρ¯ = σρ is the permutation representation of G on the right cosets of N in G. Write P = Gρ¯ = (G/M )ρ . Let ¯ = λT : G −→ N ρ¯ P λ be the embedding of G into N ρ¯ P defined in Lemma 1.1.26 and λ = λT σ : G/M −→ (N/M ) ρ P be the embedding of G/M into (N/M ) ρ P . As usual, for each x ∈ G, write ρ σ σ = (ci,g ) . ti x = ci,x tj , for some ci,x ∈ N , and ix = j. Observe that ci,g ti Write Si = S . For each i ∈ I = {1, . . . , n}, write also Ki = j∈I\{i} Sj . Then K = K1 and Ki = K ti . If we write σ ¯ : N ρ¯ P −→ (N/M ) ρ P for the epimorphism induced by σ, ¯ σ . Consider then σλ = λ¯ e¯ : N P −→ (N/K) P,

induced by e

and α ¯ : (N/K) P −→ (N/M ) P,

induced by α. ¯ eα ¯ σ = σλ and the ¯ = σ ¯ . Therefore λ¯ ¯ = λ¯ Since eα = σ|N , we find that e¯α following diagram is commutative:

16


G

σ

/ G/M

α ¯

/ (N/M ) P

¯e λ¯

λ

(N/K) P

¯

The commutativity of the diagram shows that M λ¯eα¯ = M σλ = 1 and then ¯ α). M λ¯e ≤ Ker(¯ ¯ ¯ e). Consider an element x ∈ G such that xλ = (c1,x , . . . , cn,x )xρ¯ ∈ Gλ ∩Ker(¯ Then we have 1 = (Kc1,x , . . . , Kcn,x )xρ . This means that xρ = id and n ti ci,x ∈ K, for i ∈ I. Therefore, ci,x = ti xt−1 i , for i ∈ I. Hence, x ∈ i=1 K = n ¯ λ ¯ e is a monomorphe) = 1 and then λ¯ i=1 Ki = 1. Therefore G ∩ Ker(¯ ism. Observe that Ker(α ¯ ) = (M/K) = (M/K)1 × · · · × (M/K)n and then ¯ e|M : M −→ Ker(¯ α) is an isomorphism. |Ker(¯ α)| = |M |. Thus, the restriction λ¯ Therefore, the following diagram is commutative: 1

/M

/G

1

/ Ker(¯ α)

/ (N/K) P

σ

/ G/M

α ¯

/ (N/M ) P

¯e λ¯

/1

λ

/1

Therefore G is the induced extension defined by α.

Remark 1.1.30. We are interested in the action of the group G on the normal subgroup M = S1 × · · · × Sn , when G is an induced extension. We keep the notation of Theorem 1.1.29. The action of the group N on S, ψ : N −→ Aut(S), is defined by conjugation: if x ∈ N , then xψ is the automorphism of S given by the conjugation in N by the element x: for every s ∈ S, we have ψ sx = sx . The induced extension G can be considered as a subgroup of the wreath product W = N ρ P , via the embedding ¯ = λT : G −→ N ρ¯ P λ

¯

given by xλ = (c1,x , . . . , cn,x )g ρ¯, for all x ∈ G.

If (x1 , . . . , xn ) ∈ M = S1 × · · · × Sn and x ∈ G, then, by Definition 1.1.17, x

(x1 , . . . , xn ) =

cψ cψ x11,x , . . . , xnn,x

xρ¯ = (y1 , . . . , yn ),

cψ

where xi i,x = yixρ¯ , for i ∈ {1, . . . , n}. Proposition 1.1.31. In the hypotheses 1.1.25, assume that S is a group and C acts on S by a group homomorphism ψ : C −→ Aut(S). Then the group B acts on the direct product S n = S1 × · · · × Sn by a group homomorphism ψ B : B −→ C ψ ρ P ≤ Aut(S n )


17

such that for (x1 , . . . , xn ) ∈ S n and h ∈ B, then (x1 , . . . , xn )h

ψB

cψ

= (y1 , . . . , yn ), where xi i,h = yihρ¯ , for i ∈ {1, . . . , n}. (1.3)

Moreover, Ker(ψ B ) = CoreB Ker(ψ) .

Proof. If ψ¯ : C ρ P −→ C ψ ρ P is induced by ψ and λ is the monomorphism ¯ Clearly ψ B is a group homomorphism. of Lemma 1.1.26, then ψ B = λψ. B Observe that h ∈ Ker(ψ ) if and only if hρ is the identity permutation and = ci,h for all ci,h ∈ Ker(ψ), for all i ∈ I. This means that ti ht−1 i ∈ Ker(ψ), Ker(ψ) . In other i ∈ I. And this is equivalent to saying that h ∈ Core B words, Ker(ψ B ) = CoreB Ker(ψ) . These observations motivate the following definition. Definition 1.1.32. With the notation of Proposition 1.1.31, the action ψ B is called the induced B-action from ψ, and the B-group (S n , ψ B ) is the induced B-group. The semidirect product [S n ]ψB B = [S1 × · · · × Sn ]B is called the twisted wreath product of S by B; it is denoted by S (C,ψ) B. Thus, if G is the induced extension defined by the map α : N/K −→ G/M as in Theorem 1.1.29, then the conjugacy action of G on the normal subgroup M = S1 × · · · × Sn is the induced G-action from the conjugacy action of N = NG (S1 ) on S1 . Remarks 1.1.33. 1. The structure of induced B-group does not depend, up to equivalence of B-groups, on the chosen transversal of C in B. 2. The construction of induced actions is motivated by the classical construction of induced modules. If S is a C-module, the induced B-action gives to S n the well-known structure of induced B-module: S n ∼ = S B . This explains the name and the notation. Proposition 1.1.34. Let S and B be groups and C a subgroup of B. Suppose that (S, ψ) is a C-group and consider the twisted wreath product G = S(C,ψ) B. Then 1. NB (S1 ) = C and CB (S1 ) = Ker(ψ). 2. CB (S ) = CoreB Ker(ψ) . Moreover if CoreB (C) = 1, then CG (S ) = Z(S ). ρ

Proof. 1. If h ∈ NB (S1 ), then, by (1.3), 1h = 1 and h = c1,h ∈ C. ρ ρ Conversely, if c ∈ C, then c = c1,c and 1c = 1; moreover (x, 1, . . . , 1)c = ψ (xc , 1, . . . , 1). Hence C ≤ NB (S1 ). Observe that the elements of CB (S1 ) are elements c ∈ C such that cψ = idS1 . Hence CB (S1 ) = Ker(ψ).

18


n that S1ti = Si , forall i ∈ I. Therefore CB (S ) = i=1 CB (Si ) = n 2. Observe ti B Ker(ψ) . i=1 CB (S1 ) = CoreB CB (S1 ) = Core n If (x1 , . . . , xn )h ∈ CG (S ), then h ∈ i=1 NB (Si ) = CoreB (C) = 1. There fore (x1 , . . . , xn ) ∈ Z(S ). If 1 −→ M −→ G −→ B −→ 1 is the induced extension defined by a group homomorphism α : A −→ B, then G splits over M if and only if G is acs isomorphic to the twisted wreath product S(C,ψ) B. F. Gross and L. G. Kov´ characterise when the induced extension splits. This characterisation, which will be crucial in Chapter 7, is just a consequence of a deep analysis of the supplements of M in G. Theorem 1.1.35 (([GK84])). Let G be a group in which there exists a normal subgroup M of G such that M = S1 ×· · ·×Sn , where {S1 , . . . , Sn } is the set of all conjugate subgroups of a normal subgroup S1 of M . Write N = NG (S1 ) and K = S2 × · · · × Sn . 1. Let L/K be a supplement of M/K in N/K. Then, there exists a supplement H of M in G satisfying the following: a) L = (H ∩ N )K and H ∩ M = (H ∩ S1 ) × · · · × (H ∩ Sn ). Further, {H ∩ S1 , . . . , H ∩ Sn } is a conjugacy class in H, and H ∩ S1 = L ∩ S1 . b) Suppose that H0 is a supplement of M in G such that H0 ∩ N ≤ L. Then there is an element k ∈ K such that H0k ≤ H. Moreover, H0k = H if and only if L = (H0 ∩ N )K and H0 ∩ M = (H0 ∩ S1 ) × · · · × (H0 ∩ Sn ). c) In particular, H is unique up to conjugacy under K. 2. Suppose that H is a supplement M in G such that H ∩ M = (H ∩ S1 ) × · · · ×(H ∩Sn ). Write L = (H ∩N )K. Assume further that R is a subgroup of G such that G = RM . Then the following are true: a) R is conjugate in G to a subgroup of H if and only if R ∩ N is conjugate in N to a subgroup of L. b) R is conjugate to H in G if and only if (R ∩ N )K is conjugate to L in N and also R ∩ M = (R ∩ S1 ) × · · · × (R ∩ Sn ). 3. There is a bijection between, on the one hand, the conjugacy classes in G of supplements H of M in G such that H ∩ M = (H ∩ S1 ) × · · · × (H ∩ Sn ), and, on the other hand, the conjugacy classes in N/K of supplements L/K of M/K in N/K, Moreover, under this bijection, we have the following: a) the conjugacy classes in G of supplements U of M which are maximal subgroups of G such that U ∩ M = (U ∩ S1 ) × · · · × (U ∩ Sn ) are in one-to-one correspondence with the conjugacy classes in N/K of supplements of M/K which are maximal subgroups of N/K. b) the conjugacy classes in G of complements of M , if any, are in oneto-one correspondence with the conjugacy classes in N/K of complements of M/K. Proof. By Theorem 1.1.29, the group G is the induced extension defined by α : N/K −→ G/M given by (Kx)α = M x, for all x ∈ G. Let T = {t1 =


19

1, . . . , tn } be a right transversal of N in G and write ρ : G −→ Sym(n) the permutation representation of G on the right cosets of N in G. As usual, for ρ N , and ix = j. Write Si = S ti . each x ∈ G, write ti x = ci,x tj , for some ci,x ∈ For each i ∈ I = {1, . . . , n}, write also Ki = j∈I\{i} Sj . Then K = K1 and Ki = K ti . For P = Gρ , let λ be the embedding of G into (N/K) ρ P defined by λ : G −→ (N/K) ρ P such that xλ = (Kc1,x , . . . , Kcn,x )xρ , for any x ∈ G. 1a. Define λ−1 = {x ∈ G : ci,x ∈ L, for all i ∈ II}. H = (L/K) ρ P This subgroup H satisfies the required properties. Fix an element g ∈ G. Then, for each i ∈ I, we have that ci,g ∈ N = M L and there exists mi,g ∈ M such that m−1 i,g ci,g ∈ L. −1

Observe that, if m ∈ M , then ci,m = mti . Then −1

−1

−1

−1

mλ = (Kmt1 , . . . , Kmtn ) = (Km, Kmt2 , . . . , Kmtn ). Write m = (s1 , . . . , sn ). Then, for any i ∈ I, using (1.1) in Definition 1.1.17, ρ −1 (mti )π1 = si , since 1ti = i. Therefore (s1 , . . . , sn )λ = (Ks1 , . . . , Ksn ). Since the restriction of λ to M is an isomorphism onto (M/K) , i.e. M λ = (M/K) , there exists a unique mg ∈ M such that mλg = (Km1,g , . . . , Kmn,g ). Hence λ λ −1 λ −1 ρ g = (Km−1 (m−1 g g) = (mg ) 1,g , . . . , Kmn,g )(Kc1,g , . . . , Kcn,g )g = −1 ρ = (Km−1 1,g c1,g , . . . , Kmn,g cn,g )g ∈ (L/K) ρ P,

and then m−1 g g ∈ H. Hence, G = HM .

t−1

Observe that Kmi,g = Kci,mg = Kmgi . If g ∈ L, then we can choose m1,g = 1, and then mg ∈ K. Thus m−1 g g ∈ H ∩ N . Then L ≤ K(H ∩ N ). On the other hand, if h ∈ H ∩ N , then h = c1,h ∈ L. Hence L = K(H ∩ N ). If m = (s1 , . . . , sn ) ∈ M ∩ H, then Ksi ∈ L/K, for all i ∈ I. Observe that, for any i ∈ I, we have that (1, . . . , si , . . . , 1)λ = (K, . . . , Ksi , . . . , K) ∈ (L∩M )/K and then (1, . . . , si , . . . , 1) ∈ H ∩Si . Hence, H ∩M = (H ∩ S1 )× · · · × (H ∩ Sn ). Since G = HM , we can choose the transversal T ⊆ H. Hence, for all i ∈ I, {H ∩S1 , . . . , H ∩Sn } is a conjugacy we have that H ∩Si = (H ∩S1 )ti . Therefore class in H. Moreover (L ∩ M )/K = (H ∩ N )K ∩ M /K = (H ∩ M )K/K = (H ∩ S1 )K/K ∼ = H ∩ S1 and also (L ∩ M )/K = (L ∩ S1 )K/K ∼ = L ∩ S1 . Hence |H ∩ S1 | = |L ∩ S1 |. Since H ∩ S1 = H ∩ N ∩ S1 ≤ L ∩ S1 , we have the equality H ∩ S1 = L ∩ S 1 . 1b. Assume now that H0 is a subgroup of G such that G = M H0 and H0 ∩ N ≤ L. For each i ∈ I, there must be an element mi ∈ M such that ti ∈ m−1 i H0 , i.e. mi ti ∈ H0 . We may choose m1 = 1. Now, there exists a unique k ∈ M such that

20

1 Maximal subgroups and chief factors −1

−1

(K, Km2 , . . . , Kmn ) = k λ = (Kk t1 , . . . , Kk tn ). −1 This implies that k ∈ K and ti kt−1 ∈ K, for all i ∈ I. We show that i mi k−1 H0 ≤ H. −1 Let x ∈ H0 and consider y = xk . Observe that, for all i ∈ I, M ti x = −1 ρ ρ M ti xk = M ti y and then ix = iy . Now −1 −1 −1 tixρ = ci,y = ti yt−1 ρ = ti yt xρ = ti kxk iy i −1 −1 −1 −1 −1 ρ ρ tixρ = = ti k(t−1 i mi mi ti )x(tixρ mixρ mix tix )k −1 −1 −1 −1 −1 ρ ρ tixρ ). = (ti kt−1 i mi )(mi ti xtixρ mixρ )(mix tix k −1 −1 −1 Now observe that mi ti and t−1 ρ m xρ are in H0 and then, mi ti xt xρ m xρ ∈ H0 . ix i i i −1 −1 −1 On the other hand, ti xtixρ = ci,x ∈ N , and then mi ti xtixρ mixρ ∈ N . Since −1 ∈ K and also mixρ tixρ k −1 t−1 ti kt−1 ρ ∈ K, we have that i mi ix −1 −1 −1 −1 −1 ρ ρ ci,y = (ti kt−1 tixρ ) ∈ K(H0 ∩ N )K ≤ L i mi )(mi ti xtixρ mixρ )(mix tix k

for all i ∈ I. This means that y ∈ H. Assume that H0k ≤ H, for k ∈ K. Clearly, if H0k = H, then L = (H0 ∩N )K and H0 ∩ M = (H0 ∩ S1 ) × · · · × (H0 ∩ Sn ). Conversely, suppose that L = (H0 ∩ N )K and H0 ∩ M = (H0 ∩ S1 ) × · · · × (H0 ∩ Sn ). Observe that H0k satisfies the same properties. Thus, we can assume that H0 ≤ H. As in 1a, since G = H0 M , we have that {H0 ∩ S1 , . . . , H0 ∩ Sn } is a conjugacy class in H0 , and H0 ∩ S1 = L ∩ S1 . Hence, |H ∩ S1 | = |H0 ∩ S1 |, and then H ∩ S1 = H0 ∩ S1 . Therefore, H ∩ M = H0 ∩ M . Then, from G = H0 M = HM , we deduce that |G : H0 | = |M : M ∩ H0 | = |M : M ∩ H| = |G : H|. Hence, |H| = |H0 | and then, H = H0 . Part 1c is a direct consequence of 1b. 2a. Clearly L/K is a supplement of M/K in N/K. By 1c, the subgroup H is determined, up to conjugacy in K, by L. Suppose that G = RM and R ∩ N is conjugate to a subgroup of L in N . Since N = RM ∩ N = (R ∩ N )M , there is an element m ∈ M such that (R ∩ N )m ≤ L. Write H0 = Rm . Then G = H0 M and H0 ∩ N ≤ L. It follows, from 1b, that H0 is conjugate to a subgroup of H. Hence R is conjugate to a subgroup of H. Conversely, if R is conjugate to a subgroup of H, then, since G = RM , we have that Rm ≤ H, for some m ∈ M . Then (R ∩ N )m = Rm ∩ N ≤ H ∩ N ≤ L. 2b. If G = RM and L is (R ∩ N )K in N , there is an element conjugateto m = (R ∩ N )m K = (Rm ∩ N )K. If m ∈ M such that L = (R ∩ N )K R ∩ M = (R ∩ S1 ) × · · · × (R ∩ Sn ), by 1b, we deduce that H0 = Rm is conjugate to H. The rest of 2b follows easily. 3. The bijection follows easily from 1 and 2. 3a. Let L be a maximal subgroup of N such that K ≤ L and N = LM and consider one of the supplements U of M in G determined by the conjugacy


21

class of L in N under the bijection. Suppose that U ≤ H < G. Then N = (H ∩N )M . Set L0 = (H ∩N )K. Then L0 /K is a supplement of M/K in N/K. Clearly L = (U ∩ N )K ≤ L0 . By maximality of L, we have that L = L0 . But then H ∩ N ≤ L and, by 1b, H k ≤ U , for some k ∈ K. Clearly, this implies that U = H. Hence U is maximal in G. Conversely, let U be a maximal subgroup of G which supplements M in G such that U ∩ M = (U ∩ S1 ) × · · · × (U ∩ Sn ). Write L = (U ∩ N )K. Suppose that L ≤ L0 < N . Consider a supplement R of M in G determined by L0 under the bijection. Then L0 = (R ∩ N )K. Since U ∩ N ≤ L0 , then U k ≤ R, for some k ∈ K. By maximality of U , we have that R = U k . This implies that L and L0 are conjugate in N and, since L ≤ L0 , equality holds. 3b. Observe that if L/K is a complement of M/K in N/K, then L∩S1 = 1. Hence H ∩ S1 = 1 and therefore H ∩ M = 1. This is to say that H is a complement of M in G. Conversely, if H is a complement of M in G, then (H ∩ N )K ∩ M = (H ∩ N ∩ M )K = K. The following result, due also to F. Gross and L. G. Kov´ acs, is an application of the induced extension procedure to the construction of groups which are not semidirect products. We will use it in Chapter 5. Theorem 1.1.36 ([GK84]). Let B be any finite simple group. Then there exists a finite group G with a minimal normal subgroup M such that M is a direct product of copies of Alt(6), the alternating group of degree 6, the quotient group G/M is isomorphic to B and G does not split over M . Proof. Consider the group A = Aut Alt(6) . Let D denote the normal subgroup of inner automorphisms, D ∼ = Alt(6), of A. It is well-known that the quotient group A/D is isomorphic to an elementary abelian 2-group of order 4 and A does not split over D, i.e. there is no complement of D in A (see [Suz82]). By the Odd Order Theorem ([FT63]), the Sylow 2-subgroups of B are nontrivial. By the Burnside Transfer Theorem (see [Suz86, 5.2.10, Corollary 2]), a Sylow 2-subgroup of B cannot by cyclic. By a theorem of R. Brauer and M. Suzuki (see [Suz86, page 306]), the Sylow 2-subgroups of G cannot by isomorphic to a quaternion group. Hence a Sylow 2-subgroup of B has two transpositions generating a dihedral 2-group (see [KS04, 5.3.7 and 1.6.9]). Therefore B must contain a subgroup G which is elementary abelian of order 2. Then there is a homomorphism α of A into B such that Aα = C and Ker (α) . =D Now let G be the induced extension defined by α : A −→ B. Since A does not split over D, the group G has the required properties. Let G be a group which is an induced extension of a normal subgroup M = S1 × · · · × Sn . We have presented above a complete description of those supplements of M in G whose intersection with M is a direct product of the projections in each component H ∩M = (H ∩S1 )×· · ·×(H ∩Sn ). But nothing

22


is said about those supplements H whose projections πi : H ∩ M −→ Si are surjective. Subgroups D of a direct product M such that all projections πi : D −→ Si are surjective are fully described by M. Aschbacher and L. Scott in [AS85]. In the sequel we present here an adaptation of their results suitable for our purposes. n Definition 1.1.37. Let G = i=1 Si be a direct product of groups. A subgroup H of G is said to be diagonal if each projection πi : H −→ Si , i = 1, . . . , n, is injective. If each projection πi : H −→ Si is an isomorphism, then the subgroup H is said to be a full diagonal subgroup. n Obviously if H is a full diagonal subgroup of G = i=1 Si , then all the Si are isomorphic. Observe that if x = (x1 , . . . , xn ) ∈ H, then xi = xπi , for π −1 π

π −1 π

all i = 1, . . . , n, and then x = (x1 , x1 1 2 , . . . , x1 1 n ). All ϕi = π1−1 πi are isomorphisms of S1 and then ϕ = (ϕ1 = 1, ϕ2 , . . . , ϕn ) ∈ Aut(S1 )n . Conversely, given a group S and ϕ = (ϕ1 , ϕ2 , . . . , ϕn ) ∈ Aut(S)n , it is clear that n {xϕ = (xϕ1 , xϕ2 , . . . , xϕn ) : x ∈ S} is a full diagonal subgroup n of S . More generally, given a direct product of groups G = i=1 Si such that all Si are isomorphic copies of a group S, to each pair (∆, ϕ), where ∆ = {I1 , . . . , Il } is a partition of the set I = {1, . . . , n} and ϕ = (ϕ1 , . . . , ϕn ) ∈ × · · · × Dl , where each Aut(S)n , we associate a direct product D(∆,ϕ) = D1 Dj is a full diagonal subgroup of the direct product i∈Ij Si defined by the automorphisms {ϕi : i ∈ Ij }. It is easy to see that if Γ is a partition of I refining ∆, then D(∆,ϕ) ≤ D(Γ,ϕ) . In particular, the trivial partition Ω =

{1}, . . . {n} of I gives D(Ω,ϕ) = G, for any ϕ ∈ Aut(S)n . For groups S with trivial centre, the group G can be embedded in the wreath product W = Aut(S) Sym(n). In particular, if S is a non-abelian simple group, then G ≤ Aut(S n ). In the group W the conjugacy by the ϕ , where id denotes the element ϕ ∈ W makes sense and D(∆,ϕ) = D(∆,id) n-tuple composed by all identity isomorphisms. Lemma 1.1.38. Let H be a full diagonal subgroup of the direct product G = n S , where the Si are copies of a non-abelian simple group S. Then H is i i=1 self-normalising in G. Proof. Since H is a full diagonal subgroup of G, all πi are isomorphisms of π −1 πj

H onto Si . Observe that (x1 , . . . , xn ) ∈ H if and only if xj = x1 1 j = 2, . . . , n and for all x1 ∈ S. Write ϕj = π1−1 πj , for j = 2, . . . , n. If g = (g1 , . . . , gn ) ∈ NG (H), then for all x ∈ S we have that (x, xϕ2 , . . . , xϕn )g = xg1 , (xϕ2 )g2 , . . . , (xϕn )gn ∈ H. ϕj

, for

Hence, for j = 2, . . . , n, (xϕj )gj = (xg1 )ϕj = (xϕj )g1 and the automorphism ϕ ϕ g1 j gj−1 is the trivial automorphism of Sj . Hence g1 j = gj and g ∈ H. This is to say that H is self-normalising in G.


23

Proposition 1.1.39. Suppose that H is a subgroup of the direct product G = n S , where the Si are non-abelian simple groups for all i ∈ I = {1, . . . , n}. i=1 i Assume that all projections πi : H −→ Si , i ∈ I, are surjective. 1. There exists a partition ∆ of I such that the subgroup H is the direct product H πD , H= D∈∆

where a) each H πD is a full diagonal subgroup of i∈D Si , b) the partition ∆is uniquely determined by H in the sense that if H = πD = G∈Γ H πG , for ∆ and Γ partitions of I, then ∆ = Γ , D∈∆ H and c) if H ≤ K ≤ G, then K = G∈Γ H πG , where Γ is a partition of I which refines ∆. 2. Suppose that the Si are isomorphic copies of a non-abelian simple group Then S, for all i ∈ I, i.e. G ∼ = S n . Let U be a subgroup of Aut(G). U, acting by conjugation on the simple components Si of Soc Aut(G) , is a permutation group on the set {S1 , . . . , Sn } (and therefore on I). Observe that the action of U on I induces an action on the set of all partitions of I. We can say that a partition ∆ of I is U -invariant if ∆x = ∆ for all x ∈ U . If H is U -invariant, i.e. U ≤ NAut(G) (H), then the partition ∆ is a U-invariant set of blocks of the action of U on I. 3. In the situation of 2, if Γ is a U -invariant partition of I which refines ∆ and every member of Γ is again a block for the action of U on I, then the subgroup K = G∈Γ H πG is also U -invariant. Proof. 1a. Let D be a subset of I minimal such that the subgroup D = H∩ i∈D Si is non-trivial. It is clear that D is a normal subgroup of H and then every projection of D is a normal subgroup of the corresponding projection of H. Since, by minimality of D, Dπj is non-trivial, for each j ∈ D, we have that Dπj = Sj . Moreover, for each j ∈ D, we have that Ker(πj )∩D = H∩ i∈D,i =j Si = 1, by minimality of D. Therefore D is a full diagonal subgroup of i∈D Si . Let E = H πD be the image of the projection of H in Then D = DπD is normal in E. By Lemma 1.1.38, D = E. Write i∈D Si . F = H ∩ i∈D / Si . Clearly D×F ≤ H. For each x ∈ H, we can write x = x1 x2 , wherex1 is the projection of x onto i∈D Si and x2 is the projection of x onto i∈D / Si . Observe that x1 ∈ D ≤ H and then x2 ∈ F . This is to say that H = D × F . Now the result follows by induction onthe cardinality of I. To prove 1b suppose that H = D∈∆ H πD = G∈Γ H πG , for ∆ and Γ partitions ofI. Observe that for each D ∈ ∆, since H πD is a full diagonal subgroup of i∈D Si , we have that the following statements are equivalent for a non-trivial element h ∈ H:

24


1. h ∈ H πD ; 2. hπi = 1 if and only if i ∈ D; 3. there exists an i ∈ D such that hπi = 1 and for each D ∈ ∆, with D = D, there exists a j ∈ D such that hπj = 1. Suppose that h ∈ H πD . Then hπi = 1, for all i ∈ D, and hπj = 1, for all j∈ / D. If i ∈ D, there exists G ∈ Γ such that i ∈ G. Thus h ∈ H πG and in fact D = G. Hence ∆ = Γ . 1c. Suppose finally that K is a subgroup of G containing H. Obviously, the K −→ Si are surjective. Then, by the above arguments, we projections πi : have that K = G∈Γ K πG , where Γis a partition of I, and, for each G ∈ Γ , K πG is a full diagonal subgroup of j∈G Sj . In particular, for all i ∈ G, πthe D to a non-abelian simple group S . Since H = , Si are isomorphic G D∈∆ H πG πD∩G πD∩G ∼ . If G ∩ D = ∅, then H we have H = D∩G,D∈∆ H = SG . Observe that H πG is a direct product contained in K πG ∼ = SG . This implies that the direct product has a unique component which is equal to K πG . Hence, for each G ∈ Γ , H πG = K πG , and G ⊆ D, for some D ∈ ∆, i.e. Γ is a partition of I which refines ∆. 2. By Proposition 1.1.20, we can consider that U is a subgroup of the wreath product A Sym(n), for A = Aut(S) and S a non-abelian simple group such that S ∼ = Si , for all i ∈ I. We see in Remark 1.1.18 (2) of that U acts by conjugation on the set {A1 , . . . , An } of factors of the base group. Since S is the unique minimal normal subgroup of A, the group U acts by conjugation on {S1 , . . . , Sn }. Suppose that H is U -invariant. Then, for any x ∈ U , by Remark 1.1.18 (4), we have x π x (H πD ) = (H x ) D = H πD x H = Hx = D∈∆

D∈∆

D x ∈∆x

and then ∆ = ∆x , by 1b. Hence ∆ is U -invariant. Moreover Dx is an element of the partition ∆. Therefore either D = Dx or D∩Dx = ∅. Hence the elements of ∆ are blocks for the action of U on I. 3. This follows immediately from Remark 1.1.18 (4): for any x ∈ U , we have x π x (H πG ) = (H x ) G = H πG = K, Kx = G∈Γ

and therefore K is U -invariant.

G∈Γ

G∈Γ

The purpose of the following is to present a proof of the Theorem of O’Nan and Scott classifying all primitive groups of type 2. The first version of this theorem, stated by Michael O’Nan and Leonard Scott at the symposium on Finite Simple Groups at Santa Cruz in 1979, appeared in the proceedings in [Sco80] but one of the cases, the primitive groups whose socle is complemented by a maximal subgroup, is omitted. In [Cam81], P. J. Cameron presented an outline of the proof of the O’Nan-Scott Theorem again with the same omission. Finally, in [AS85] a corrected and expanded version of the theorem


25

appears. Independently, L. G. Kov´ acs presented in [Kov86] a completely different approach to the same result. We are indebted to P. Jiménez-Seral for her kind contributions in [JS96]. These personal notes, written for a doctoral course at the Universidad de Zaragoza and adapted for her students, are motivated mainly by the selfcontained version of the O’Nan-Scott Theorem which appears in [LPS88]. To study the structure of a primitive group G of type 2 whose socle Soc(G) is non-simple, we will follow the following strategy. We observe that in general, for any supplement M of Soc(G) in G, we have that M is a maximal subgroup of G if and only if M ∩ Soc(G) is a maximal M -invariant subgroup of Soc(G). We will focus our attention in the structure of the intersection U ∩ Soc(G) of a core-free maximal subgroup U of G with the socle. General remarks and notation 1.1.40. We fix here the main notation which is used in our study of primitive groups of type 2 in the sequel. We also review some previously known facts and make some remarks. All these observations give rise to the first steps of the classification theorem of O’Nan and Scott. Let G be a primitive group of type 2. 1. Write Soc(G) = S1 × · · · × Sn where the Si are copies of a non-abelian simple group S, for i ∈ I = {1, . . . , n}. Write also Kj = i∈I,i=j Si , for each j ∈ I. 2. Write N = NG (S1 ) and C = CG (S1 ). Let ψ : N −→ Aut(S1 ) denote the conjugacy action of N = NG (S1 ) on S1 . Sometimes we will make the identification S1ψ = Inn(S1 ) = S1 . 3. The quotient group X = N/C is an almost simple group with Soc(X) = S1 C/C. 4. Suppose that U is a core-free maximal subgroup of G. 5. The subgroup U ∩ Soc(G) is maximal with respect to being a proper U -invariant subgroup of Soc(G). 6. By Proposition 1.1.19, the group G, acting by conjugation on the elements of the set {S1 , . . . , Sn }, induces the structure of a G-set on I. Write ρ : G −→ Sym(n) for this action. The kernel of this action is Ker(ρ) = n ρ i=1 NG (Si ) = Y . Therefore G/Y is isomorphic to a subgroup G = Pn ρ of Sym(n). For any g ∈ G, we write g for the image of g in Pn . Moreover, since Soc(G) is a minimal normal subgroup, the conjugacy action of G on {S1 , . . . , Sn }, and on I, is transitive. Observe that Sixρ = Six and Kixρ = Kix , for x ∈ G and i ∈ I. It is worth remarking here that the action of Soc(G) on I is trivial. Therefore if H is a supplement of Soc(G) in G and ∆ is a partition of I, then ∆ is H-invariant if and only if ∆ is G-invariant. Also, a subset D ⊆ I is block for the action of H if and only if D is a block for the action of G. Since the set {S1 , . . . , Sn } is a conjugacy class of subgroups of G, we have that Y = CoreG (N ). In particular Soc(G) ≤ Y . Now U is core-free and maximal in G and therefore G = U Y . This means that if τ is a permutation of I in Pn , there exists an element x ∈ U such that

26


the conjugation by x permutes the Si in the same way τ does: Siτ = Six , for all i ∈ I. In other words, xρ = τ . This is to say that the projection of U onto Pn is surjective. 7. The stabiliser of the element 1 for the action of G on I is N = NG (S1 ). Therefore |G : N | = n. Observe that N = NG (S1 ) = NG (K1 ). Let T = {1 = t1 , t2 , . . . , tn } be a right transversal of N in G such that S1ti = Si , for i ∈ I. 8. Observe that Soc(G) ≤ N and then G = U N . For this reason the transversal T can be chosen such that T ⊆ U . 9. Write V = U ∩ N = NU (S1 ). Then T is a right transversal of V in U . Observe that N = N ∩ U Soc(G) = (N ∩ U ) Soc(G) = V Soc(G) = V CS1 . 10. The conjugation in S1 by the elements of V induces a group homomorphism ϕ : V −→ Aut(S1 ). It is clear that Ker(ϕ) = CU (S1 ). 11. For any i ∈ I, we have ρ

ti g = ai,g tj , with ai,g ∈ N and ig = j. Moreover, since T ⊆ U , if g ∈ U , then ai,g ∈ V . 12. Denote with a star (∗) the projection of N in X: if a ∈ N , then a∗ = aC ∈ X. 13. The group G is the induced extension defined by α : N/K1 −→ G/ Soc(G). Hence, the action of G on Soc(G) is the induced G-action from ψ: ψ G : G −→ X ρ Pn ≤ Aut(S n ), given by g ψ = (a∗1,g , . . . , a∗n,g )g ρ , for any g ∈ G. Observe that Ker(ψ G ) = CoreG Ker(ψ) = CoreG CG (S1 ) = 1. Hence ψ G is injective. In other words, ψ G is an embedding of G into the wreath product X ρ Pn , and then into G Aut(S n ). We identify G and Gψ . With this identification, NG (S1 ) = G ∩ (X1 × [X2 × · · · × Xn ]Pn−1 ), where Pn−1 is the stabiliser of 1. If g ∈ NG (S1 ), then g ρ fixes 1, i.e. g ρ ∈ Pn−1 . G ∗ )g ρ ∈ (X1 × [X2 × · · · × Moreover a1,g = g. Hence g ψ = (g ∗ , a∗2,g , . . . , an,g Xn ]Pn−1 ). Hence the projection of NG (S1 ) on X1 is surjective. written as 14. Observe that, for each i ∈ I, any element xi of Si can be t−1 ti j xi = ei , for certain ei ∈ S1 . For any j = i, we have that xi ∈ Sk , for G

t−1

some k = 1 and therefore xij ∈ CG (S1 ). This implies that a∗j,xi = 1 for any j = i. Moreover ai,xi = ei . Also it is clear that xi normalises all the Sj , for G j = 1, . . . , n and then xi ρ = 1. Hence xψ = e∗i . This is to say that, with the i G G ψ identification of 2, Si = Si , for all i ∈ I, and then Soc(G)ψ = S . 15. For each i ∈ I, the quotient Y CG (Si )/ CG (Si ) is isomorphic to a group n subgroup of Aut(Si ) and then Y / i=1 CY (Si ) ∼ = Y is embedded in Aut(S1 ) × · · · × Aut(Sn ). Observe that the kernel of the homomorphism which assigns to each n-tuple of Aut(S1 ) × · · · × Aut(Sn ) the n-tuple of the corresponding projections of Out(S1 ) × · · · × Out(Sn ) is Soc(G). Hence the quotient group Y / Soc(G) is isomorphic to a subgroup of Out(S1 ) × · · · × Out(Sn ). Hence,


27

by the Schreier’s conjecture ([KS04, page 151]), the group Y / Soc(G) = Y ∩ U Soc(G) / Soc(G) = (Y ∩ U ) Soc(G)/ Soc(G) ∼ = (U ∩ Y )/ U ∩ Soc(G) is soluble. 16. As in Remarks 1.1.18, if S ⊆ I, then we write Sj πS : Soc(G) −→ j∈S

for the projection of Soc(G) onto j∈S Sj . If S = {j}, then the projection πj . onto Sj is denoted simply πj 17. Write Rj = U ∩ Soc(G) . Since the action of G on I is transitive and G = U Soc(G), then all projections Rj , j = 1, . . . , n are conjugate by elements of U . Hence U ∩ Soc(G) ≤ R1 × · · · × Rn = R1 × R1t2 × · · · × R1tn . 18. By Remark 1.1.18 (4), if y ∈ U ∩ Soc(G) and g ∈ V , then (y g )π1 = π1 g (y ) . This is to say that R1 is a V -invariant subgroup of S1 . Therefore R1 × · · · × Rn = R1 × R1t2 × · · · × R1tn is a V -invariant subgroup of Soc(G). 19. By 5 and 18, we have two possibilities for each Ri : a) either Ri is a proper subgroup of Si ; in this case, U ∩ Soc(G) = R1 × · · · × Rn = (U ∩ S1 ) × · · · × (U ∩ Sn ), b) or Ri = Si , i.e. the projections of U ∩ Soc(G) on each Si are surjective. 20. Let us deal first with the Case 19a: suppose that R1 is a proper subgroup of S1 . Suppose that R1 ≤ T1 < S1 and T1 is a V -invariant subgroup of S1 . Then T1 × T1t2 · · · × T1tn is U -invariant in Soc(G) and, by 5, we have that T1 × T1t2 · · · × T1tn = U ∩ Soc(G) = R1 × · · · × Rn . Hence R1 = T1 . This means that if R1 is a proper subgroup of S1 , then R1 is maximal with respect to being a proper V -invariant subgroup of S1 . 21. If the projection π1 of U ∩ Soc(G) on S1 is not surjective, then two possibilities arise: a) either R1 = 1, i.e. U ∩ Soc(G) = 1: the core-free maximal subgroup U complements Soc(G); b) or 1 = R1 < S1 . 22. Suppose that 1 = R1 < S1 . Then, by 19a, R1 = U ∩ S1 and then R1 ≤ V . Hence R1 = V ∩ S1 . Moreover, if we suppose that there exists a proper subgroup M of N such that V C ≤ M < N , then M ∩ S1 is a V -invariant subgroup of S1 and R1 ≤ M ∩ S1 . Observe that if S1 ≤ M , then N = V CS1 ≤ M and N = M , against our assumption. Hence, R1 ≤ M ∩ S1 = S1 . By maximality of R1 , we have that R1 = M ∩ S1 and then M = M ∩ CV S1 = CV (M ∩ S1 ) = CV R1 = CV . Therefore V C is a maximal subgroup of N . 23. Now we consider the Case 19b. Assume that H is a supplement of Soc(G) in G and we suppose that the projection of H ∩ Soc(G) on each component Si of Soc(G) is surjective. Then, by Proposition 1.1.39, there exists an H-invariant partition ∆ of I into blocks for the action of H on I such that

28


H ∩ Soc(G) =

πD , H ∩ Soc(G)

D∈∆

πD and, for each D ∈ ∆, the is a full diagonal subgroup projection H ∩Soc(G) of the direct product i∈D Si . Now we prove that H is maximal in G if and only if ∆ is a minimal non-trivial G-invariant partition of I in blocks for the action of G on I. Suppose 1 < Γ < ∆, where all are H-invariant partitions of I into blocks for the action of H on I. Then by Proposition 1.1.39 (3), the product of projections of H ∩ Soc(G) obtained from Γ is an H-invariant subgroup J of Soc(G). By Proposition 1.1.39 (1b), H ∩ Soc(G) < J < Soc(G). But if H is maximal in G, then H ∩ Soc(G) is maximal as an H-invariant subgroup of Soc(G) as in 5. Hence H is not maximal in G. Now suppose H < L < G. Then H Soc(G) = L Soc(G) = G implies H ∩ Soc(G) < L ∩ Soc(G) < Soc(G). Then, by Proposition 1.1.39 (1c), L ∩ Soc(G) is the product of projections of H ∩ Soc(G) (which are the same as the projections of L ∩ Soc(G)) obtained from a non-trivial proper refinement Γ of ∆. Then by Proposition 1.1.39 (2), Γ is L-invariant so, like ∆, it is an H-invariant set of blocks for the action of H on I. Thus if ∆ is a minimal such partition of I, then H is maximal in G. Finally, any H-invariant block is G-invariant, by 6. 24. If the projection of U ∩ Soc(G) on each component Si of Soc(G) is 1 ≤ l < n, and each Di is surjective, then U ∩ Soc(G) = D1 × · · · × Dl , with isomorphic to S. Hence Soc(G) = U ∩ Soc(G) K1 and then G = U K1 . 25. In this study we have observed three different types of core-free maximal subgroups U of a primitive group G of type 2 according to the image of the projection π1: U ∩ Soc(G) −→ S1 . π1 = S1 , i.e. the projection π1 of U ∩ Soc(G) on S1 is sura) U ∩ Soc(G) jective. π1 < S1 , i.e. the image of the projection π1 of b) 1 = R1 = U ∩ Soc(G) U ∩ Soc(G) on S1 is a non-trivial proper subgroup of S1 . In this case 1 = U ∩ Soc(G) = R1 × · · · × Rn = (U ∩ S1 ) × · · · × (U ∩ Sn ). π1 c) U ∩ Soc(G) = 1, i.e. U is a complement of Soc(G) in G. 26. With all the above remarks, we have a first approach to the O’NanScott classification of primitive groups of type 2. We have the following five situations: a) Soc(G) is a simple group, i.e. n = 1: the group G is almost simple; b) n > 1 and U ∩ Soc(G) = D is a full diagonal subgroup of Soc(G); c) n > 1 and U ∩ Soc(G) = D1 × · · · × Dl , a direct product of l subgroups, with 1 < l < n, such that, for each j = 1, . . . , l, the subgroup Dj is a full diagonal subgroup of a direct product i∈Ij Si , and {I1 , . . . , Il } is a minimal non-trivial G-invariant partition of I in blocks for the action of U on I;


29

π1 d) n > 1 and the projection R1 = U ∩ Soc(G) is a non-trivial proper subgroup of S1 ; here, R1 = V C ∩ S1 and V C/C is a maximal subgroup of X. e) U ∩ Soc(G) = 1. This enables us to describe all configurations of primitive groups of type 2. Proposition 1.1.41. Let S be a non-abelian simple group and consider an almost simple group X such that S ≤ X ≤ Aut(S). Let Pn be a primitive group of permutations of degree n. Construct the wreath product W = X Pn and consider the subgroups DX = {(x, . . . , x) : x ∈ X} ≤ X and DS = {(s, . . . , s) : s ∈ S} ≤ S . Clearly Pn ≤ CW (DX ). Suppose that U is a subgroup of W such that DS ≤ U ≤ DX × Pn , and the projection of U on Pn is surjective. Then the group G = S U is a primitive group of type 2 and U is a core-free maximal subgroup of G. Proof. It is clear that S is a minimal normal subgroup of G and CG (S ) = 1. Hence G is a primitive group of type 2 and Soc(G) = S . Observe that DS = Soc(G) ∩ DX = Soc(G) ∩ U . Since Pn is a primitive group, the action of U on the elements of the set {S1 , . . . , Sn } is primitive and there are no non-trivial blocks. By 1.1.40 (23), U is a maximal subgroup of G. Definition 1.1.42. A primitive pair (G, U ) constructed as in Proposition 1.1.41 is called a primitive pair with simple diagonal action. A detailed and complete study of these primitive groups of simple diagonal type appears in [Kov88]. Remarks 1.1.43. In a primitive pair (G, U ) with simple diagonal action, we have the following. 1. U ∩ Soc(G) = DS = 1: this is the case 26b in 1.1.40 2. DS ∩ (S2 × · · · × Sn ) = 1 and Soc(G) = DS (S2 × · · · × Sn ). Hence NG (S1 ) = NU (S1 ) Soc(G) = NU (S1 )(S2 × · · · × Sn ), and analogously for the centraliser. Hence NG (S1 )/ CG (S1 ) ∼ = NU (S1 )/ CU (S1 ). Proposition 1.1.44. Let (Z, H) be a primitive pair such that either Z is an almost simple group or (Z, H) is a primitive pair with simple diagonal action. Write T = Soc(Z). Given a positive integer k > 1, let Pk be a transitive group of degree k and construct the wreath product W = Z Pk . Write Pk−1 for the stabiliser of 1. Consider a subgroup G ≤ W such that 1. Soc(W ) = T = T1 × · · · × Tk ≤ G, 2. the projection of G onto Pk is surjective,

30


3. the projection of NG (T1 ) = NW (T1 ) ∩ G = (Z1 × [Z2 × · · · × Zk ]Pk−1 ) ∩ G onto Z1 is surjective. Put U = G ∩ (H Pk ). Then G is a primitive group of type 2 and U is a core-free maximal subgroup of G. Proof. Set M = H ∩ T ; clearly NZ (M ) = H. With the obvious notation, write M = M1 × · · · × Mk . Then clearly H Pk ≤ NW (M ). Moreover if (z1 , . . . , zk )x ∈ NW (M ), then zi ∈ NZi (Mi ) = Hi for any i = 1, . . . , k. Hence H Pk = NW (M ) and therefore U = NG (M ). Notice that T1 × · · · × Tk is a minimal normal subgroup of G and CG (T1 × · · ·×Tk ) = 1. Hence G is a primitive group of type 2 and Soc(G) = T1 ×· · ·×Tk . Clearly G = U Soc(G). Since W is a semidirect product, every element of W can be written uniquely as a product of an element of Z and an element of Pk . Hence, if (h1 , . . . , hk )x ∈ T , for x ∈ Pk and hi ∈ Hi , i = 1, . . . , k, then x = 1 and hi ∈ Ti ∩ Hi = Mi . Hence U ∩ Soc(G) = M . In particular, U is core-free in G. Let us see that U is a maximal subgroup of G. Observe that NG (T1 ) = NW (T1 ) ∩ U Soc(G) = NU (T1 ) Soc(G). Let V1 be the projection of NU (T1 ) on Z1 . It is clear that V1 is contained in the projection of U on Z1 , i.e. V1 ≤ H1 . Since the projection of NG (T1 ) onto Z1 is surjective and the projection of Soc(G) on Z1 is T1 , then Z1 = V1 T1 . Since clearly M1 ≤ NG (T1 ), then M1 ≤ V1 ≤ H1 , so V1 ∩ T1 = M1 and by easy order calculations, V1 = H1 . Let L be an intermediate subgroup U ≤ L < G. By the above arguments, the projection of NL (T1 ) on Z1 is an intermediate subgroup between H1 and Z1 . By maximality of H in Z, we have that this projection is either H1 or Z1 . Write Qi for the projection of L ∩ Soc(G) on Ti , for i = 1, . . . , k. Since L acts transitively by conjugation on the elements of the set {T1 , . . . , Tk }, we have that all Qi are isomorphic to a subgroup Q such that M ≤ Q ≤ T and L ∩ Soc(G) ≤ Q1 × · · · × Qk . The subgroup L ∩ Soc(G) is normal in L and then in NL (T1 ). Hence Q1 is normal in the projection of NL (T1 ) on Z1 . If this projection is H1 , then Q is normal in H and then M ≤ Q ≤ H ∩ T = M , i.e. Q = M . In this case L = L ∩ U Soc(G) = U L ∩ Soc(G) = U . Suppose that the projection of NL (T1 ) on Z1 is the whole of Z1 . Then Q is a normal subgroup of Z and therefore Q = T . If for each i = 1, . . . , k we write Ti = Si1 × · · · × Sir , where all the Sij are isomorphic copies of a non-abelian simple group S, then we can put Soc(G) = (S11 × · · · × S1r ) × · · · × (Sk1 × · · · × Skr ). The projection of L ∩ Soc(G) on is surjective. By Reπ D eachsimple component is a direct product of full mark 1.1.40 (23), L∩Soc(G) = D∈∆ L∩Soc(G) diagonal subgroups and the partition ∆ of the set {11, . . . , 1r, . . . , k1, . . . , kr} associated with L ∩ Soc(G) is a set of blocks for the action of L. Observe that M1 × 1 × · · · × 1 ≤ L ∩ Soc(G). If Z is an almost simple group, then r = 1 and D = {1} is a block of ∆. Hence, in this case, ∆ is the trivial partition of


31

{1, . . . , k}. If (Z, H) is a primitive pair of simple diagonal action, then M is a full diagonal subgroup of T . Hence the set {11, . . . , 1r} is the union set of some members D1 , . . . , Dl of the partition ∆. Since the projection of L ∩ Soc(G) on π D l T1 is surjective, then T1 = i=1 L ∩ Soc(G) i ∼ = S1 × · · · × Sl (here the Si ’s are simply the names of the projections). Hence l = r. Since L is transitive on the Ti ’s, so that because the blocks corresponding to T1 have one element, all the blocks do. In other words, L ∩ Soc(G) = T1 × · · · × Tk . Hence L = G. Definition 1.1.45. A primitive pair (G, U ) constructed as in Proposition 1.1.44 is called a primitive pair with product action. A detailed and complete study of these primitive groups in product action appears in [Kov89]. Remarks 1.1.46. 1. If (Z, H) is a primitive pair, then Z is a permutation group on the set of right cosets of H in Z and the cardinality of Ω is |Z : H| (the degree of the permutation group Z). Now, if (G, U ) is a primitive pair with product action, as in Proposition 1.1.44, then the degree of the permutation group G is |G : U | = |G : G ∩ (H Pk )| = |W : H Pk | = |Z : H|k . 2. Observe that we have two different types of primitive pairs with product action: a) If Z is an almost simple group, T = Soc(Z), π1 and R = H ∩ T , then 1 = R < T and the projection R1 = U ∩ Soc(G) is a non-trivial proper subgroup of T1 , by Proposition 1.1.16; this is Case 26d in 1.1.40. b) If (Z, H) is a primitive pair with simple diagonal action, then U ∩ Soc(G) = D1 × · · · × Dk a direct product of k full diagonal subgroups, with 1 < k < n; here we are in Case 26c of 1.1.40. Examples 1.1.47. 1. Let S be a non-abelian simple group and H a maximal subgroup of S. If C is a cyclic group of order 2, construct the wreath product G = S C with respect to the regular action. The group G is a primitive group of type 2 and Soc(G) = S = S1 × S2 . Consider the diagonal subgroup D = {(x, x) : x ∈ S}. Then U = D×C is a core-free maximal subgroup of G and (G, U ) is a primitive pair with diagonal action. Consider now the subgroup U ∗ = H C = [H1 × H2 ]C. Then U ∗ is also a core-free maximal subgroup of G and the pair (G, U ∗ ) is a primitive pair with product action. 2. Let G be the primitive group of Example 1 and construct the wreath product W = G Z with respect to the regular action of the cyclic group Z of order 2. Then, the socle of W is isomorphic to the direct product of four copies of S: Soc(W ) = S1 × S2 × S3 × S4 . Moreover Soc(W ) is complemented by a 2-subgroup P isomorphic to the wreath product C2 C2 , that is, isomorphic to the dihedral group of order 8. The group W is a primitive group of type 2.

32


If we consider the maximal subgroup U of G and construct M = U Z, we obtain a core-free maximal subgroup of index |W : M | = |S|2 such that M ∩ Soc(W ) = D1 × D2 . Taking now the maximal subgroup U ∗ of G, then the subgroup M ∗ = U ∗ Z is another core-free maximal subgroup of W of index |S : H|4 such that M ∗ ∩ Soc(G) = H1 × H2 × H3 × H4 . Therefore the pairs (W, M ) and (W, M ∗ ) are non-equivalent primitive pairs of type 2 with product action. Write DS = {(s, s, s, s) : s ∈ S}, the full diagonal subgroup of Soc(W ). Observe that M contains properly the subgroup M0 = DS × P and therefore M0 is non-maximal in W . According to Remark 1.1.40 (26), there still remains another structure of primitive group of type 2 to describe: those primitive groups of type 2 with the special property that the core-free maximal subgroup is a complement of the socle. This new configuration is in fact a twisted wreath product. Theorem 1.1.48. 1. If (G, U ) is a primitive pair of type 2 and U ∩Soc(G) = 1, then, with the notation of Definition 1.1.32, G ∼ = S (V,ϕ) U . 2. Conversely, let S be a non-abelian simple group and a group U with a subgroup V such that there exists a group homomorphism ϕ : V −→ Aut(S). Construct the twisted wreath product G = S (V,ϕ) U . If CoreU (V ) = 1 then G is a primitive group of type 2. Moreover, if U is maximal in G, then (G, U ) is a primitive pair of type 2. By construction, U ∩ Soc(G) = 1. Proof. 1. Recall that G is the induced extension defined by α : N/K1 −→ G/ Soc(G). Hence Soc(G) is the induced U -group from the action ϕ of V on S (see Remark 1.1.40 (10)). Since G splits on Soc(G), then G is isomorphic to the twisted wreath product G ∼ = S (V,ϕ) U . 2. To prove the converse, it is enough to recall that in the twisted wreath product G = S (V,ϕ) U , we have that CG (Z ) = Z(S ) = 1, by Proposition 1.1.34, and the conclusion follows. Definition 1.1.49. A primitive pair (G, U ) constructed as in Theorem 1.1.48 is called a primitive pair with twisted wreath product action. Maximal subgroups of a primitive group G of type 2 complementing Soc(G) are called by some authors small maximal subgroups. Obviously one can wonder about the existence of primitive groups of type 2 with small maximal subgroups. P. F¨ orster, in [F¨ or84a], gives sufficient conditions for U , V , and S to obtain a primitive group with small maximal subgroups. Theorem 1.1.50 ([F¨ or84a]). Let U be a group with a non-abelian simple non-normal subgroup S such that whenever A is a non-trivial subgroup of U such that S ≤ NU (A), then S ≤ A. Write V = NU (S) and ϕ : V −→ Aut(S) for the obvious group homomorphism induced by the conjugation. Construct the twisted wreath product G = S (V,ϕ) U .


33

Then G is a primitive group of type 2 such that Soc(G) = S , the base group, is complemented by a maximal subgroup of G isomorphic to U . Proof. First we see that if CU (S) = 1, then, by hypothesis, we have that S ≤ CU (S) and this contradicts the fact that S is a non-abelian simple group. Hence CU (S) = 1 and ϕ is in fact a monomorphism of V into Aut(S) and V is an almost simple group such that Soc(V ) = S. Write n = |U : V | and S = S1 × · · · × Sn . Since U acts a transitive permutation group by right multiplication on the set of right cosets of V in U , and then on the set I = {1, . . . , n}, S is a minimal non-abelian subgroup of G. Moreover, if C = CoreU (V ) = 1, then S ≤ NU (C) = U . Now C is an almost simple group with Soc(C) = S. Hence S is normal in U , giving a contradiction. Hence C = CoreU (V ) = 1. Therefore, to prove that (G, U ) is a primitive pair of type 2 with twisted wreath product action by Theorem 1.1.48, it only remains to prove U is a maximal subgroup of G. To do this, let M be a maximal subgroup U ≤ M . Observe that M = M ∩ G = of G such that πj M ∩ U Soc(G) = U M ∩ Soc(G) . All projections Rj = M ∩ Soc(G) , for j ∈ I, are conjugate by elements of M , that is, all Ri are isomorphic to the subgroup R1 and S1 ∩ U ≤ R1 ≤ S1 and M ∩ Soc(G) ≤ R1 × · · · × Rk . Observe that V ≤ NG (S1 ) by (1.3) in Proposition 1.1.31, since v = v1,v , for all v ∈ V , Since the and 1v = 1. By 1.1.18 (4), (y v )π1 = (y π1 )v , for all y ∈ M ∩ Soc(G). π1 subgroup S normalises M ∩ Soc(G), then S normalises R1 = M ∩ Soc(G) . The automorphisms induced in S1 by S are the inner automorphisms. Hence R1 is a normal subgroup of S1 , and, since S1 is a simple group, we have that R1 = 1 or R1 = S1 . In the first case, we have that M ∩ Soc(G) = 1 and then M = U . Thus, assume that the projections πj are surjective, for all j ∈ I. By 1.1.40 (23), there exists a minimal non-trivial M -invariant partition ∆ of I in blocks for the action of M on I such that πD M ∩ Soc(G) , M ∩ Soc(G) = D∈∆

πD is a full diagonal subgroup and, for each D ∈ ∆, the projection M ∩Soc(G) of the direct product i∈D Si . For each y ∈ M ∩ Soc(G) ad x ∈ M , we have that (y x )πDx = (y πD )x for any D ∈ ∆. Suppose that ∆0 is an orbit of the action of M on ∆. Then the subgroup π D M ∩ Soc(G) T = D∈∆0

is normal in M . If ∆0 is a proper subset of ∆, then there exists some j which is not in a member of ∆0 . Then Sj centralises T and then T is normal in M, Sj . Since T is a proper subgroup of Soc(G), we have that Sj ≤ M , by maximality of M . But this implies that Soc(G) ≤ M , and this is not true. Hence, M acts transitively on ∆. And so does U , since M = U M ∩ Soc(G) . Assume that each member D of ∆ has m elements of I and |∆| = l, i.e. n = lm. Since ∆ is a non-trivial partition, then m > 1.

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Suppose that l = 1. This means that M ∩Soc(G) is a full diagonal subgroup of Soc(G). Hence M = [M ∩ Soc(G)]U and M ∩ Soc(G) is a normal subgroup of M which is isomorphic to S (π1 is an isomorphism between M ∩ Soc(G) and S1 ). This gives a homomorphism ψ : U −→ Aut(S) whose restriction to V is the monomorphism ϕ. Notice that Ker(ψ) is a normal subgroup of U and, by hypothesis, if Ker(ψ) = 1, then S ≤ Ker(ψ). This contradicts the fact that ϕ is a monomorphism. Therefore Ker(ψ) = 1 and ψ is a monomorphism. Since S ψ = Inn(S) is normal in U ψ ≤ Aut(S), then S is normal in U . But this contradicts the fact that CoreU (S) = 1. Hence l > 1. The partition ∆ has l members which are blocks for the action of M (or U ) on I. Write ∆ = {D1 , . . . , Dl }. The subgroup U acts transitively on ∆. We can assume without loss of generality that 1 ∈ D1 . Let U1 denote the stabiliser of D1 by the action of U on ∆. Clearly |U : U1 | = l. x For any x ∈ V , since V ≤ NG (S1 ), then 1x = 1 and 1 ∈ D1 ∩ Dπ1 D. 1Hence x ∼ D1 = D1 and x ∈ U1 . Therefore V ≤ U1 . Since D1 = M ∩ Soc(G) = S, ∼ there exists a group homomorphism ψ : U1 −→ Aut(D1 ) = Aut(S) whose restriction to V is the monomorphism ϕ. Repeating the arguments of the above paragraph, we obtain that S ψ is normal in U1ψ and then U1 ≤ NU (S) = V . Therefore V = U1 . But now we have that l = |U : U1 | = |U : V | = n, and then m = 1. This is the final contradiction. Thus we deduce that U is a maximal subgroup of G. Remarks 1.1.51. 1. Examples of pairs U , S satisfying the conditions of the hypothesis of Theorem 1.1.48 are S = Alt(n) and U = Alt(n + 1), for n ≥ 5. In this case S is maximal in U . Also S = PSL(2, pn ) and U = PSL(2, p2n ), for pn ≥ 3 satisfies the hypothesis. Here NU (S) ∼ = PGL(2, pn ) is maximal in U . 2. In [Laf84b], J. Lafuente proved that if G is a primitive group of type 2 and U is a small maximal subgroup of G, then U is also a primitive group of type 2 and each simple component of Soc(U ) is isomorphic to a section of the simple component of Soc(G). The O’Nan-Scott Theorem proves that these are all possible configurations of primitive groups of type 2. Theorem 1.1.52 (M. O’Nan and L. Scott). Let G be a primitive group of type 2 and U a core-free maximal subgroup of G. Then one of the following holds: 1. G is an almost simple group; 2. (G, U ) is equivalent to a primitive pair with simple diagonal action; in this case U ∩ Soc(G) is a full diagonal subgroup of Soc(G); 3. (G, U ) is equivalent to a primitive pair with product action such that U ∩ Soc(G) = D1 × · · · × Dl , a direct product of l > 1 subgroups such that, for each j =1, . . . , l, the subgroup Dj is a full diagonal subgroup of a direct product i∈Ij Si , and {I1 , . . . , Il } is a minimal non-trivial G-invariant partition of I in blocks for the action of U on I.


35

4. (G, U ) is equivalent pair with product action such that the to a primitive π1 is a non-trivial proper subgroup of S1 ; in projection R1 = U ∩ Soc(G) this case R1 = V C ∩ S1 and V C/C is a maximal subgroup of X; 5. (G, U ) is equivalent to a primitive pair with twisted wreath product action; in this case U ∩ Soc(G) = 1. Proof. Recall that by 1.1.40 we can distinguish five different cases. Case 1. If n = 1, then G is an almost simple group. Thus we suppose that n > 1. Case 2. Assume that n > 1 and U ∩Soc(G) = D is a full diagonal subgroup Then there exist automorphisms ϕi ∈ Aut(S), i ∈ I, such that D = U ∩ Soc(G) = {(xϕ1 , xϕ2 , . . . , xϕn ) : x ∈ S}. Since D is normal in U and U is maximal in G, we have that U = NG (D). Let Pn be the permutation group induced by the conjugacy action of G on the simple components of Soc(G): Pn = G/Y (see 1.1.40 (13)). By 1.1.40 (23), the group Pn is transitive and primitive. We embed G in X Pn as in 1.1.40 (13) and then in Aut(S) Pn . −1 n Consider ϕ = (ϕ−1 1 , . . . , ϕn ) ∈ Aut(S) ≤ Aut(S) Pn . By conjugation by ϕ ϕ in Aut(S) Pn , we have that D = DS = {(x, . . . , x) : x ∈ S} and U ϕ = NGϕ (DS ) = Gϕ ∩ (DX × Pn ), where DX = {(x, . . . , x) : x ∈ X}. Then Gϕ = U ϕ S and, since Siϕ = Si , for all i ∈ I, the action of U ϕ and of U on I are the same. Hence, the projection of U ϕ onto Pn is surjective. By Proposition 1.1.41, we have that (Gϕ , U ϕ ) is a primitive pair with simple diagonal action and is equivalent to (G, U ). Case 3. Assume that n > 1 and U ∩ Soc(G) = D1 × · · · × Dl , a direct product of l > 1 subgroups such that, for each j= 1, . . . , l, the subgroup Dj is a full diagonal subgroup of a direct product i∈Ij Si , and {I1 , . . . , Il } is a minimal non-trivial U -invariant partition of I in blocks for the action of U on I. Suppose that the Si are ordered in such a way that I1 = {1, . . . , m}. Write K = S1 × · · · × Sm , N ∗ = NG (K), C ∗ = CG (K). Observe that I1 is a minimal block for the action of G on I. Then N ∗ acts transitively and primitively on I1 . Hence, X ∗ = N ∗ /C ∗ is a primitive group whose socle is Soc(X ∗ ) = KC ∗ /C ∗ . Put V ∗ = U ∩N ∗ . Since Soc(G) ≤ N ∗ , then N ∗ = N ∗ ∩U Soc(G) = V ∗ Soc(G) = V ∗ C ∗ K. Moreover K ∩ V ∗ = K ∩ N ∗ ∩ U = K ∩ U = D1 . Let {g1 = 1, . . . , gl } be a right transversal of V ∗ in U (and of N ∗ in G). We can assume that this transversal is ordered in such a way that D1gi = Di , for i = 1, . . . , l, and put Ki = K gi , for i = 1, . . . , l. Then G acts transitively, by conjugation of the Ki ’s, on the set {K1 , . . . , Kl }. Clearly D1 is a V ∗ -invariant subgroup of K. Suppose that D1 ≤ T1 < K1 and T1 is a V ∗ -invariant subgroup of K1 . Then T1 ×T1g2 · · ·×T1gl is U -invariant in Soc(G) and, by maximality of U , we have that T1 × T1g2 · · · × T1gl = U ∩ Soc(G) = D1 ×· · ·×Dl . Hence D1 = T1 . In other words, D1 is maximal as V ∗ invariant subgroup of K and then a maximal V ∗ C ∗ -invariant subgroup of K. Suppose that s ∈ S1 ∩V ∗ C ∗ . There exist v ∈ V ∗ and c ∈ C ∗ , such that s = vc. Now v = sc−1 ∈ CG (Si ), for i = 2, . . . , m and v ∈ S1 CG (S1 ) ≤ NG (S1 ).

36


Consider the element (t, tϕ2 . . . , tϕm ) ∈ D1 associated with some t ∈ S1 ; then (t, tϕ2 . . . , tϕm )v = (tv , tϕ2 . . . , tϕm ) ∈ D1 , since D1 is normal in V ∗ . Hence tv = t. This happens for any t ∈ S1 and therefore v ∈ CG (S1 ). Hence s ∈ CS1 (S1 ) = 1. Therefore S1 ∩ V ∗ C ∗ = 1 and then K = K ∩ V ∗ C ∗ . Since D1 ≤ V ∗ C ∗ ∩ K ≤ K and D1 is maximal as V ∗ C ∗ -subgroup of K, we have that D1 = V ∗ C ∗ ∩ K. And, finally, if M is a maximal subgroup of N ∗ such that V ∗ C ∗ ≤ M , then M ∩ K is a V ∗ C ∗ -invariant subgroup of K containing D1 . Hence D1 = V ∗ C ∗ ∩ K = M ∩ K. Now M = M ∩ N ∗ = M ∩ V ∗ C ∗ K = V ∗ C ∗ (M ∩ K) = V ∗ C ∗ . Therefore V ∗ C ∗ /C ∗ is a core-free maximal subgroup of X ∗ . Observe that (V ∗ C ∗ /C ∗ ) ∩ Soc(X ∗ ) = D1 C ∗ /C ∗ is a full diagonal subgroup of Soc(X ∗ ). Thus X ∗ is a group of Case 2. Hence (X ∗ , V ∗ C ∗ /C ∗ ) is a primitive pair with simple diagonal action. l Write Pl = G/ i=1 NG (Ki ) for the permutation group induced by the action of G by conjugation of the Ki ’s. For any g ∈ G, we write g ρ for the projection of g in Pl . On the other hand, for each g ∈ G and each i ∈ {1, . . . , l}, let ai,g be the element of N ∗ such that gi g = ai,g gj , for some j. For any a ∈ N ∗ , write a ¯ = aC ∗ for the projection of a on X ∗ . Consider the conjugacy action ∗ ψ : N −→ Aut(K) and the induced G-action on (X ∗ ) : ψ G : G −→ X ∗ Pl

G

given by g ψ = (¯ a1,g , . . . , a ¯l,g )g ρ ,

for any g ∈ G.

Arguing as in 1.1.40 (13–14), we have that 1. the map ψ G is a group homomorphism and is injective; the projection of G Gψ on Pl is surjective; G G 2. NG (K1 )ψ = Gψ ∩(X1∗ ×[X2∗ ×· · ·×Xl∗ ]Pl−1 ), where Pl−1 is the stabiliser of 1. The image of NG (K1 ) by the projection on the first component of (X ∗ ) is the whole of X1∗ ; 3. the elements of Soc(G) can be written as (e1 , eg22 , . . . , egl l ), for certain e1 , . . . , el ∈ K1 . The image by ψ G of the elements of the socle is G

e1 , e¯2 , . . . , e¯l ), (e1 , eg22 , . . . , egl l )ψ = (¯ and then (KC ∗ /C ∗ ) = Soc(X ∗ Pl ) ≤ Gψ . G

Now, for any g ∈ U , since the gi ∈ U , we have that ai,g ∈ N ∗ ∩ U = V ∗ . G G Hence U ψ ≤ Gψ ∩ (V ∗ C ∗ /C ∗ ) Pl . Since V ∗ C ∗ /C ∗ is maximal in X ∗ and G G G G U ψ is maximal in Gψ , we have that U ψ = Gψ ∩ (V ∗ C ∗ /C ∗ P1 . By G G Proposition 1.1.44, this means that (G, U ) is equivalent to (Gψ , U ψ ) which is a primitive pair with product action. π1 is a Case 4. Suppose now n > 1 and the projection R1 = U ∩ Soc(G) non-trivial proper subgroup of S1 . Moreover, R1 = V C ∩ S1 and V C is a maximal subgroup of N . Consider the embedding ψ G : G −→ X Pn of 1.1.40 (13). Then X is G almost simple and G is isomorphic to a subgroup Gψ of X Pn satisfying


37

G G all conditions of Proposition 1.1.44. Hence U ψ ≤ Gψ ∩ (V C/C) Pn . G G Since V C/C is maximal in X and U ψ is maximal in Gψ , we have that U ψ = Gψ ∩ (V C/C) Pn . Therefore (G, U ) is equivalent to a primitive pair with product action. Case 5. Assume finally that U ∩ Soc(G) = 1. Then, by Theorem 1.1.48, G ∼ = S (V,ϕ) U and the pair (G, U ) is equivalent to a primitive pair with twisted wreath product action.

If U is a core-free maximal subgroup of a primitive group G of type 2, then there are exactly three different possibilities as we saw in 1.1.40 (25): π1 = S1 , i.e. the projection π1 of U ∩ Soc(G) on S1 is sur1. U ∩ Soc(G) jective. π1 < S1 , i.e. the image of the projection π1 of 2. 1 = R1 = U ∩ Soc(G) U ∩ Soc(G) on S1 is a non-trivial proper subgroup of S1 . 1 = U ∩ Soc(G) = R1 × · · · × Rn = (U ∩ S1 ) × · · · × (U ∩ Sn ). π1 3. U ∩ Soc(G) = 1, i.e. U is a complement of Soc(G) in G. As we saw in 1.1.35, in a primitive group G of type 2, there exists a bijection between 1. the set of all conjugacy classes π1 of maximal subgroups U of G such that is a proper subgroup of S1 , the projection U ∩ Soc(G) 2. the set of all conjugacy classes of maximal subgroups of N/(S2 × · · · × Sn ) supplementing Soc(G)/(S2 × · · · × Sn ). Under this bijection, the complements, if any, of Soc(G) in G are in correspondence with the complements of Soc(G)/K1 in N/K1 . Thus, this bijection works in Cases 2 and 3. Since core-free maximal subgroups of Case 2 occur in every primitive group of type 2, these are called frequent maximal subgroups by some authors. We complete this study in the following way. Proposition 1.1.53. Let G be a primitive group of type 2. There exist bijections between the following sets: 1. the set of all of maximal subgroups U of G such that the conjugacy classes π1 is a non-trivial proper subgroup of S1 , projection U ∩ Soc(G) 2. the set of all conjugacy classes of maximal subgroups of N/(S2 × · · · × Sn ) supplementing but not complementing Soc(G)/(S2 × · · · × Sn ), and 3. the set of all conjugacy classes of core-free maximal subgroups of X. Proof. We only have to see the bijection between the sets in 2 and 3. Write K = S2 × · · · × Sn and observe that if L/C is core-free maximal subgroup of X, then obviously L/K is a maximal subgroup of N/K and N = L Soc(G). If L/K complements Soc(G)/K in N/K, then K = L ∩ Soc(G); in particular L ∩ S1 = 1. But L ∩ S1 C = C(L ∩ S1 ) = C and this contradicts the fact

38


that (L/C) ∩ (S1 C/C) is non-trivial by Proposition 1.1.16. Thus L does not complement Soc(G)/K in N . Conversely, let L/K be a maximal subgroup of N/K such that N = L Soc(G) and K < Soc(G) ∩ L. Let us see that C ≤ L. Consider L0 /K = CoreN/K (L/K). Since Soc(G)/K is a minimal normal subgroup of N/K, then L0 /K ≤ CN/K Soc(G)/K = C/K and L0 ≤ C. If L0 = C, then C ≤ L and we are done. Suppose that C/L0 is nontrivial. Since L/L0 is a core-free maximal subgroup of N/L0 , it is clear that N/L0 is a primitive group. Observe that Soc(G)L 0 /L0 is a minimal normal subgroup of N/L0 and CN/L0 Soc(G)L0 /L0 = C/L0 . Since we are assuming that C/L0 is nontrivial, the primitive group N/L0 is of type 3. Hence L/L0 complements Soc(G)L0 /L0 . This is to say that L ∩ Soc(G) ≤ L0 , i.e. L ∩ Soc(G) = L0 ∩ Soc(G). Therefore L ∩ Soc(G) is a normal subgroup of N between K and Soc(G). Since Soc(G)/K ∼ = S, a non-abelian simple group, and L supplements Soc(G) in N , we have that K = Soc(G) ∩ L. This is not possible. As we saw in 1.1.35, the existence of complements of the socle in a primitive group G of type 2 is characterised by the existence of complements of Soc(G)/(S2 × · · · × Sn ) in NG (S1 )/(S2 × · · · × Sn ). We wonder whether it is possible to obtain a characterisation of the existence of complements of Soc(G) in G in terms of complements of Soc(X) in X as we saw in 1.1.53 for supplements. The answer is partially affirmative. Corollary 1.1.54. With the notation of 1.1.40, let G be a primitive group of type 2 such that Soc(X) is complemented in X. Then Soc(G) is complemented in G. The converse does not hold in general. Proof. Suppose that there exists a subgroup Y ≤ N such that C ≤ Y and N = Y S1 and Y ∩ S1 C = C. Then it is clear that S2 × · · · × Sn ≤ Y ∩ Soc(G) ≤ Y ∩ S1 C ∩ Soc(G) = C ∩ Soc(G) = S2 × · · · × Sn and therefore Y is a complement of Soc(G)/(S2 ×· · ·×Sn ) in N/(S2 ×· · ·×Sn ). The conclusion follows by Theorem 1.1.35. It is well-known that if S = Alt(6), the alternating group of degree 6, the automorphism group A = Aut(S) is an almost simple group whose socle is non-complemented. With the cyclic group C ∼ = C2 we consider the regular wreath product H = A C. In H we consider the diagonal subgroups DS = {(x, x) : x ∈ S)} and DA = {(x, x) : x ∈ A}. Then NH (DS ) = DA C. Since DS ∼ = S, the conjugacy action of NH (DS ) on DS gives a group homomorphism ϕ : NH (DS ) −→ Aut(S). We construct the twisted wreath product G = S (NH (DS ),ϕ) H. Then Soc(G) = S1 × · · · × Sn is a minimal normal subgroup of G and it is the direct product of n = |H : NH (D)| copies of S. Moreover since CoreH NH (DS ) = 1, then CG Soc(G) = 1 by Proposition 1.1.34 (2). Hence G is a primitive group of type 2. Clearly Soc(G) is


39

complemented in G. NH (S1 ) = NH (DS ) = DA C and CH (S1 ) = Ker(ϕ) = CH (DS ) = C. Hence, X ∼ = DA ∼ = A and Soc(X) is not complemented in X. Primitive pairs (G, U ) of diagonal type, i.e. core-free maximal subgroups U of primitive groups G of type 2 such that the projection π1 of U ∩ Soc(G) on S1 is surjective, appear in Cases (2) and (3) of the O’Nan-Scott Theorem. In this case U ∩ Soc(G) is a direct product of l full diagonal subgroups, with 1 ≤ l < n, and U = NG (D). Proposition 1.1.55. Let G be a primitive group of type 2. Given a minimal non-trivial partition ∆ = {I1 , . . . , Il } of I in blocks for the action of G on I and a subgroup D = D1 × · · · × Dl , where Dj is a full diagonal subgroup of i∈Ij Si , for each j = 1, . . . , l, associated with ∆. The following statements are pairwise equivalent: 1. there exists a maximal subgroup U of G such that U ∩ Soc(G) = D; 2. NG (D) is a maximal subgroup of G; 3. G = NG (D) Soc(G). Proof. 1 implies 2. Suppose that there exists a maximal subgroup U of G such that U ∩ Soc(G) = D. Then U ≤ NG (D) and, by maximality of U in G, we have that U = NG (D). 2 implies 3. Observe that NG (D) ∩ Soc(G) = NSoc(G) (D) = D, by Lemma 1.1.38, and then Soc(G) ≤ NG (D). Therefore G = NG (D) Soc(G). 3 implies 1. Let H be a maximal subgroup of G such that NG (D) ≤ H. Then D = NSoc(G) (D) = Soc(G) ∩ NG (D) ≤ Soc(G) ∩ H. Then H ∩ Soc(G) is a direct product of full diagonal subgroups associated with a partition of I which refines {I1 , . . . , Il }, by Proposition 1.1.39. By minimality of the blocks, we have that H ∩ Soc(G) = D and therefore H = NG (D). Example 1.1.56. We construct a primitive group G of type 2 with no maximal subgroup of diagonal type. Consider the symmetric group of degree 5, H ∼ = Sym(5) and denote with S the alternating group of degree 5. If C is a cyclic group of order 2, let G be the regular wreath product G = H C. Then Soc(G) = S1 × S2 ∼ = Alt(5) × Alt(5). Any full diagonal subgroup of Soc(G) is isomorphic to Alt(5) and its normaliser N is isomorphic to Sym(5) × C2 . Observe that |G/ Soc(G)| = 8 > 4 = |N Soc(G)/ Soc(G)|. Hence N does not satisfy 3. Clearly N Soc(G) is a normal maximal subgroup of G containing N . Proposition 1.1.57. Let G be a primitive group of type 2. Two maximal subgroups U , U ∗ of G, such that U ∩Soc(G) and U ∗ ∩Soc(G) are direct products of full diagonal subgroups, are conjugate in G if and only if U ∩ Soc(G) and U ∗ ∩ Soc(G) are conjugate in Soc(G). g ∗ Proof. Suppose that U = U for some ∗g ∈ G. Then g = xh, with x ∈ g NG U ∩Soc(G) and h ∈ Soc(G). Hence U ∩Soc(G) = U ∩Soc(G) = U ∩

40


h h Soc(G) . Conversely, if U ∗ ∩ Soc(G) = U ∩ Soc(G) for some h ∈ Soc(G), h h then U ∗ = NG U ∗ ∩ Soc(G) = NG U ∩ Soc(G) = NG U ∩ Soc(G) = U h.

1.2 A generalisation of the Jordan-H¨ older theorem In the first book dedicated to Group Theory, the celebrated Traité des substitutions et des équations algébriques ([Jor70]), published in Paris in 1870, the author, C. Jordan, presents the first version of a theorem known as the Jordan-H¨ older Theorem: The length of all composition series of a finite group is an invariant of the group and the orders of the composition factors are uniquely determined by the group. Nineteen years later, in 1889, O. H¨ older ([H¨ ol89]) completed his contribution to the theorem proving that not only the orders but even the composition factors are uniquely determined by the group. In recent years a number of generalisations of the classic Jordan-H¨ older Theorem have been done. For example it has been proved that given two chief series of a finite group G, there is a one-to-one correspondence between the chief factors of the series, corresponding factors being G-isomorphic, such that the Frattini chief factors of one series correspond to the Frattini chief factors of the other (see [DH92, A, 9.13]). This result was first published by R. W. Carter, B. Fischer, and T. O. Hawkes (see [CFH68]) for soluble groups, and for finite groups in general by J. Lafuente (see [Laf78]). A further contribution is given by D. W. Barnes (see [Bar72]), for soluble groups, and again by J. Lafuente [Laf 89] for finite groups in general, describing the bijection in terms of common supplements. But if we restrict our arguments to a proper subset of the set of all maximal subgroups, we find that this is no longer true. For instance, in the elementary abelian group G of order 4, there are three maximal subgroups, say A, B, and C. If we consider the set X = {A, B}, the maximal subgroup B is a common complement in X for the chief factors A and C. Also G/A is complemented by A ∈ X. However G/C has no complement in X. In general, the key of the proof of these Jordan-H¨ older-type theorems is to prove the result in the particular case of two pieces of chief series of a group G of the form 1 < N1 < N1 × N 2

1 < N 2 < N 1 × N2

where N1 and N2 are minimal normal subgroups of G. It is not difficult to prove that if N1 N2 /N1 is supplemented by a maximal subgroup M , then M also supplements N2 (see Lemma 1.2.16), but the converse is not true. The particular case in which N1 and N2 are supplemented and either N1 N2 /N1 or N1 N2 /N2 is a Frattini chief factor is the hardest one (see [DH92, A, 9.12]) and,

1.2 A generalisation of the Jordan-H¨ older theorem

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in fact, proving the generalised Jordan-H¨ older Theorem is reduced to proving that, in the above situation, N1 N2 /N1 and N1 N2 /N2 are simultaneously Frattini chief factors of G. For this reason J. Lafuente, in [Laf89], wonders about the precise condition on a set X of maximal subgroups of a group G which allows a proof that, in the above situation, if N1 and N2 have supplements in X, then N1 N2 /N1 and N1 N2 /N1 possess simultaneously supplements in X, or, in other words, which is the precise condition on X to prove a Jordan-H¨ older-type Theorem. In this section we present, among other related results, an answer to this question. Definition 1.2.1. Given a group G and two normal subgroups K, H of G such that K ≤ H, we say that the section H/K is a chief factor of G if there is no normal subgroup of G between K and H, i.e. if N is a normal subgroup of G and K ≤ N ≤ H, then either H = N or K = N . Equivalently, H/K is a chief factor of G if H/K is a minimal normal subgroup of G/K. Hence H/K is a direct product of copies of a simple group and we have two possibilities: 1. either H/K is abelian, and there exists a prime p such that H/K is an elementary abelian p-group, or 2. H/K is non-abelian, and there exists a non-abelian simple group S such that H/K ∼ = S for all i = 1, . . . , n. = S1 × · · · × Sn , where Si ∼ Given a group G and two normal subgroups K, H of G such that K ≤ H, the group G acts by conjugation on the cosets of the section H/K: for h ∈ H and g ∈ G, then (hK)g = hg K. This action of G on H/K defines a group homomorphism ϕ : G −→ Aut(H/K) such that Ker(ϕ) = CG (H/K) = {g ∈ G : hg K = hK for all h ∈ H}. We say that C G (H/K) is the centraliser of H/K in G. We write AutG (H/K) = Im(ϕ) ∼ = G CG (H/K) for the group of automorphisms of H/K induced by the conjugation of the elements of G. The set of G composed of all elements which induce inner automorphisms on H/K is the subset C∗G (H/K) = H CG (H/K). Definition 1.2.2. Given a chief factor H/K of a group G, the inneriser of H/K in G is the subgroup C∗G (H/K) = H CG (H/K). It is clear that if H/K is abelian, then C∗G (H/K) = CG (H/K) Definition 1.2.3. Let G be a group and let F1 and F2 two chief factors of G. A map γ : F1 −→ F2 is a G-isomorphism if γ is a group isomorphism and (xg )γ = (xγ )g , for any x ∈ F1 and any g ∈ G. Two chief factors F1 , F2 of G are G-isomorphic if there exists a Gisomorphism γ : F1 −→ F2 . If two chief factors F1 , F2 of G are G-isomorphic, then write F1 ∼ =G F2 .

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Proposition 1.2.4. Let G be a group and let H1 /K1 and H2 /K2 be two chief factors of G. 1. If H1 /K1 and H2 /K2 are G-isomorphic, then CG (H1 /K1 ) = CG (H2 /K2 ). 2. In general, the converse of 1 is not true. 3. Suppose that H1 /K1 and H2 /K2 are non-abelian. Then H1 /K1 and H2 /K2 are G-isomorphic if and only if CG (H1 /K1 ) = CG (H2 /K2 ). Proof. Since clearly 1 is true, we prove 3 and give a counterexample to prove 2. Suppose that H1 /K1 and H2 /K2 are non-abelian chief factors of G such that C = CG (H1 /K1 ) = CG (H2 /K2 ). We have that Ki ≤ C ∩ Hi ≤ Hi , for i = 1, 2. Since the chief factors are non-abelian, Hi is not contained in C. Therefore Ki = C ∩ Hi , for i = 1, 2. Hence, Hi /Ki ∼ =G Hi C/C, for i = 1, 2. Observe that H1 C/C is a minimal normal subgroup of the group G/C with trivial centraliser. This means that G/C is a primitive group of type 2, by Proposition 1.1.14. Since H2 C/C is also a minimal normal subgroup of G/C, then H1 C = H2 C. Hence H1 /K1 and H2 /K2 are G-isomorphic. To see that this does not hold when the chief factors are abelian, let P be an extraspecial p-group, p an odd prime, of order p3 . Let F be a field of characteristic q, with q = p, such that F contains a primitive p-th root of unity. Then there exist p − 1 non-equivalent irreducible and faithful P -modules over F of dimension p (see [DH92, B, 9.16]). Since p − 1 > 1, we can consider two non-isomorphic such P -modules, V1 , V2 . If V is the direct sum V = V1 ⊕ V2 , construct the semidirect product G = [V ]P . The group G has two isomorphic minimal normal subgroups V1 , V2 such that CG (Vi ) = V , for i = 1, 2. But V1 and V2 are not G-isomorphic. Observe that in a primitive group G of type 3, the two minimal normal subgroups are not G-isomorphic. In other words, G-isomorphism is an equivalence relation in the set of all chief factors of G which is too “narrow” to include the case of the relation between the two minimal normal subgroups of a primitive group of type 3. J. Lafuente and P. F¨ orster [För83] propose two equivalent “enlargements” of G-isomorphism. Here we follow Lafuente’s definition. Definition 1.2.5. Let G be a group. We say that two given chief factors of G are G-connected if either they are G-isomorphic or there exists a normal subgroup N of G such that G/N is a primitive group of type 3 whose minimal normal subgroups are G-isomorphic to the given chief factors. Obviously, in a group G, two abelian chief factors are G-connected if and only if they are G-isomorphic. Proposition 1.2.6 ([Laf84a]). In a group G, the relation of being G-connected is an equivalence relation on the set of all chief factors of G.


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Proof. The only non-obvious property to prove is transitivity. Let F1 , F2 , F3 be chief factors of G such that F1 is G-connected to F2 and F2 is G-connected to F3 . We may suppose that no two are G-isomorphic. Therefore 1. there exists a normal subgroup N of G such that G/N is group of type 3 whose minimal normal subgroups are A/N B/N ∼ =G F2 , and 2. there exists a normal subgroup M of G such that G/M is group of type 3 whose minimal normal subgroups are C/M D/M ∼ =G F3 .

a primitive ∼ =G F1 and a primitive ∼ =G F2 and

Observe that CG (F2 ) = CG (B/N ) = A and also CG (F2 ) = CG (C/M ) = D. Hence A = D. Moreover N ≤ N M ≤ A and A/N is a chief factor. If N = N M , then M ≤ N ≤ A and M = N . This implies that F1 ∼ =G F3 and, in particular, F1 and F3 are G-connected. Now suppose that A = M N . Then the group G/A is isomorphic to (G/N ) (A/N ), which is the quotient group of a primitive group of type 3 over one of its minimal normal subgroups. Therefore G/A is a primitive group of type 2 by Corollary 1.1.13. On the other hand BA/A ∼ =G B/(B ∩ A) = B/N ∼ =G C/(C ∩ A) = =G F2 and, since M = A ∩ C, we have that CA/A ∼ C/M ∼ =G F2 , so BA/A and CA/A are minimal normal subgroups of G/A. Hence AC = AB. Analogously, working with G/B, we obtain that AB = BC. Note that if C is contained in B, then AB = B and then A = B, giving a contradiction. If B is contained in C, then AB = C. Since M < A ≤ AB = C and C/M is a chief factor of G, we have that A = C and then A = B, which gives again a contradiction. Hence the subgroup E = B ∩ C is a proper subgroup of B and of C. Consider the group G/E. We have that B/E ∼ =G A/(A ∩ C) = A/M ∼ =G F3 =G BC/C = AC/C ∼ and then CG (B/E) = CG (F3 ) = CG (A/M ) = C. Also

C/E ∼ =G A/(A ∩ B) = A/N ∼ =G F 1 =G BC/B = AB/B ∼

and then CG (C/E) = CG (F1 ) = CG (A/N ) = B. On the other hand, let U , V be maximal subgroups of G such that N ≤ U and U is a common complement of A/N and B/N and M ≤ V and V is a common complement of A/M and C/M . Consider the subgroup X = (U ∩ V )E. If X = G, then U = U ∩ X = (U ∩ V )(U ∩ E) = (U ∩ V )(N ∩ C) = (U ∩ V )(N ∩ M ) = U ∩ V . This contradicts the fact that U = V . Hence X is a proper subgroup of G. Now we have: XB = (U ∩ V )B = (U ∩ V N )B = U B = G

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and XC = (U ∩ V )C = (U M ∩ V )C = V C = G. Moreover B ∩ X is a normal subgroup of X and (B ∩ X)/E is centralised by CG (B/E) = C. Hence B ∩ X is a normal subgroup of XC = G. Since B/E is a chief factor of G and X is a proper subgroup of G, then B ∩ X = E. Analogously C ∩ X = E. In other words, the subgroup X is a common complement of B/E and C/E. By Corollary 1.1.13, the group G/(B ∩ C) is a primitive group of type 3. Consequently, F1 is G-connected to F3 . Definition 1.2.7. Let H/K be a chief factor of a group G. 1. We say that H/K is a Frattini chief factor of G if H/K ≤ Φ(G/K). 2. If there exists a proper subgroup M of G such that G = M H and K ≤ H ∩ M , we say that H/K is a supplemented chief factor of G and M is a supplement of H/K in G. If H/K in a non-Frattini chief factor of G, then H/K is supplemented in G by a maximal subgroup of G. 3. If H/K is a chief factor of G supplemented by a subgroup M of G and K = H ∩ M , then we say that H/K is a complemented chief factor of G and M is a complement of H/K in G. Remarks 1.2.8. Let G be a group and H/K a supplemented chief factor of G. Consider a maximal subgroup M of G supplementing H/K in G. Clearly, in the quotient group G/MG , the maximal subgroup M/MG is core-free. Therefore G/MG is a primitive group. We get K = H ∩ MG and then note that if MG < X < HMG and X is normal in G, then X = MG (X ∩ H), where K ≤ X ∩ H ≤ H. Hence X ∩ H = K or H. In both cases we have a contradiction. Thus HMG /MG is a minimal normal subgroup of the primitive group G/MG . 1. Note that if M is a maximal subgroup of type 1 or 3 of a group G, then each chief factor of G supplemented by M is in fact complemented by M . In these cases, HMG /MG is a minimal normal subgroup of the primitive group G/MG , which is of type 1 or 3, and then M ∩ HMG = MG . Therefore M ∩ H = MG ∩ H = K, as claimed. 2. Observe that HMG /MG ∼ =G H/K. Write C = CG (H/K) = CG (HMG /MG ). a) If H/K is abelian, then the primitive group G/MG is of type 1; in this case C = HMG and M/MG ∼ = G/C; therefore G/MG is isomorphic to the semidirect product [H/K](G/C). b) if H/K is non-abelian, then two cases arise: i. If C = MG , then G/MG is a primitive group of type 2; clearly Soc(G/C) = HC/C ∼ =G H/K. ii. If MG is contained in C, then G/MG is a primitive group of type 3 whose minimal normal subgroups are HMG /MG and C/MG ; in this case G/C is a primitive group of type 2 and Soc(G/C) = HC/C ∼ =G


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H/K. If S is a maximal subgroup supplementing HC/C in G, then G = HS and K = H ∩ C = H ∩ SG . Hence S is also a supplement of H/K in G and SG = C as in 2(b)i. Hence for any supplemented chief factor H/K of G, there exists a maximal subgroup M of G supplementing H/K in G such that G/MG is a monolithic primitive group. We say then that M is a monolithic supplement of H/K in G. This observation leads us to two definitions. Definition 1.2.9. For any chief factor H/K of a group G, we define the primitive group associated with H/K in G to be 1. the semidirect product [H/K] G CG (H/K) , if H/K is abelian, or 2. the quotient group G CG (H/K), if H/K is non-abelian. Notation 1.2.10. The primitive group associated with H/K is denoted by [H/K] ∗ G. It is easy to see that if H/K is a supplemented chief factor of a group G, and M is a monolithic supplement of H/K in G, then [H/K] ∗ G ∼ = G/MG . Definition 1.2.11. Let H/K be a supplemented chief factor of the group G. Assume that M is a maximal subgroup G supplementing H/K in G such that G/MG is a monolithic primitive group. We say that the chief factor Soc(G/MG ) = HMG /MG is the precrown of G associated with M and H/K, or simply, a precrown of G associated with H/K. Remarks 1.2.12. 1. If H/K is a non-abelian chief factor of the group G, then for each maximal subgroup M of G supplementing H/K in G such that G/MG is a monolithic primitive group, we have that MG = CG (H/K). Therefore the unique precrown of G associated with H/K is Soc(G/MG ) = HMG /MG

= H CG (H/K) CG (H/K) = C∗G (H/K) CG (H/K).

2. If H/K is a complemented abelian chief factor of G and M is a complement of H/K in G, then the precrown of G associated with M and H/K is Soc(G/MG ) = HMG /MG = CG/MG (HMG /MG ) = CG (H/K)/MG . For this reason it is interesting to know how many different precrowns are associated with a particular abelian chief factor. The answer, in a soluble group, is particularly elegant. Proposition 1.2.13. Let H/K be a complemented chief factor of a soluble group G. Then the function which assigns to each conjugacy class of complements of H/K in G, {M g : g ∈ G} say, the common core MG of its elements

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induces a bijection between the set of all conjugacy classes of complements of H/K in G and the set of all normal subgroups of G which complement H/K in CG (H/K). Therefore there exists a bijection between the set of all precrowns of G associated with H/K and the set of all conjugacy classes of complements of H/K in G. Proof. Write C = CG (H/K). Let N be a normal subgroup of G such that C = HN and H ∩ N = K. Then HN/N ∼ =G H/K and HN/N is a selfcentralising minimal normal subgroup of the group G/N . By Theorem 1.1.10, HN/N is complemented in G/N and all complements are conjugate. If M/N is one of these complements, then N = MG . Hence the correspondence is surjective. Let M and S be two complements of H/K in G such that N = MG = SG . Then G/N is a soluble primitive group such that and S/N , M/N are complements of Soc(G/N ) = HN/N . By Theorem 1.1.10, there exists an element g ∈ G such that S g = M . Hence the correspondence is injective. Finally observe that, since H/K is abelian, the precrowns of G associated with H/K have a common numerator CG (H/K) and different denominators MG , one for each conjugacy class of complements of H/K in G. Our next goal is to give a characterisation of the property of being Gconnected. Observe that in a primitive group G of type 3, if A and B are the minimal normal subgroups, then C∗G (A) = C∗G (B) = AB = Soc(G). This means that two G-connected chief factors have the same inneriser. But this cannot be a characterisation as we can see from the example in Proposition 1.2.4 (2). To characterise the property of being G-connected in terms of the inneriser we have to be more precise. But before that we have to include here a technical lemma, which will be crucial in our presentation. Lemma 1.2.14 (see [F¨ or88]). , . . . , Nn be normal subgroups of a group G (n ≥ 2), and consider 1. Let N1 n n n N = i=1 Ni . Suppose that i=1 Ni = 1 and that |N | = i=1 |N/Ni |. For i = 1, . . . , n, write pi : G/Ni −→ G/N for the natural projection: (gNi )pi = gN , for all g ∈ G. Then the following statements are equivalent: a) There exists a subgroup U of G which complements all the Ni ’s in G. b) There exist group isomorphisms ϕi : G/N1 −→ G/Ni , for i = 2, . . . , n, such that ϕi pi = p1 , for all i = 2, . . . , n. 2. Let N1 and N2 be two normal subgroups of a group G such that N1 ∩ N2 = 1. Write N = N1 N2 . Suppose that, for i = 1, 2, there exist group isomorphisms γi between G/Ni and a semidirect product X = [Z]Y , where Z is a normal subgroup of X, such that (N/Ni )γi = Z. Then there exists a subgroup H of G such that G = HN and H ∩ N = 1. For such H the following statements are equivalent:


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a) there exists a subgroup U of G such that H ≤ U and U is a common complement of N1 and N2 in G, and b) N1 ∼ =H N2 . If, moreover, the Ni , i = 1, 2, are abelian, then each of the previous statements is equivalent to c) N1 ∼ =G N2 . Proof. 1. Define ϕ : N −→ N/N1 × · · · × N/Nn , by xϕ = (xN1 , . . . , xNn ), for every x ∈ N . It is clear that ϕ is a group homomorphism. If x ∈ Ker(ϕ), n n then x ∈ i=1 Ni = 1. Moreover, since |N | = i=1 |N/Ni |, we have that ϕ is an isomorphism. Suppose that there exist group isomorphisms ϕi : G/N1 −→ G/Ni , for i = 2, . . . , n, such that ϕi pi = p1 , for all i = 2, . . . , n. Given g1 N1 ∈ G/N1 , we consider gi Ni = (g1 N1 )ϕi , for i = 2, . . . , n. Then (g1 N1 )ϕi pi = gi N and (g1 N1 )p1 = g1 N . Hence g1−1 gi ∈ N , for all i = 1, . . . , n. Since ϕ is an isomorphism, there exists a unique element x0 ∈ N such that (N1 , g1−1 g2 N2 , . . . , g1−1 gn Nn ) = (x0 N1 )ϕ = (x0 N1 , . . . , x0 Nn ) −1 and then x0 ∈ N1 and x−1 0 g1 gi ∈ Ni , for i = 2, . . . , n. Therefore gi Ni = g1 x0 Ni , for all i = 2, . . . , n. Then, (g1 x0 N1 )ϕi = (g1 N1 )ϕi = gi Ni = g1 x0 Ni . For the element g = g1 x0 ∈ g1 N1 ∩ g2 N2 ∩ · · · ∩ gn Nn , we have that (gN1 )ϕi = gNi , for i = 2, . . . , n. For each i = 1, . . . , n, we choose a system of coset representatives Ui = {x1i , . . . , xri } of Ni in G, such that (xk1 N1 )ϕi = xki Ni for all i = 2, . . . , n and all k = 1, . . . , r. The above arguments show that there exist zk ∈ xk1 N1 ∩ xk2 N2 ∩ · · · ∩ xkn Nn such that (zk N1 )ϕi = zk Ni , for all i = 2, . . . , n. Thus we obtain a common system of coset representatives U = {z1 , . . . , zk } of all the Ni ’s in G. Let us prove that U is a subgroup of G. If we suppose that x11 N1 = N1 , , we obtain Ni = N1ϕi = (z1 N1 )ϕi = z1 Ni , for all which forces z1 N1 = N1 n i = 2, . . . , n. Hence z1 ∈ i=1 Ni = 1 and 1 ∈ U . −1 n Suppose that (zk N1 ) −1= zt N1 for some t. Then zk zt ∈ N1 . Hence zk zt ∈ i=1 Ni = 1. Therefore zk = zt ∈ U . For zk , zj ∈ U , we have that zk zj N1 =zt N1 for some t. Then zt−1 zk zj ∈ n N1 . As above this implies that zt−1 zk zj ∈ i=1 Ni = 1 and zk zj = z t ∈ U . Therefore U is a subgroup of G and is the required common complement of all the Ni ’s in G. To prove the converse, let U be a common complement of the Ni ’s in G and define ϕi : G/N1 −→ G/Ni by (gN1 )ϕi = uNi , where g = un, u ∈ U , and n ∈ Ni . This is a well-defined homomorphism and it is injective. Since all the Ni have a common complement, they have, in particular, the same order and |G/N1 | = |G/Ni |, for all i = 2, . . . , n. Then the ϕi are group isomorphisms. Finally note that, for all i = 2, . . . , n and all g ∈ G, (gN1 )ϕi pi = uN = gN = (gN1 )p1 , i.e. ϕi pi = p1 .

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2. Since Y is a complement of Z in X and γi is a group isomorphism, −1 −1 then Hi /Ni = Y γi is a complement of Z γi = N/Ni in G/Ni , for each i = 1, 2. Consider the subgroup H = H1 ∩ H2 . Observe that G = H1 N = H1 N2 = H2 N1 . Then HN = (H1 ∩ H2 )N = (H1 ∩ H2 N1 )N2 = H1 N2 = G, and H ∩ N = H1 ∩ H2 ∩ N = N1 ∩ N2 = 1. Suppose that there exists a subgroup U of G such that H ≤ U and U is a common complement of N1 and N2 in G. Consider the isomorphisms ϕi between G/Ni = U Ni /Ni and U defined by (uNi )ϕi = u. Then (N/Ni )ϕi = U ∩ N . Write τi for the restriction of ϕi to N/Ni . Consider also the isomorphisms ρi : N/Ni −→ N3−i , i = 1, 2, given by (nNi )ρi = n3−i , for all n ∈ N , where n = n1 n2 , n1 ∈ N1 and n2 ∈ N2 . −1 Consider the isomorphism ψ = ρ−1 1 τ1 τ2 ρ2 between N2 and N1 . It is not ψ difficult to see that if n2 ∈ N2 , then n2 = n−1 1 , where n2 = un1 for u ∈ U and n1 ∈ N1 . The fact that ψ is H-invariant is an easy consequence of the fact that U is H-invariant. Therefore 2a implies 2b. Conversely, if ϕ is an H-isomorphism between N1 and N2 , then T = {aaϕ : a ∈ N1 } is a subgroup of N = N1 N2 , and H ≤ NG (T ). Consider U = HT . Since N = T Ni , then G = U Ni , for i = 1, 2. Moreover, U ∩ Ni ≤ HT ∩ N = T (H ∩ N ) = T , and then U ∩ Ni ≤ T ∩ Ni = 1, for i = 1, 2. Hence U is a common complement of N1 and N2 in G. Therefore 2b implies 2a. If, moreover, the Ni , i = 1, 2, are abelian and 2a is true, then it is easy to see that any H-isomorphism between N1 and N2 is a G-isomorphism. Proposition 1.2.15. Let G be a group and Hi /Ki , i = 1, 2, two supplemented chief factors of G. Then the following are equivalent. 1. H1 /K1 and H2 /K2 are G-connected; 2. for each i = 1, 2, there exists a precrown Ci /Ri associated with Hi /Ki , such that a) C1 = C2 , and b) there exists a common complement U of the factors Ri /(R1 ∩ R2 ) in G, i = 1, 2. Proof. 1 implies 2. If the Hi /Ki , i = 1, 2, are abelian, then H1 /K1 ∼ =G H2 /K2 . In this case C1 = CG (H1 /K1 ) = C2 = CG (H2 /K2 ) = C. Hence the numerators of the precrowns coincide. For each i = 1, 2, let Mi be a complement of Hi /Ki in G. Then C = H1 (M1 )G = H2 (M2 )G . If R = (M1 )G = (M2 )G , then both chief factors have the same precrown C/R and we can take U = G. Otherwise R1 = (M1 )G = (M2 )G = R2 . We can assume without loss of generality that R1 ∩ R2 = 1. In particular, C = R1 × R2 and R1 ∼ = G R2 ∼ =G H1 /K1 . Note that G/R1 ∼ = G/R2 ∼ = [H1 /K1 ](G/C) and the isomorphisms map the C/Ri onto H1 /K1 . By the previous lemma, there exists a common complement to R1 and R2 in G.


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∼G Suppose now that the Hi /Ki , i = 1, 2, are non-abelian and H1 /K1 = H2 /K2 . Then they have the same precrown and we can take G as complement of the trivial factor. Assume finally that Hi /Ki , i = 1, 2, are non-abelian and there exists a normal subgroup N of G such that G/N is a primitive group of type 3 with minimal normal subgroups A1 /N and A2 /N such that A1 /N ∼ =G H1 /K1 and A2 /N ∼ =G H2 /K2 . Clearly CG (A1 /N ) = A2 and CG (A2 /N ) = A1 . Hence the precrown of G associated with H1 /K1 and with A1 /N is A1 A2 /A2 and the precrown of G associated with H2 /K2 and with A2 /N is A1 A2 /A1 . Since A1 ∩ A2 = N and G/N is a primitive group of type 3, the conclusion follows easily from Theorem 1 (3c). 2 implies 1. Suppose that there exist normal subgroups C, R1 , R2 of G such that C/Ri is a precrown associated with Hi /Ki and there exists a common complement U of the factors Ri /(R1 ∩ R2 ) in G, i = 1, 2. If H1 /K1 and H2 /K2 are non-abelian, then Ri = CG (Hi /Ki ) and G/Ri is a primitive group of type 2, i = 1, 2. If R1 = R2 , then H1 /K1 and H2 /K2 are G-isomorphic and then G-connected. If R1 = R2 , we apply Corollary 1.1.13 to conclude that G/(R1 ∩ R2 ) is a primitive group of type 3 whose minimal normal subgroups are Ri /(R1 ∩ R2 ) ∼ =G Hi /Ki , i = 1, 2. Therefore H1 /K1 and H2 /K2 are G-connected. Assume that H1 /K1 and H2 /K2 are abelian. If R1 = R2 , then H1 /K1 and H2 /K2 are G-isomorphic and if R1 = R2 , then both factors are G-isomorphic to Soc(G/UG ). In both cases, they are G-connected. Lemma 1.2.16 ([Bra88]). Let G be a group and suppose that Z, Y , X, W are normal subgroups of G such that Z = XY and X ∩ Y = W . 1. If Z/X is complemented in G by M , then Y /W is complemented in G by M. 2. Moreover, if M complements Z/X and S complements X/W , then (M ∩ S)Y complements Z/Y ; in this case M ∩ S complements Z/W in G. 3. Parts 1 and 2 hold in terms of supplements. When Y /W is a non-abelian chief factor of G, we can say even more: 4. the set of monolithic supplements of Y /W in G coincides with the set of monolithic supplements of Z/X in G; 5. moreover, if X/W is an abelian chief factor of G then the (possibly empty) set of complements of X/W in G coincides with the set of complements of Z/Y in G. Proof. 1, 3. If G = M Z and X ≤ Z ∩ M, then G = M Y . Moreover W = X ∩ Y ≤ M ∩ Z ∩ Y = M ∩ Y . Then M is a supplement of Y /W in G. 2, 3. If G = M Z with X ≤ Z ∩ M and G = SX with W ≤ S ∩ X, then (M ∩ S)Y Z = (M ∩ S)Z = (M ∩ S)XY = (M ∩ SX)Y = M Y = M (XY ) = M Z = G. Moreover (M ∩ S)Y ∩ Z = (M ∩ S ∩ Z)Y contains (X ∩ S)Y and Y = W Y ≤ (X ∩ S)Y . Hence (M ∩ S)Y is a supplement of

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Z/Y in G. Moreover, in this case, G = (M ∩ S)Z and W is contained in S ∩ X = S ∩ X ∩ Z ≤ M ∩ S ∩ Z. This is to say that M ∩ S supplements Z/W in G. A substitution of the above inequalities by equalities gives the result in terms of complements. For the remainder of the proof we can suppose without loss of generality that W = 1 and then Y is a non-abelian minimal normal subgroup of G centralising X. 4. If M is a monolithic supplement of Y in G then G = M Y and the group G/MG is a monolithic primitive group of non-abelian socle. Then we have Soc(G/MG ) = MG Y /MG and CG (Y ) = CG (MG Y /MG ) = MG . Hence X ≤ CG (Y ) = MG ≤ M . Then G = M Z with X ≤ Z ∩ M and M is a monolithic supplement of Z/X in G. Conversely, if M is a monolithic supplement of Z/X in G, then, by Statement 3, M supplements Y in G. 5. Suppose that X is an abelian minimal normal subgroup of G complemented by M . Then CG (X) = XMG and then Z = X × (Z ∩ MG ). Since Y is non-abelian, this implies that Y = Z is contained in Z ∩ MG . Then Y is contained in M and M complements Z/Y . Note that the roles of X and Y in the original hypothesis can be interchanged without loss. Hence, by Statement 1, the (possibly empty) set of complements in G of X coincides with the set of complements of Z/Y in G. Lemma 1.2.17 (see [Haw67]). Let U and S be two maximal subgroups of a group G such that UG = SG . Suppose that U and S supplement the same chief factor H/K of G. Then M = (U ∩ S)H is a maximal subgroup of G such that MG = H(UG ∩ SG ). 1. Assume that H/K is abelian. Then M is a maximal subgroup of type 1 and complements the chief factors UG /(UG ∩SG ) and SG /(UG ∩SG ). Moreover M ∩ U = M ∩ S = U ∩ S. 2. Assume that H/K is non-abelian. Then either U or S is of type 3. Suppose that U is of type 3 and S is monolithic. Then UG < SG = CG (H/K). Moreover M is a maximal subgroup of type 2 of G such that M supplements the chief factor SG /UG . 3. Assume that U and S are of type 3. Then M is a maximal subgroup of type 3 of G such that M complements the chief factors HSG /MG and HUG /MG . Moreover M ∩ U = M ∩ S = U ∩ S. Proof. 1. Assume that H/K is abelian and denote C = CG (H/K). First observe that M ∩U = H(U ∩S)∩U = (H ∩U )(U ∩S) = K(U ∩S) = U ∩S, since H ∩U = K, by the abelian nature of H/K. Analogously M ∩S = U ∩S. Hence M is a proper subgroup of G. Note also that C = UG H = SG H = UG SG and UG /(UG ∩SG ) is a G-chief factor which is G-isomorphic to the precrown C/SG . Hence UG /(UG ∩ SG ) is G-isomorphic to H/K. Now, M UG = (U ∩ S)HUG = (U ∩ SUG )H = U H = G and UG ∩ SG ≤ M ∩ UG and then we deduce that M is a maximal subgroup of G which complements UG /(UG ∩ SG ). The same


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argument holds for the chief factor SG /(UG ∩ SG ). Since M also complements the chief factor C/(UG ∩ SG )H, we have that MG = H(UG ∩ SG ). 2. Assume that H/K is non-abelian. If U and S were both monolithic, of type 2, then UG = SG = CG (H/K). This is not true by hypothesis and then either U or S is of type 3. Assume that U is of type 3 and S is monolithic. It is clear that SG = CG (H/K). Observe that HUG /UG is a chief factor of G which is G-isomorphic to H/K. Then HUG /UG and SG /UG are the two minimal normal subgroups of the primitive group G/UG of type 3. Both are complemented by U ; in particular, G = U SG . Observe that M SG = H(U ∩ S)SG = H(U SG ∩ S) = HS = G and M ∩ SG = (U ∩ S)H ∩ SG contains UG H ∩ SG = UG (H ∩ SG ) = UG K = UG and then M supplements the chief factor SG /UG . Now the group G/UG H = (M/UG H)(SG H/UG H) is primitive of type 2. If the normal subgroup MG /UG H were non-trivial, then SG H would be contained in MG and so SG ≤ M . This is not possible. Hence MG = UG H. Consider a subgroup T such that U ∩ S ≤ T ≤ U . Then S = (U ∩ S)SG ≤ T SG ≤ U SG = G. By maximality of S in G we have that either S = T SG or G = T SG . Observe that T ∩ SG = U ∩ SG = UG , and then, U ∩ T SG = T (U ∩SG ) = T (T ∩SG ) = T , so U ∩S = T or U = U ∩G = T . This means that U ∩ S is a maximal subgroup of U . In the isomorphism U/(U ∩ H) ∼ = G/H, the image of (U ∩ S)/(U ∩ H) is M/H. Hence M is a maximal subgroup of G of type 2. 3. Assume now that U and S are maximal subgroups of type 3: the quotient groups G/UG and G/SG are primitive groups of type 3. If C = CG (H/K), then U complements the chief factors HUG /UG and C/UG . Analogously, S complements the chief factors HSG /SG and C/SG . In particular, UG ≤ SG and SG ≤ UG . Therefore G = U SG = SUG . Now, by an analogous argument to that presented at the end of 2, we have that M = (U ∩ S)H is a maximal subgroup of G. On the other hand, since C/SG and C/UG are chief factors of G and UG = SG , then C = UG SG . Write L = UG ∩ SG . Observe that HUG /HL ∼ =G UG /(UG ∩ HL) = UG /L ∼ =G C/SG and then HUG /HL is a chief factor of G and CG (HUG /HL) = CG (C/SG ) = HSG . Similarly HSG /HL is a chief factor of G and CG (HSG /HL) = HUG . Hence the quotient group G∗ = G/HL has two minimal normal subgroups, namely N = HSG /HL and CG∗ (N ) = HUG /HL. Observe that M (SG H) = (U ∩ S)SG H = (USG ∩ S)H = SH = G. Because U complements C/UG , we have that U ∩ UG SG = UG , so U ∩SG = UG ∩ SG = L and M ∩ HSG = (U ∩ S ∩ HSG )H = (U ∩ SG)H = HL. Analogously G = M (UG H) and M ∩ HUG = HL. Therefore, the maximal subgroup M ∗ = M/HL of G∗ complements N and CG∗ (N ). By Proposition 1.1.12, the group G∗ is a primitive group of type 3. Hence MG = HL. Finally observe that M ∩ U = H(U ∩ S) ∩ U = (H ∩ U)(U ∩ S) = K(U ∩ S) = U ∩ S. Analogously M ∩ S = U ∩ S.

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Definitions 1.2.18. Let X be a set of maximal subgroups of a group G. 1. If X is non-empty, then the X-Frattini subgroup of G is defined to be the intersection of the cores of all members of X. It is denoted by ΦX (G). If X = ∅, we define ΦX (G) = G. 2. Let H/K be a chief factor of G. We say that H/K is an X-supplemented (respectively, X-complemented) chief factor if it has a supplement (respectively, complement) in X; otherwise H/K is said to be an X-Frattini chief factor. 3. If C ∗ /N is a precrown of G associated with an X-supplemented chief factor H/K of G, we shall say that C ∗ /N is an X-precrown of G associated with H/K. Notation 1.2.19. Let N be a normal subgroup of a group G and let X be a set of maximal subgroups of G. We write X/N = {Z/N : Z ∈ X and N is contained in Z} and if ϕ : G −→ H is a group homomorphism, we write Xϕ = {S ϕ : S ∈ X}. The following lemma will be used frequently in the sequel. Lemma 1.2.20. Let X be a set of maximal subgroups of a group G. Let H/K be a chief factor of a group G. 1. H/K is an X-Frattini chief factor of G if and only if H/K ≤ ΦX/K (G/K). 2. If A is a normal subgroup of G contained in K, then H/K is X-Frattini in G/K if and only if (H/A) (K/A) is X/A-Frattini in G/A. Furthermore, if H/K is X-supplemented in G, then a maximal subgroup U ∈ X is a supplement of H/K in G if and only if U/A is an X/A-supplement of (H/A) (K/A) in G/A. Definition 1.2.21. A set X of maximal subgroups of a group G is said to be solid for the Jordan-H¨ older theorem, or simply JH-solid, if it satisfies the following condition: SG and both supplement a chief factor H/K of (JH) If U , S ∈ X with UG = G, then there exists M ∈ X such that MG = (UG ∩ SG )H. Applying Lemma 1.2.17, the set of all maximal subgroups of a group G that supplement a single chief factor, the set Max(G) of all maximal subgroups of a group G, and the set Max∗ (G) of all monolithic maximal subgroups of a group G are JH-solid. Note that Φ(G) = {M ∈ Max(G)} = {M ∈ Max∗ (G)}. We will use the following results in inductive arguments.


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Theorem 1.2.22. Let G be a group factorised as G = M N , where M is a subgroup of G and N is a normal subgroup of G. Then G/N ∼ = M/(N ∩ M ), and we have the following. 1. If N = H n < · · · < H0 = G

(1.4)

is a piece of chief series of G, then M ∩ N = M ∩ H n < · · · < M ∩ H0 = M

(1.5)

is a piece of chief series of M . If S is a maximal subgroup of G which supplements a chief factor Hi /Hi+1 in (1.4), then M ∩ S is a maximal subgroup of M which supplements the chief factor (Hi ∩ M )/(Hi+1 ∩ M ) in (1.5). Moreover, the core of M ∩ S in M is (M ∩ S)M = M ∩ SG . 2. Conversely, if M ∩ N = Mn < · · · < M 0 = M (1.6) is a piece of chief series of M , then N = Mn N < · · · < M0 N = M N = G

(1.7)

is a piece of chief series of G. If U is a maximal subgroup of M which supplements a chief factor Mi /Mi+1 in (1.6), then U N is a maximal subgroup of G which supplements the chief factor Mi N/Mi+1 N in (1.7). Moreover, the core of U N in G is (U N )G = UM N . Lemma 1.2.23. Let X be a JH-solid set of maximal subgroups of a group G and N a normal subgroup of G. 1. The set X/N is a JH-solid set of maximal subgroups of G/N. 2. Suppose that the subgroup M supplements N in G: G = M N. Then the set (X ∩ M )/(N ∩ M ) = {(S ∩ M )/(N ∩ M ) : N ≤ S ∈ X} is a JH-solid set of maximal subgroups of M/(N ∩ M ). Moreover, if ϕ is the isomorphism between G/N and M/(N ∩ M ) then we have that (X/N )ϕ = (X ∩ M )/(M ∩ N ). Now we can prove the announced strengthened form of the Jordan-H¨ older theorem for chief series of finite groups and give an answer to Lafuente’s question. To do this we proceed following Lafuente’s arguments in [Laf89]. It must be observed that these arguments deal with the modular lattice of all normal subgroups of a group in which we can use the Duality Principle (see [Bir69, Chapter 1, Theorem 2]). Notation 1.2.24. If A/B and C/D are sections of a group G, then we write A/B C/D (or C/D A/B) if C = AD and B = A ∩ D.

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Observe that if A/B C/D, then A/B ∼ =G C/D. In particular, A/B is a chief factor of G if and only if C/D is a chief factor of G. Lemma 1.2.25. Let K and H be normal subgroups of a group G and let K = Y0 < Y1 < · · · < Ym−1 < Ym = H be a piece of chief series between K and H. Suppose that X ∗ /X is a chief factor of G between H and K. 1. If X ∗ Yj = XYj , then X ∗ Yk = XYk for j ≤ k ≤ m. 2. If X ∗ ∩ Yj−1 = X ∩ Yj−1 , then X ∗ ∩ Yk−1 = X ∩ Yk−1 , for 1 ≤ k ≤ j. 3. If X ∗ Yj−1 > XYj−1 , then X ∗ Yk−1 > XYk−1 , for 1 ≤ k ≤ j and X ∗ ∩ Yj−1 = X ∩ Yj−1 . In this case, X ∗ Yj−1 /XYj−1 X ∗ Yk−1 /XYk−1 X ∗ /X. 4. If X ∗ ∩ Yj > X ∩ Yj , then X ∗ ∩ Yk > X ∩ Yk , for j ≤ k ≤ m and X ∗ Yj = XYj . Moreover X ∗ /X (X ∗ ∩ Yk )/(X ∩ Yk ) (X ∗ ∩ Yj )/(X ∩ Yj ).

Proof. Note that Statement 1 and its dual, which is Statement 2, are obvious. 3. By Statement 1, if X ∗ Yj−1 > XYj−1 , then X ∗ Yk−1 > XYk−1 , for 1 ≤ k ≤ j. On the other hand, we have (X ∗ Yk−1 )(XYj−1 ) = X ∗ Yj−1

X ∗ Yk−1 = X ∗ (XYk−1 ).

Moreover X ≤ X(X ∗ ∩ Yj−1 ) = X ∗ ∩ XYj−1 ≤ X ∗ . Since X ∗ /X is a chief factor of G, then either X = X(X ∗ ∩Yj−1 ) = X ∗ ∩XYj−1 or X ∗ ∩XYj−1 = X ∗ . In the last case X ∗ ≤ XYj−1 and then X ∗ Yj−1 = XYj−1 , contrary to our supposition. Hence X ∗ ∩ Yj−1 ≤ X and then X ∗ ∩ Yj−1 = X ∩ Yj−1 . By Statement 2, X ∗ ∩ Yk−1 = X ∩ Yk−1 . Hence X ∗ ∩ XYk−1 = X(X ∗ ∩ Yk−1 ) = X(X ∩ Yk−1 ) = X and XYj−1 ∩ X ∗ Yk−1 = (XYj−1 ∩ X ∗ )Yk−1 = X(Yj−1 ∩ X ∗ )Yk−1 = X(X ∩ Yj−1 )Yk−1 = XYk−1 . Statement 4 is dual of Statement 3.

Definition 1.2.26. Let A/B, A/C and C/D be chief factors of a group G such that A/B C/D. If X is a set of maximal subgroups of G, such that A/B is X-Frattini and C/D is X-supplemented, we will say that the situation A/B C/D is an X-crossing. We write [A/B C/D] to denote an X-crossing.


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Remark 1.2.27. 1. If A/B C/D and A/B is X-supplemented, then C/D is X-supplemented, by Lemma 1.2.16. 2. If [A/B C/D] is an X-crossing, then C/D is abelian. If C/D is a non-abelian X-supplemented chief factor, then A/B is also X-supplemented, by Lemma 1.2.16 (4), against our supposition. Next we see a characterisation of JH-solid sets of monolithic maximal subgroups in terms of X-crossing situations. Theorem 1.2.28. Let X be a set of maximal subgroups of a group G. 1. Assume that X is JH-solid. Let Z/Y , Y /W and X/W be chief factors of G. If [Z/X Y /W ] is an X-crossing, then [Z/Y X/W ] is an Xcrossing. Moreover, in this case, a maximal subgroup U ∈ X supplements Y /W if and only if U supplements X/W . 2. Conversely, assume that X is a monolithic set of maximal subgroups of G such that whenever we have chief factors Z/Y , Y /W and X/W of G such that [Z/X Y /W ] is an X-crossing, then [Z/Y X/W ] is an X-crossing. Then X is JH-solid. Proof. 1. We can assume that W = 1. We have to prove that if X and Y are minimal normal subgroups of G, Z/X is X-Frattini chief factor and Y is X-suplemented, then Z/Y is X-Frattini and X is X-supplemented. Assume that U is an X-supplement of Y . If X ≤ U , then G = U Z and X ≤ U ∩ Z, so U supplements Z/X. This contradiction yields that X is not contained in U and then U supplements X. Suppose that, in this case, there exists S ∈ X supplementing Z/Y . Then S also supplements X. Since Y ≤ UG and Y ≤ SG , by the property (JH), there exists M ∈ X such that MG = (UG ∩ SG )X. If Z ≤ M , then Z = Z ∩ MG = X(UG ∩ SG ∩ Z) = X(UG ∩ Y ) = X, which is a contradiction. Hence M supplements Z/X, which we have supposed to be X-Frattini. We deduce that Z/Y must be an XFrattini chief factor of G. 2. Suppose that we have U , S ∈ X, both supplementing the same chief factor H/K of G and UG = SG . Since U and S are monolithic, the chief factor H/K must be abelian, by Lemma 1.2.17 (2). Therefore K = U ∩ H = UG ∩ H = SG ∩ H = S ∩ H. Observe that C = CG (H) = HSG = HUG = UG SG . Write A = UG ∩ SG . Then C/HA = HUG /HA ∼ =G UG /(UG ∩ HA) = UG /A ∼ =G C/SG and then C/HA is a chief factor of G and C/HA UG /A. Observe that UG /A is X-complemented by S. Suppose that C/HA is X-Frattini. Then [C/HA UG /A] is an X-crossing. By hypothesis, [C/UG HA/A] is an Xcrossing. But C/UG is obviously X-complemented by U . This contradiction yields that C/HA is X-complemented in G, i. e. there exists M ∈ X such that G = M C and HA = MG . Therefore X is JH-solid.

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Proposition 1.2.29. With the hypotheses of Lemma 1.2.25, assume that X ∗ /X is an X-supplemented chief factor of G. Let j = max{j : X ∗ Yj−1 /XYj−1 is X-supplemented chief factor of G} and set Y ∗ = Yj and Y = Yj −1 . Then Y ∗ /Y is X-supplemented. Furthermore the following conditions are satisfied: 1. If X ∗ Y ∗ = XY ∗ , then X ∗ Y ∗ = XY ∗ = X ∗ Y . Write R∗ = X ∗ Y ∗ and R = XY . Then X ∗ /X R∗ /R Y ∗ /Y . Moreover X ∗ ∩Y = X ∩Y = X ∩Y ∗ . Write S = X ∩ Y and S ∗ = X ∗ ∩ Y ∗ , then X ∗ /X S ∗ /S Y ∗ /Y . 2. If X ∗ Y ∗ = XY ∗ , then [X ∗ Y ∗ /XY ∗ X ∗ Y /XY ] is an X-crossing and X ∗ /X X ∗ Y /XY and XY ∗ /XY Y ∗ /Y . In particular, in both cases X ∗ Y /XY and XY ∗ /XY are X-supplemented chief factors of G. Proof. Observe that X ∗ Y0 /XY0 = X ∗ /X is X-supplemented. Hence j is well-defined. Assume that XY ∗ = XY . Then X ∗ Y ∗ = X ∗ Y . So X ∗ Y ∗ /XY ∗ = ∗ X Y /XY is X-supplemented, giving a contradiction to the election of j . Therefore XY ∗ /XY Y ∗ /Y and XY ∗ /XY is a chief factor. 1. Assume that X ∗ Y ∗ = XY ∗ . Then XY ≤ X ∗ Y ≤ X ∗ Y ∗ = XY ∗ . Therefore X ∗ Y ∗ = X ∗ Y because X ∗ Y > XY by hypothesis. From part 3 of Lemma 1.2.25, it follows that X ∗ /X R∗ /R Y ∗ /Y . On the other hand, X ∗ = XY ∗ ∩ X ∗ = (X ∗ ∩ Y ∗ )X. Hence X ∗ /X (X ∗ ∩ Y ∗ )/(X ∩ Y ∗ ). Now, from part 3 of Lemma 1.2.25, X ∗ ∩ Y = X ∩ Y = X ∩ Y ∗ . Thus, X ∗ /X S ∗ /S Y ∗ /Y . In this case R∗ /R = X ∗ Y ∗ /XY = XY ∗ /XY = X ∗ Y /XY is Xsupplemented, by definition of j . 2. Now consider X ∗ Y ∗ = XY ∗ . From the choice of j , it follows that ∗ ∗ X Y /XY ∗ is an X-Frattini chief factor of G. Then XY ≤ XY ∗ ∩ X ∗ Y ≤ X ∗ Y . If XY ∗ ∩ X ∗ Y = X ∗ Y , it follows that X ∗ Y ∗ = XY ∗ contrary to our assumption. Hence XY = XY ∗ ∩ X ∗ Y and [X ∗ Y ∗ /XY ∗ X ∗ Y /XY ] is an X-crossing. Moreover X ∗ /X X ∗ Y /XY and XY ∗ /XY Y ∗ /Y . Since [X ∗ Y ∗ /XY ∗ X ∗ Y /XY ] is an X-crossing, we have that X ∗ Y /XY and XY ∗ /XY are X-supplemented chief factors of G. Proposition 1.2.30. With the hypotheses of Lemma 1.2.25, assume that X ∗ /X is an X-Frattini chief factor of G. Let j = min{j : (X ∗ ∩ Yj )/(X ∩ Yj ) is an X-Frattini chief factor of G} and set Y ∗ = Yj and Y = Yj −1 . Then Y ∗ /Y is X-Frattini. Furthermore the following conditions are satisfied: 1. If X ∗ ∩Y = X∩Y , then X∩Y = X∩Y ∗ = X ∗ ∩Y . Write S ∗ = X ∗ ∩Y ∗ and S = X ∩ Y . Then X ∗ /X S ∗ /S Y ∗ /Y . Moreover X ∗ Y = X ∗ Y ∗ = XY ∗ . Write R = XY and R∗ = X ∗ Y ∗ , then X ∗ /X R∗ /R Y ∗ /Y .


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2. If X ∗ ∩ Y = X ∩ Y , then [(X ∗ ∩ Y ∗ )/(X ∩ Y ∗ ) (X ∗ ∩ Y )/(X ∩ Y )] is an X-crossing and X ∗ /X (X ∗ ∩ Y ∗ )/(X ∩ Y ∗ ) and (X ∗ ∩ Y ∗ )/(X ∗ ∩ Y ) Y ∗ /Y . In particular, in both cases (X ∗ ∩ Y ∗ )/(X ∗ ∩ Y ) and (X ∗ ∩ Y ∗ )/(X ∩ Y ∗ ) are X-Frattini chief factors of G. Proof. This is the dual statement of Proposition 1.2.29.

Definition 1.2.31. Given a set X of maximal subgroups of a group G, we say that two chief factors of G, say X ∗ /X and Y ∗ /Y , are X-related if one of these properties is satisfied: 1. There exists an X-supplemented chief factor R∗ /R such that X ∗ /X R∗ /R Y ∗ /Y , 2. There exists an X-crossing [A/Z T /B] such that X ∗ /X Z/B and T /B Y ∗ /Y . 3. There exists an X-Frattini chief factor S ∗ /S such that X ∗ /X S ∗ /S Y ∗ /Y , 4. There exists an X-crossing [A/Z T /B] such that X ∗ /X A/Z and A/T Y ∗ /Y . The importance of the X-relation becomes clear in the following theorem. Theorem 1.2.32. Let X be a JH-solid set of maximal subgroups of a group G. If the chief factors X ∗ /X and Y ∗ /Y are X-related, then 1. X ∗ /X and Y ∗ /Y are G-connected, and 2. X ∗ /X is X-Frattini if and only if Y ∗ /Y is X-Frattini. 3. If X ∗ /X and Y ∗ /Y are X-supplemented, there exists a common X-supplement to both. Furthermore, if X is composed of monolithic maximal subgroups of G then any two X-related chief factors are G-isomorphic. Proof. 1. Observe that in Cases 1 and 3 of the definition of X-relation, we have that X ∗ /X is G-isomorphic to Y ∗ /Y . Suppose that there exists an X-crossing [A/Z T /B] such that X ∗ /X Z/B and T /B Y ∗ /Y . Since X is JH-solid, there exists a common X-supplement U of Z/B and T /B, by Theorem 1.2.28. Then T UG /UG and ZUG /UG are minimal normal subgroups of the primitive group G/UG . If ZUG = T UG , then Z/B ∼ =G T /B; in this case X ∗ /X ∼ =G Y ∗ /Y . Otherwise G/UG is a primitive group of type 3 whose minimal normal subgroups are T UG /UG and ZUG /UG . Since X ∗ /X ∼ =G Z/B and Y ∗ /Y ∼ =G T /B, then X ∗ /X and Y ∗ /Y are G-connected. The analysis of Case 4 is analogous. Observe that if all elements of X are monolithic maximal subgroups of G, then necessarily ZUG = T UG in the above analysis. Therefore X ∗ /X ∼ =G Y ∗ /Y .

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2. If X ∗ /X is X-Frattini, then we are not in Case 1 of the definition of X-relation. Suppose that there exists an X-crossing [A/Z T /B] such that X ∗ /X Z/B and T /B Y ∗ /Y . Then [A/T Z/B] is an Xcrossing by Theorem 1.2.28. Then Z/B is X-complemented. This implies that X ∗ /X is X-supplemented by Lemma 1.2.16. Therefore we are not in Case 2 of Definition 1.2.31 either. If we are in Case 3, then Y ∗ /Y is X-Frattini by Lemma 1.2.16. In Case 4, [A/T Z/B] is an X-crossing by Theorem 1.2.28 and again Y ∗ /Y is X-Frattini by Lemma 1.2.16. 3. If X ∗ /X and Y ∗ /Y are X-supplemented, we are either in Case 1 or in Case 2 of Definition 1.2.31. In Case 1, if U is an X-supplement of R∗ /R, then U supplements X ∗ /X and Y ∗ /Y , In Case 2, there exists an X-crossing [A/Z T /B] such that X ∗ /X Z/B and T /B Y ∗ /Y . By Theorem 1.2.28, we know that there exists a common X-supplement U to Z/B and T /B. By Lemma 1.2.16, U also X-supplements X ∗ /X and Y ∗ /Y , Lemma 1.2.33. Under the hypotheses of Lemma 1.2.25, assume that X ∗ /X and Yj /Yj−1 are X-related. 1. X ∗ /X and Yj /Yj−1 are X-supplemented in G if and only if X ∗ Yj−1 /XYj−1 is X-supplemented in G. 2. X ∗ /X and Yj /Yj−1 are X-Frattini if and only if (X ∗ ∩ Yj )/(X ∩ Yj ) are X-Frattini. Proof. 1. Set Y ∗ = Yj , Y = Yj−1 and assume that there exists an Xsupplemented chief factor R∗ /R such that X ∗ /X R∗ /R Y ∗ /Y . Since (X ∗ Y )R = R∗ and XY ≤ R, then XY < X ∗ Y . By part 3 of Lemma 1.2.25, X ∗ Y /XY X ∗ /X and in particular, X ∗ Y /XY is a chief factor. On the other hand, XY ≤ X ∗ Y ∩ R ≤ X ∗ Y . As X ∗ Y is not contained in R, then R∗ /R X ∗ Y /XY . Therefore X ∗ Yj−1 /XYj−1 is X-supplemented in G. Now suppose that there exists an X-crossing [A/Z T /B] such that Z/B X ∗ / Xy T /B Y ∗ /Y . Since XY ≤ B and (X ∗ Y )B = Z, we have that X ∗ Y > XY and, as above, X ∗ /X X ∗ Y /XY . Now Z = (X ∗ Y )B and X ∗ Y ∩ B = Y (X ∗ ∩ B) = XY . Hence Z/B X ∗ Y /XY . Therefore X ∗ Yj−1 /XYj−1 is X-supplemented in G. The converse follows from part 3 of Lemma 1.2.25. 2. This is the dual statement of 1. Theorem 1.2.34. Let G be a group and X a JH-solid set of maximal subgroups of G. For any pair K, H of normal subgroups of G such that K < H and two pieces of chief series of G between K and H K = X 0 ≤ X1 ≤ · · · ≤ Xn = H and K = Y0 ≤ Y1 ≤ · · · ≤ Ym = H, then n = m and there exists a unique permutation σ ∈ Sym(n) such that Xi /Xi−1 and Yiσ /Yiσ −1 are X-related, for 1 ≤ i ≤ n. Furthermore


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iσ = max{j : Xi Yj−1 /Xi−1 Yj−1 is X-supplemented} if Xi /Xi−1 is X-supplemented, and iσ = min{j : (Xi ∩ Yj )/(Xi−1 ∩ Yj ) is X-Frattini} if Xi /Xi−1 is X-Frattini. Proof. We can assume without loss of generality that m ≤ n. Write X ∗ = Xi , X = Xi−1 , Y ∗ = Yiσ and Y = Yiσ −1 . By Proposition 1.2.29, if X ∗ /X is X-supplemented, then so is Y ∗ /Y . Furthermore, if X ∗ Y ∗ = XY ∗ , then X ∗ /X R∗ /R Y ∗ /Y , where R∗ = X ∗ Y ∗ = X ∗ Y and R = XY , by part 1 of Proposition 1.2.29. Hence R∗ /R is X-supplemented by definition of iσ . So, this is Case 1 of the definition of X-relation. And if X ∗ Y ∗ = XY ∗ , then we are in Case 2 of Definition 1.2.31 by part 2 of Proposition 1.2.29. Dually, by Proposition 1.2.30, if X ∗ /X is X-Frattini, then so is Y ∗ /Y . Furthermore, if X ∗ ∩ Y ∗ = X ∩ Y ∗ , then X ∗ /X S ∗ /S Y ∗ /Y , where S ∗ = X ∗ ∩ Y ∗ and S = X ∩ Y , by part 1 of Proposition 1.2.30. Hence S ∗ /S is X-Fratttini by definition of iσ . So, this is Case 3 of the definition of X-relation. and if X ∗ ∩ Y ∗ = X ∩ Y , then we are in Case 4 of Definition 1.2.31. Therefore, in any case, Xi/Xi−1 and Yiσ /Yiσ −1 are X-related, for 1 ≤ i ≤ n. Now we prove that the map σ : {1, . . . , n} → {1, . . . , m} defined above is injective. Write Z ∗ = Xk and Z = Xk−1 , where i < k and iσ = k σ . Suppose that X ∗ /X is X-supplemented; then so are Y ∗ /Y and Z ∗ /Z. Assume that X ∗ Y ∗ = XY ∗ . From X ∗ ≤ Z we get that ZY ∗ = ZY . Since Z ∗ /Z is X-supplemented and k σ = j, Z ∗ Y /ZY is a chief factor of G and then ZY = ZY ∗ < Z ∗ Y = Z ∗ Y ∗ . By part 2 of Proposition 1.2.29, ZY ∗ /ZY Y ∗ /Y . In particular ZY ∗ > ZY and yields a contradiction. Hence X ∗ Y ∗ > XY ∗ . Then [X ∗ Y ∗ /XY ∗ X ∗ Y /XY ] is an X-crossing by part 2 of Proposition 1.2.29. The chief factor X ∗ Y ∗ /X ∗ Y is X-Frattini. Since k σ = j, then Z ∗ Y /ZY and ZY ∗ /ZY are X-supplemented chief factors of G. As X ∗ ≤ Z gives X ∗ Y ≤ ZY and X ∗ Y ∗ ≤ ZY ∗ . Observe that ZY ∗ = (ZY )(X ∗ Y ∗ ). Moreover ZY ∩ X ∗ Y ∗ = X ∗ (Z ∩ Y ∗ )Y In the situation Y ≤ (Z ∩ Y ∗ )Y ≤ Y ∗ and Y ∗ /Y chief factor of G, we cannot have ZY ∩Y ∗ = Y ∗ , since this would imply Y ∗ ≤ ZY and then ZY ∗ = ZY and this contradicts the fact that ZY ∗ /ZY is a chief factor. Hence ZY ∩X ∗ Y ∗ = X ∗ Y . In other words, ZY ∗ /ZY X ∗ Y ∗ /X ∗ Y and we deduce that ZY ∗ /ZY is X-Frattini by Lemma 1.2.16. This is a contradiction. We have shown that the restriction to σ to the subset I of {1, . . . , n} composed of all indices i corresponding to X-supplemented chief factors Xi /Xi−1 , is injective. Applying dual arguments we show that the restriction of σ to the subset of {1, . . . , n} \ I composed of all indices i corresponding to X-Frattini chief factors Xi /Xi−1 , is injective. By the arguments at the beginning of the proof, σ is injective. Therefore n = m and σ is a permutation of the set {1, . . . , n}.

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Finally if τ is any permutation with the above properties, then the definition of σ requires that iτ ≤ iσ for all i ∈ I and iτ ≥ iσ for all i ∈ {1, . . . , n} \ I by Lemma 1.2.33. Consequently σ = τ . Remark 1.2.35. By Theorem 1.2.32, the bijection constructed in Theorem 1.2.34 satisfies that if Xi /Xi−1 and Yiσ /Yiσ −1 are X-supplemented, there exists a common X-supplement to both. Clearly when Xi /Xi−1 , and Yiσ /Yiσ −1 , is abelian we can change the X-supplementation by X-complementation. But we can go further and say the same even for non-abelian X-complemented chief factors. We know, by Theorem 1.1.48, the existence of non-abelian chief factors complemented by maximal subgroups. Observe that if Xi /Xi−1 and Yiσ /Yiσ −1 are X-complemented non-abelian chief factors, then we are in Case 1 of Definition 1.2.31, since Case 2 is not possible by Remark 2 of 1.2.27. If U is an X-complement of the non-abelian chief factor Xi /Xi−1 and Yiσ /Yiσ −1 R∗ /R Xi /Xi−1 , then U also supplements R∗ /R, by of Lemma 1.2.16 (4), and the same for Yiσ /Yiσ −1 . Observe that U/UG is a small maximal subgroup of the primitive group G/UG of type 2. Then, Soc(G/UG ) = Xi UG /UG = R∗ UG /UG = Yiσ UG /UG and UG = U ∩ Yiσ UG . Thus, U ∩ Yiσ = UG ∩ Yiσ = Yiσ −1 and U complements Yiσ /Yiσ −1 . Theorem 1.2.36. Let G be a group and X a set of monolithic maximal subgroups of G. Then the following conditions are equivalent: 1. X is a JH-solid set. 2. For any pair K, H of normal subgroups of G such that K < H and two pieces of chief series of G between K and H K = X0 ≤ X 1 ≤ · · · ≤ X n = H

and

K = Y0 ≤ Y1 ≤ · · · ≤ Ym = H,

then n = m and there exists σ ∈ Sym(n) such that a) Xi /Xi−1 ∼ =G Yiσ /Yiσ −1 ; σ b) Xi /Xi−1 is X-Frattini if and only if Yiσ /Y i −1 is X-Frattini; c) if Xi /Xi−1 is X-supplemented (respectively, complemented) in G, there exists a maximal subgroup U ∈ X of G such that G supplements (respectively, complements) both Xi /Xi−1 and Yiσ /Yiσ −1 . Proof. After Theorem 1.2.34 we have only to see that 2 implies 1. Suppose that we have U , S ∈ X, both supplementing the same chief factor H/K of G and UG = SG . Since U and S are monolithic, the chief factor H/K must be abelian, by Lemma 1.2.17 (2). Therefore K = U ∩ H = UG ∩ H = SG ∩ H = S ∩ H. Observe that C = CG (H) = HSG = HUG = UG SG . Write A = UG ∩ SG . Then C/HA = HUG /HA ∼ =G UG /(UG ∩ HA) = UG /A ∼ =G C/SG and then C/HA is a chief factor of G and C/HA UG /A. Observe that UG /A is X-complemented by S and C/UG is obviously X-complemented by U . By


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Statement 2, all chief factors of G between C and A are X-complemented. In particular C/HA is X-complemented in G, i. e. there exists a maximal subgroup M ∈ X such that G = M C and HA = M ∩ C. This implies that MG = HA. Therefore X is JH-solid. Corollary 1.2.37. If X is a JH-solid set of maximal subgroups of a group G and H is a normal subgroup of G such n that all chief factors H/Ki , i = 1, . . . , n, of G are X-supplemented, and i=1 Ki = K, then every chief factor between K and H is X-supplemented. j Proof. Denote K j = i=1 Ki and K 0 = H. Then K = K n ≤ K n−1 ≤ · · · ≤ K 0 = H is a piece of a chief series of G. Assume that K i = K i+1 . Then H = K i Ki+1 , K i /K i+1 is a chief factor of G and K i /K i+1 ∼ =G H/Ki+1 . If M is an Xsupplement of H/Ki+1 in G, then M is an X-supplement of K i /K i+1 in G by Lemma 1.2.16 (1). We deduce that all chief factors in the above series are X-supplemented. Now apply Theorem 1.2.36 to conclude the proof. Corollary 1.2.38. Let X be a JH-solid set of monolithic maximal subgroups of a group G and write R = ΦX (G). Suppose that N is a normal subgroup of G such that N = N1 × · · · × Nn, where Ni is a minimal normal subgroups of G, 1 ≤ i ≤ n. If R ∩ N = 1, then every chief factor of G below N is X-supplemented in G. Proof. We use induction on n. If n = 1, the result is obvious. Thus we assume that n ≥ 2. If N1 is X-Frattini, then N1 ≤ R ∩ N = 1, giving a contradiction. Hence there exists M ∈ X such that G = M N1 . The quotient group G/MG is a monolithic primitive group and then N MG /MG = N1 MG /MG = Soc(G/MG ). Then N = N1 × (N ∩ MG ). By Theorem 1.2.36, every piece of chief series of G between N1 and N has exactly n − 1 chief factors and so every piece of chief series of G below N0 = N ∩ MG has exactly n − 1 chief factors. Since the normal subgroup N0 is contained in Soc(G), we have that N0 can be written as a direct product of n−1 minimal normal subgroups of G. Since R∩N0 = 1, it follows that every chief factor of G below N0 is X-supplemented by induction. Since clearly M supplements N/N0 , we have that all chief factors of G below N are X-supplemented, by Theorem 1.2.36. Observe that in a primitive group G of type 3 with minimal normal subgroups N1 and N2 , if M is a core-free maximal subgroup, then X = {M } is a JH-solid set of maximal subgroups of G, R = MG = 1, and N = Soc(G) satisfies that R ∩ N = 1. However neither N/N1 nor N/N2 are X-supplemented. Remarks 1.2.39. 1. Given a modular lattice L, J. Lafuente in [Laf89] introduced the concept of M-set in L and he proved a general Jordan-H¨ older theorem in modular lattices with an M-set.

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In fact, Theorem 1.2.28 shows that, for a set of maixmal subgroups X of a group G, the set MX of all X-supplemented chief factors of G is an M-set in the modular lattice N of all normal subgroups of G if and only if X is JH-solid. 2. For JH-solid sets containing some maximal subgroups of type 3, a converse of Theorem 1.2.34, giving an equivalence analogous to Theorem 1.2.36, does not hold. Let T be a non-abelian simple group and consider the group G which is the direct product of three copies of T : G = T1 × T2 × T3 . Suppose that X is the set whose elements are three monolithic maximal subgroups M1 , M2 , and M3 , such that (Mi )G = Tj × Tk , where {i, j, k} = {1, 2, 3}. Consider the subgroups U1 = ∆23 × T1 = {(x, y, y) : x, y ∈ T }, which is a maximal subgroup of type 3 of G such that (U1 )G = T1 , and U2 = ∆13 × T2 = {(x, y, x) : x, y ∈ T }, a maximal subgroup of type 3 of G such that (U2 )G = T2 . The set X ∪ {U1 , U2 } is not a JH-solid set of maximal subgroups: the minimal normal subgroup T3 is supplemented by U 1 and U2 but no maximal subgroup of X ∪ {U1 , U2 } has core (U1 )G ∩ (U2 )G T3 = T3 . On the other hand, it is easy to see that no chief factor of G is X-Frattini, and that any two G-isomorphic chief factors are supplemented by exactly one element of X, so the conditions of Theorem 1.2.36 (2) hold. In other words, X is a JH-solid set of maximal subgroups of G.

1.3 Crowns The concept of crown of a soluble group was introduced in [Gas62]. In this seminal paper, W. Gasch¨ utz analyses the structure of the chief factors of a soluble group G as G-modules. Associated with a G-module a, there exists a section of the group, called a-Kopf, or crown in English, such that, viewed as a G-module, is completely reducible and homogeneous with a composition series of length the number of complemented chief factors G-isomorphic to a in any chief series of G. These crowns are complemented sections of G. The study of non-soluble chief factors made by J. Lafuente in [Laf84a], and, in particular, the introduction of the concept of G-connected chief factors, allowed him to discover that some sections associated with non-abelian chief factors can be constructed enjoying similar properties to Gasch¨ utz’s crowns. This originated the concept of crown of a non-abelian chief factor. Given a group G, fixing a JH-solid set of maximal subgroups X of G and restricting ourselves to X-supplemented chief factors, we can presume, after the results of Section 1.2, that most of the known results on crowns hold for the so-called X-crowns. The aim of this section is to present results in this direction. Let us start with the following observations. Let G be a group and H/K a non-abelian chief factor of G. If there exists a maximal subgroup M of G of type 3 complementing H/K, then the primitive group G/MG has two minimal

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normal subgroups, namely HMG /MG and C/MG , where C = CG (H/K) and HMG ∩ C = MG . In this case C∗G (H/K) = HC. By Remark 1.2.8 (2b), there exists a monolithic maximal supplement S of HMG /MG such that SG = C. Analogously, since CG (C/MG ) = HMG , there exists a monolithic maximal supplement T of C/MG such that TG = HMG . This means that, although the sets E1 = {N : C ∗ /N is a precrown associated with a chief factor G-connected to H/K} E2 = {MG : M is a maximal subgroup of G supplementing a chief factor G-connected to H/K}, and E3 = {MG : M is a maximal subgroup of G supplementing a chief factor G-isomorphic to H/K} in general are different, in fact {N : N ∈ E1 } = {N : N ∈ E2 } = {N : N ∈ E3 }. If we replace the set of all maximal subgroups for a proper JH-solid subset, the above equalities are not longer true. Let G be a primitive group of type 3 with minimal normal subgroups A and B. If M and S are monolithic maximal subgroups with MG = A and SG = B, and X = {M, S}, then X is JH-solid and E4 = {MG : M is a maximal subgroup in X supplementing a chief factor G-connected to A} = {A, B} and E5 = {MG : M is a maximal subgroup in X supplementing a chief factor G-isomorphic to A} = {B}. Then

{N : N ∈ E4 } = 1 < B =

{N : N ∈ E5 }.

These observations motivate the following definitions. Definitions 1.3.1. 1. Let H/K be a supplemented chief factor of a group G and consider the set E composed of all cores of the monolithic maximal subgroups of G which supplement chief factors G-connected to H/K. Write R = {N : N ∈ E} and C ∗ = C∗G (H/K). Then we say that the factor C ∗ /R is the crown of G associated with H/K.

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2. Let X be a JH-solid set of monolithic maximal subgroups of a group G and H/K an X-supplemented chief factor of G. Write C ∗ = C∗G (H/K) and consider the normal subgroup RX =

{MG : M ∈ X and M supplements a chief factor G-connected to H/K}.

Then C ∗ /RX is the X-crown of G associated with H/K. Obviously a crown of G associated with a supplemented chief factor of G is just an X-crown of G for the set X = Max∗ (G) of all monolithic maximal subgroups of G. Theorem 1.3.2. Let X be a JH-solid set of monolithic maximal subgroups of a group G and H/K an X-supplemented chief factor of G. Write C ∗ /RX for the X-crown of G associated with H/K. Then C ∗ /RX = Soc(G/RX ). Furthermore 1. every minimal normal subgroup of G/RX is an X-supplemented chief factor of G which is G-connected to H/K, and 2. no X-supplemented chief factor of G over C ∗ or below RX is G-connected to H/K. In other words, there exist m normal subgroups A1 , . . . , Am of G such that C ∗ /RX = A1 /RX × · · · × Am /RX where Ai /RX is an X-supplemented chief factor G-connected to H/K, for i = 1, . . . , m, and m is the number of X-supplemented chief factors G-connected to H/K in each chief series of G. Moreover, Φ(G/RX ) = Oq (G/RX ) = 1, for each prime q dividing the order of |H/K|. Proof. We can write RX = R = N1 ∩ · · · ∩ Nr , such that C ∗ /Ni are Xprecrowns associated with chief factors G-connected to H/K and r is minimal with this property. Consider the group monomorphism ψ : C ∗ /R = C ∗ /(N1 ∩ · · · ∩ Nr ) −→ C ∗ /N1 × · · · × C ∗ /Nr c(N1 ∩ · · · ∩ Nr ) −→ (cN1 , . . . , cNr ) for any c ∈ C ∗ . Observe that ψ is compatible with the action of G: g c(N1 ∩ · · · ∩ Nr )ψ = (cN1 , . . . , cNr )g = (cg N1 , . . . , cg Nr ) ψ = cg (N1 ∩ · · · ∩ Nr ) .

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From minimality of r, we have that C ∗ = Ni (N1 ∩ · · · ∩ Ni−1 ), for i ≤ r, and then (N1 ∩ · · · ∩ Ni−1 )/(N1 ∩ · · · ∩ Ni ) ∼ =G C ∗ /Ni . Therefore the chain R = (N1 ∩ · · · ∩ Nr ) ≤ (N1 ∩ · · · ∩ Nr−1 ) ≤ · · · ≤ N1 ≤ C ∗ is a piece of chief series of G and each chief factor is G-connected to H/K. Hence the order |C ∗ /R| = |H/K|r and ψ is an isomorphism. By Corollary 1.2.37, every chief factor of G between R and C ∗ is X-supplemented in G. Therefore, there exist r normal subgroups A1 , . . . , Ar of G such that C ∗ /R = A1 /R × · · · × Ar /R, where Ai /R is a X-supplemented chief factor G-connected to H/K, i = 1, . . . , r. Suppose that H0 /K0 is a X-supplemented chief factor of G which is Gconnected to H/K and let M ∈ X be a supplement of H0 /K0 in G. Then H0 ≤ C ∗ . Observe that since R ≤ MG , then H0 ≤ R. Therefore no Xsupplemented chief factor of G over C ∗ or below R is G-connected to H/K. By Theorem 1.2.36, the number of X-supplemented chief factors Gconnected to H/K in each chief series of G is an invariant of the group and coincides with the length of any piece of chief series of G between R and C ∗ . If B/R is a minimal normal subgroup of G/R and B ∩ C ∗ = R, then B ≤ CG (A1 /R) which is contained in C ∗ by Proposition 1.2.15. This contradiction implies that C ∗ /R = Soc(G/R). Since every minimal normal subgroup of G/R is supplemented in G/R, we have that Φ(G/R) = 1 = Oq (G/R), for each prime q dividing the order of |H/K|. Corollary 1.3.3 ([Laf84a]). Two supplemented chief factors of a group G define the same crown of G if and only if they are G-connected. Let C ∗ /R be the X-crown of G associated with an X-supplemented chief factor H/K. Applying Theorem 1.3.2, we have that C ∗ /R = (RX /R) × (C0 /R), and the X-crown of G associated to H/K is isomorphic to C0 /R which is a direct product of X-supplemented components of C ∗ /R. Corollary 1.3.4. Let X be a JH-solid set of monolithic maximal subgroups of a group G. Let H/K be an X-supplemented chief factor of a group G and write C ∗ /R for the X-crown of G associated with H/K. Then 1. if H/K is abelian and p is the prime dividing |H/K|, then C ∗ = CG (H/K) = C and C/R = Soc(G/R) = F(G/R) = Op (G/R) is a completely reducible and homogeneous G-module over GF(p) whose composition factors are G-isomorphic to H/K and the length of a composition series of C/R, as G-module, is the number of X-complemented G-chief factors G-isomorphic to H/K in each chief series of G;

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2. if H/K is non-abelian, then {Aj /R : j = 1, . . . , m} is the set of all minimal normal subgroups of G/R; in particular, if C ∗ /R is a chief factor of G, then R = CG (H/K) and G/R ∼ = [H/K] ∗ G is a primitive group of type 2. Proof. Applying Theorem 1.3.2, C ∗ /R = Soc(G/R). 1. If H/K is abelian, then H/K is a p-group for some prime p and C ∗ = C = CG (H/K) is the common centraliser of the chief factors of G between C and R. Then CG/R (C/R) = C/R = F (G/R) and Statement 1 follows from Theorem 1.3.2. 2. Suppose now that H/K is non-abelian. Then {Aj /R : j = 1, . . . , m}, as in Theorem 1.3.2, are the minimal normal subgroups of G/R. Finally observe that if C ∗ /R is a chief factor, then C ∗ /R is the X-precrown of G associated with H/K and R = CG (H/K). Our main goal is now to prove that in every group G, we can order in some sense the X-crowns of G to obtain a chief series of G in which some G-isomorphic images of the X-crowns are placed one after the other, possibly separated by X-Frattini chief factors, and all X-supplemented chief factors which are G-connected are consecutive. We need a technical proposition to explore how the crowns of the quotient group are related to the crowns of the original group. We will use it in inductive arguments. Proposition 1.3.5. Let X be a JH-solid set of monolithic maximal subgroups of a group G and let N be a normal subgroup of G contained in some maximal subgroup of G in X. 1. For any X-crown C ∗ /R of G, either a) C ∗ ≤ RN or ) is an X/N -crown of G/N . b) RN < C ∗ and (C ∗ /N ) (RN/N 2. For any X/N -crown (C0∗ /N ) (R0 /N ) of G/N , there is an X-crown C ∗/R of G such that C0∗ = C ∗ and R0 = RN . Proof. 1. Assume that C ∗ is not contained in RN . Then, applying Corollary 1.3.4, there exists a minimal normal subgroup A/R of G/R such that A is not contained in RN . Therefore AN/RN is a chief factor of G which is G-isomorphic to A/R. Hence RN < AN ≤ C∗G (AN/RN ) = C∗ . Applying Theorem 1.3.2, AN/RN is X-supplemented and clearly (C ∗ /N ) (N R/N ) is the X/N -crown of G/N associated with the chief factor AN/N/RN/N of G/N . 2. Let (C0∗ /N ) (R0 /N ) be the X/N -crown of G/N associated with an X/N -supplemented chief factor (H/N ) (K/N ) of G/N . Then (H/N ) (K/N ) is G-isomorphic to the chief factor H/K of G and H/K is X-supplemented with H/K. It follows that in G. Consider the X-crown C ∗/R of G associated C0∗ /N = C∗G/N (H/N ) (K/N ) = C∗G (H/K) N and then C0 = C ∗ .

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On the other hand, it is clear that RN ≤ R0 . In addition, every chief factor of a given chief series of G between RN and R0 is X-supplemented in G and G-connected to H/K. Since, by Theorem 1.3.2, the number of X/N -supplemented chief factors of each chief series of G/N which are G/N connected to (H/N ) (K/N ) is exactly the number of chief factors of G/N between R0 /N and C ∗ /N , we have that RN = R0 . Lemma 1.3.6. Let G be a group with Φ(G) = 1. There exists a crown C ∗ /R and a non-trivial normal subgroup D of G such that C ∗ = R × D. Proof. We argue by induction on the order of G. Let M be a minimal normal subgroup of G. Since Φ(G) = 1, it follows that M is supplemented in G and we can consider the crown C0∗ /R0 and a precrown C0∗ /N0 associated with M in G. We know that C0∗ = N0 × M . If N0 = R0 , then the normal subgroup D = M and the crown C ∗ /R = C0∗ /R0 fulfils our requirements. Assume that R0 < N0 . This means that R0 × M < C0∗ . Write F/M = Φ(G/M ). By Proposition 1.3.5, (C0∗ /M ) (R0 M/M ) is a crown of G/M associated with the chief factors of G/M , i.e. the chief factors ofG over M , which are G-connected to M . Since, by Theorem 1.3.2, Φ (G/M ) (R0 M/M ) = 1, we have that F ≤ R0 M and then F = M × (F ∩ R0 ). Put N = F ∩ R0 . Suppose that N = 1, and let A be a minimal normal subgroup of G contained in N . Recall that all monolithic maximal subgroups of G form a JH-solid set and their intersection is Φ(G). Since obviously M A ∩ Φ(G) = 1, we can apply Corollary 1.2.38 and deduce that the chief factor M A/M is supplemented in G. But this contradicts the fact that M A/M ≤ F/M = Φ(G/M ). Therefore F = M and Φ(G/M ) = 1. By induction, there exists a crown (C1∗ /M ) (R1 /M ) and a non-trivial normal subgroup D1 /M of G/M , such that C1∗ /M = (R1 /M ) × (D1 /M ). Suppose first that (C1∗ /M ) (R1 /M ) is the crown associated with the chief factors G-connected to M . Then C1∗ = C0∗ and R1 = R0 × M . In this case, we take D = D1 and C ∗ /R = C0∗ /R0 . Note that M = D1 ∩R1 = D1 ∩(R0 ×M ) = (D1 ∩ R0 ) × M . Hence D1 ∩ R0 = 1. Suppose now that the chief factors of G between M and D1 are not Gconnected to M . If C0∗ /M ≤ (R0 M/M )(D1 /M ), then C0∗ = R0 (C0∗ ∩D1 ). Then C0∗ /R0 ∼ =G (C0∗ ∩D1 )/(R0 ∩D1 ) and M ≤ C0∗ ∩D1 . Hence all chief factors of G between (R0 ∩ D1 ) × M and C0∗ ∩ D1 are G-connected to M by Theorem 1.3.2. Since no chief factor of G between M and D1 is G-connected to M , we deduce that C0∗ ∩ D1 = (R0 ∩ D1 ) × M . Then C0∗ = R0 M , against our assumption. Hence, by Proposition 1.3.5, we have that (R0 M/M )(D1 /M ) < C0∗ /M and then R0 ≤ R0 M ≤ R0 D1 ≤ C0∗ . Applying Theorem 1.3.2, every chief factor of G between R0 M and R0 D1 is G-connected to M . Since D1 R0 /M R0 ∼ =G D1 /M (D1 ∩ R0 ) and we are assuming that all chief factors of G between M and D1 are not G-connected to M , we have that D1 = M (D1 ∩ R0 ). In this case, take D = D1 ∩ R0 = 1 and C ∗ = C1∗ . This completes the proof.

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We prove now the corresponding result for a JH-solid set X of monolithic maximal subgroups of a group G. Proposition 1.3.7. Let G be a group and X a JH-solid set of monolithic maximal subgroups of G such that ΦX (G) = 1. There exists an X-crown C ∗ /RX of G and a non-trivial normal subgroup D of G such that C ∗ = RX × D. Proof. Observe first that Φ(G) ≤ ΦX (G) = 1. By Lemma 1.3.6, there exists a crown C ∗ /R and a non-trivial normal subgroup D of G such that C ∗ = R×D. Consider the G-isomorphism ϕ : C ∗ /R −→ D. If C ∗ /R = (A1 /R) × · · · × (Ar /R), then all the images (Ai /R)ϕ = Ni are minimal normal subgroups of G below D, the Ni are G-connected, C ∗ /R is the crown of G associated with them and D = N1 × · · · × Nr . Moreover, by Theorem 1.2.38, every chief factor of G below D is X-supplemented in G. Hence R = RX and C ∗ /R = C ∗ /RX is the X-crown of G associated with the Ni . Theorem 1.3.8 (see [F¨ or88]). Let X be a non-empty JH-solid set of monolithic maximal subgroups of a group G and. 1. Let C1∗ /R1 , . . . , Cn∗ /Rn denote the X-crowns of G. Then there exists a permutation σ ∈ Sym(n) and a chain of normal subgroups of G 1 = C(0) ≤ R(1) < C(1) ≤ R(2) < C(2) ≤ · · · < C(n−1) ≤ R(n) < C(n) ≤ G such that G/C(n) = ΦX/C(n) (G/C(n) ) (including the case G = C(n) ) and for i = 1, . . . , n, we have R(i) /C(i−1) = ΦX/C(i−1) (G/C(i−1) ),

Ci∗σ = Riσ C(i) ,

Riσ ∩ C(i) = R(i) .

2. Moreover, if N is a normal subgroup of G and C(k−1) ≤ N ≤ R(k) , for some k ∈ {1, . . . , n}, 1 = N/N = C(k−1) N/N ≤ R(k) /N < C(k) /N ≤ R(k+1) /N < · · · < C(n) /N ≤ G/N is a chain of G/N enjoying the corresponding property. Proof. 1. We use induction on |G|. Clearly ΦX (G) is contained in each Ri . Moreover, every X-supplemented chief factor of G is G-isomorphic to an X/ΦX (G)-supplemented chief factor of G/ΦX (G). Hence, by Proposition 1.3.5 (2), we can assume without loss of generality that ΦX (G) = R(1) = 1. By Proposition 1.3.7, there exists an X-crown Ck∗ /Rk of G and a normal subgroup C(1) of G such that Ck∗ = Rk × C(1) . If G = C(1) , the result is trivial. If C(1) is a proper subgroup of G and X/C(1) = ∅, then ΦX/C(1) (G/C(1) ) = G/C(1) or, in other words, no maximal subgroup of G in X contains C(1) . Hence no chief factor of G over C(1) is X-supplemented.

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In this case there exists exactly one X-crown of G and the theorem holds trivi¯ = X/C(1) is non-empty, i.e. C(1) is contained in some ally. Assume that X maximal subgroup of G in X. Then we can apply the inductive hypothesis to ¯ = G/C(1) . Observe that if C ∗ ≤ Rj C(1) , for some j = k, the quotient group G j ∗ ∗ ∼ then Cj /Rj =G (Cj ∩ C(1) )/(Rj ∩ C(1) ) and the chief factors of G between Cj∗ and Rj are G-connected to some chief factors of G below C(1) and therefore to the chief factors in Ck∗ /Rk , which is not possible by Theorem 1.3.2. Hence, by Proposition 1.3.5, Rj C(1) < Cj∗ , for all j ∈ {1, . . . , n} \ {k}. Therefore ¯ j : j = k} are the X-crowns of G ¯ and, by induction, there exists a {C¯j∗ /R bijection τ : {2, . . . , n} −→ {1, . . . , n} \ {k}, and a chain of normal subgroups ¯ of G ¯ (2) < C¯(2) ≤ R ¯ (3) < C¯(3) ≤ · · · < C¯(n−1) ≤ R ¯ (n) < C¯(n) ≤ G ¯ 1 = C¯(1) ≤ R ¯ C¯(n) ), and for i = 2, . . . , n + 1, we have ¯ C¯(n) = ΦX/ such that G/ ¯ C ¯(n) (G/ ¯ (i) /C¯(i−1) = ΦX/C¯ ¯ C¯(i−1) ), R (G/ (i−1)

¯ iτ C¯(i) , C¯i∗τ = R

¯ iτ ∩ C¯(i) = R ¯ (i) . R

¯ (j) and C(j) /C(1) = C¯(j) , Now, just take the inverse images R(j) /C(1) = R for j = 2, . . . , n. The required permutation is σ such that 1σ = k and iσ = iτ , for i = 2, . . . , n. 2. Assume that N is a normal subgroup of G such that C(k−1) ≤ N ≤ R(k) . Every X-supplemented chief factor H/K of G such that N ≤ K is G-isomorphic to an X/N -supplemented chief factor of G/N and therefore is G-connected to some chief factor between R(j) /N and C(j) /N , for some j ≥ k. The X/N -crown of G/N associated with (H/N ) (K/N ) is (Cj∗σ /N ) (Rj σ /N ) and clearly we have that Cj∗σ /N = (Rj σ /N )(C(j) /N ) and (Rj σ /N ) ∩ (C(j) /N ) = R(j) /N . In addition, R(i) /C(i−1) is equal to ΦX/C(i−1) (G/C(i−1) ). Hence (R(i) /N ) (C(i−1) /N ) = Φ(X/N )/(C(i−1) /N ) (G/N ) (C(i−1) /N ) for all i = k + 1, . . . , n. Now R(k) /C(k−1) = ΦX/C(k−1) (G/C(k−1) ) implies that ΦX/N (G/N ) = R(k) /N . Now, the result we were looking for becomes clear. Corollary 1.3.9 (see [Gas62] and [F¨ or88]). Let X be a JH-solid set of monolithic maximal subgroups of a group G. If C1∗ /R1 , . . . , Cn∗ /Rn are the X-crowns of G, there exists a permutation σ ∈ Sym(n) and a chief series of G 1 = F1,0 < F1,1 < · · · < F1,m1 = N1,0 < N1,1 < · · · < N1,k1 = F2,0 < F2,1 < · · · < F2,m2 = N2,0 < N2,1 < · · · < N2,k2 ... = Fn,0 < Fn,1 < · · · < Fn,mn = Nn,0 < Nn,1 < · · · < Nn,kn = Fn+1,0 < Fn+1,1 < · · · < Fn+1,mn+1 = G

70


such that 1. the Fi,j /Fi,j−1 are X-Frattini chief factors of G, 2. the Ni,j /Ni,j−1 are X-supplemented chief factors of G satisfying that Ni,j /Ni,j−1 is G-connected to Ni ,j /Ni ,j −1 if and only if i = i ; moreover Ciσ /Riσ is the X-crown associated with Ni,j /Ni,j−1 ; 3. Fi,mi / Fi,j = ΦX/Fi,j (G/Fi,j ), for each i = 1, . . . , n+1 and j = 1, . . . , mi −1. Let X be a JH-solid set of monolithic maximal subgroups of a group G. Then if C ∗ /RX is the X-crown of G associated with a chief factor H/K, and RX = G0 < G1 < · · · < Gn = C ∗ is a piece of chief series of G, then the subgroup V =

n

{Mi : Mi is an X-supplement of Gi /Gi−1 }.

i=1

is a supplement (if H/K is abelian, then V is a complement) of C ∗ /RX in G, by repeated applications of Lemma 1.2.16 (2). However, this supplement depends on the choice of the chief series and on the choice of the maximal subgroups and it is not preserved by epimorphic images. The following example is illustrative of these problems. Example 1.3.10. Denote by N the elementary abelian group of order 32 . The cyclic group Z of order 2 acts on N by inversion. Form the semidirect product G = [N ]Z and write A = a, N = a, b, and Z = z. Consider the JH-solid set of maximal subgroups X = {M1 = a, z, M2 = b, az, M3 = ab, z, M4 = a2 b, z}. The X-crown of G associated with any of the chief factors below N is N = CG (A). All subgroups of the form Vij = Mi ∩Mj , i = j, are complements 4 of N in G. Note that i=1 Mi = 1. Consider now the group G/A. Observe that X/A = {M1 /A} and the X/Acrown of G/A associated with N/A is N/A itself. Notice that the subgroup V23 A/A = a, bz/A is a complementchief factor!complemented of N/A in G/A which does not belong to X/A. Proposition 1.3.11. Let G be a group and X a JH-solid set of monolithic maximal subgroups of G. Assume that if U and S are two distinct elements of X, then UG = SG . Let C ∗ /RX be the X-crown of G associated with the X-supplemented chief factor F . Consider the set XF = {M ∈ X : M supplements a chief factor G-connected to F }. We define the subgroup T = T(G, X, F ) = {M : M ∈ XF }. Clearly TG = RX . 1. Assume that if U and S are two distinct elements of X and both supplement a chief factor H/K of G, then M = (U ∩ S)H ∈ X. Then the subgroup T satisfies the following properties.

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71

a) For any piece of chief series of G, RX = G0 < G1 < · · · < Gn = C ∗ and any family {Mi ∈ X : i = 1, . . . , n} such that Mi is a supplement of Gi /Gi−1 , for each i = 1, . . . , n, we have T(G, X, F ) =

n

Mi

i=1

and T(G, X, F ) is a supplement (a complement, if F is abelian) of C ∗ /RX in G. b) For any normal subgroup N of G such that F is G-connected with an X/N -supplemented chief factor F1 of G/N , then T(G/N, X/N, F1 ) = T N/N . 2. Conversely, assume that the subgroup T satisfies the above Conditions 1a and 1b. Then, if U and S are elements of XF such that UG = SG , and both supplement a chief factor H/K of G, then M = (U ∩ S)H ∈ XF . Proof. 1. a) Fix a piece of chief series of G, RX = G0 < G1 < · · · < Gn = C ∗ and a family {Mi ∈ X : i = 1, . . . , n} such n that Mi is a supplement of Gi /Gi−1 , for each i = 1, . . . , n and write D = i=1 Mi . If XF = {Mi ∈ X : i = 1, . . . , n}, then there is nothing to prove. Assume that there exists U ∈ XF \ {Mi ∈ X : i = 1, . . . , n}. Then U supplements Gj /Gj−1 , for some j = 1, . . . , n. Since U and Mj are distinct monolithic XF -supplements of the same chief factor Gj /Gj−1 and UG = Mj G , we have that Gj /Gj−1 is abelian by Lemma 1.2.17 (2), and so is C ∗ /RX . Therefore U and Mj complement Gj /Gj−1 . By hypothesis, M = (U ∩Mj )Gj ∈ XF . Now we have that Mj ∩M = (Mj ∩U )(Mj ∩Gj ) = (Mj ∩U )Gj−1 = Mj ∩U and analogously U ∩ M = Mj ∩ U . Then D ∩ U = D ∩ M . Observe that M complements a chief factor Gk /Gk−1 , for some k > j. If M = Mk , then D ∩ U = D. If M = Mk , repeat the previous argument replacing U by M and Mj by Mk . Observe also that Gn /Gn−1 is self-centralising in G/Gn−1 and so the latter group is primitive. Hence (Mn )G = Gn−1 . Therefore Mn is the unique maximal subgroup of G in ∈ XF complementing the last chief factor. Since the other possible maximal subgroups in XF do not change the n intersection, it follows that T(G, X, F ) = i=1 Mi . Moreover, ifwe apply repeatedly Lemma 1.2.16 (2), we deduce that the n subgroup T = i=1 Mi is a supplement (complement if the crown is abelian) ∗ of C /RX in G. b) Let N be a minimal normal subgroup of G such that G/N has an X/N to F . By Proposition 1.3.5, supplemented chief factor F1 which is G-connected we have that RX N < C ∗ and (C ∗ /N ) (RX N/N ) is the X/N -crown of G/N associated with F1 . If N ≤ RX , it is clear that T /N = T(G/N, X/N, F1 ). Assume that N is G-connected to F , i.e. RX < RX N . We consider a piece of chief series of G RX = G 0 < G 1 = R X N < G 2 · · · < G n = C ∗ .

72


By Statement 1a we have that n

T = T(G, X, F ) =

{Mi : Mi is an X-supplement of Gi /Gi−1 }.

i=1

Since N ≤

n

Mi and G = M1 N , we have that n n

n TN = Mi N = M1 N ∩ Mi = Mi ,

i=2

i=1

and T N/N =

i=2 n

i=2

(Mi /N ) = T(G/N, X/N, F ).

i=2

An inductive argument proves the validity of the Statement 1b for any normal subgroup N of G such that F is G-connected with a chief factor of G/N . 2. Assume that the subgroup T satisfies Statement 1a and Statement 1b and suppose that U and S are elements of XF such that UG = SG , and both supplement the same chief factor H/K of G. Since U and S are monolithic and UG = SG , H/K is abelian by Lemma 1.2.17 (2). Observe that C ∗ = C = CG (H/K) = HUG = HSG and K = UG ∩ H = SG ∩H. Suppose that RX < UG ∩SG . Let N/R X be a chief factor of G such that N ≤ UG ∩SG . It is clear that F2 = (HN/N ) (KN/N ) is a chief factor of G/N and the X/N -crown of G/N associated with F2 is C/N . We see that in the group G/N all hypotheses hold for X/N and T(G/N, X/N, F2 ) = T N/N . To see that T N/N satisfies Statement 1a, let 1 = N/N = G1 /N < · · · < Gn /N = C/N be a piece of chief series of G/N and Mi /N an X/N -complement of (Gi /N ) (Gi−1 /N ) for i = 2, . . . , n. Let M1 be an X-complement of N/RX . Then RX < N = G1 < · · · < Gn = C is a piece of chief series of G and Mi is Statement an X-complement ofGi /Gi−1 , for i = 1, . . . , n. Since G satisfies n n 1a, n ∩ ( i=2 Mi ) = i=2 Mi . we have that T = i=1 Mi and then T N = M1 N n Since T satisfies Statement 1b, we have T N/N = i=2 (Mi /N ) and T N/N satisfies Statement 1a. Clearly, T N/N satisfies Statement 1b. Arguing by induction, we have that the maximal subgroup M/N = (U/N ) ∩ (S/N ) (HN/N ) = (U ∩ S)H /N ∈ XF /N and then M = (U ∩S)H ∈ XF . Hence, we can assume that UG ∩SG = RX = 1. This implies that K = 1 and H is a minimal normal subgroup of G. Observe that also UG and SG are minimal normal subgroups of G. We can consider these three different pieces of chief series of G below C: 1 0, π a set of primes, and G a group. 1. An irreducible KG-module V is called π-special if the dimension of V is a π-number and whenever S is a subnormal subgroup of G and U is a composition factor of VS , then det(x on U ) = 1 for all π -elements x of S. 2. Suppose that P = {πi : i ∈ I} is a partition of P, the set of all primes. An irreducible KG-module V is called P-factorable if V = Uj1 ⊗ · · · ⊗ Ujn for some πji -special modules Uji , πji ∈ P, i = 1, . . ., n, and ji = jk when i = k.

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121

3. An irreducible KG-module V is called π-factorable if V is P-factorable for P = {π, π }. It turns out that if U and W are respectively π-special and π -special irreducible KG-modules, then U ⊗W is irreducible. Moreover, if U and W are respectively π-special and π -special irreducible KG-modules, and U ⊗ W ∼ = U ⊗ W , then U ∼ = U and W ∼ = W (see [CK87, 2.4] for more details and notation). It is also true the following: Lemma 2.5.8 ([CK87, 2.2]). Let G be a group, K a field, π a set of primes and V be π-special KG-module. If S is a subnormal subgroup of G, then every irreducible constituent of VS is π-special. The next lemma equips us with the basic arguments to prove closure properties of “Fitting type.” Its proof is rather technical and can be seen in [CK87]. Lemma 2.5.9. Let G be a group, K an algebraically closed field and P = {πi : i ∈ I} a partition of the set P of all primes. 1. If V is a P-factorable KG-module and N is a normal subgroup of G, then any irreducible KN -submodule of VN is a P-factorable KN -module. 2. Suppose that M and N are normal subgroups of G such that G = M N . Let V be an irreducible KG-module such that all irreducible KM -submodules of VM and all irreducible KN -submodules of VN are P-factorable. Then V is a P-factorable KG-module. And now we prove the main result. Theorem 2.5.10. Let K be an algebraically closed field of characteristic a prime p. Let P = {πi : i ∈ I} be a partition of the set P of all primes. For each i ∈ I, let Xi be a Fitting formation. Denote X = {Xi : i ∈ I}. For every soluble group G, denote by M(G) the class of all irreducible P-factorable KG-modules V such that V = V1 ⊗ · · · ⊗ Vn(V ) . Suppose further that 1. Vj is a πi(j) -special KG-module, and 2. G/ CG (Vj ) ∈ Xi(j) , for j = 1, . . ., n(V ). Then M = M(K, P, X ) = G M(G) is a Fitting family in the universe S. Proof. 1. Let G be a group, let N be a normal subgroup of G, and let V be a KG-module in M(G) such that N ≤ CG (V ). Suppose that V = V1 ⊗ · · · ⊗ Vn is a P-factorisation of V where Vj is a πi(j) -specialKG-module n and G/ CG (Vj ) ∈ Xi(j) , for each j = 1, . . ., n. Then CG (V ) = j=1 CG (Vj ), and so N ≤ CG (Vj ) for each j. Consider Vj as a K(G/N )-module. It is clear that Vj is also a πi(j) -special K(G/N )-module. Therefore the K(G/N )module V is P-factorable. Since (G/N ) CG/N (Vj ) ∼ = G/ CG (Vj ), we have that V ∈ M(G/N ).

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2 Classes of groups and their properties

2. Let G and H be two groups such that ϕ : G −→ H is an epimorphism, and consider an KH-module V ∈ M(H). Suppose that V = V1 ⊗ · · · ⊗ Vn is a P-factorisation of V where Vj is a πi(j) -special KH-module and H/ CH (Vj ) ∈ Xi(j) , for each j = 1, . . ., n. Each Vj is considered as a KG-module via ϕ. Let S be a subnormal subgroup of G. Since ϕ is an epimorphism, the image S ϕ is a subnormal subgroup of H. Moreover, U is a composition factor of (Vj )S if and only if U is a composition factor of (Vj )S ϕ . For any π(j) -element x of S, we have that xϕ is a π(j) -element of S ϕ and det(x on U ) = det(xϕ on U ) = 1. Therefore the KG-module Vj is πi(j) -special and V is a P-factorable KGn module. Finally, observe that Ker(ϕ) ≤ CG (V ) = j=1 CG (Vj ). Then ∼ G/ Ker(ϕ) C H/ C (V )/ Ker(ϕ) (V G/ CG (Vj ) ∼ = = G j H j ) ∈ Xi(j) . Therefore V ∈ M(G). 3. Let G be a group, N be a normal subgroup of G and V a KG-module in M(G). Let U be an irreducible KN -submodule of VN . Then U is a Pfactorable KN -module by Lemma 2.5.9 (1). In fact, if V = V1 ⊗ · · · ⊗ Vn is a P-factorisation of V where Vj is a πi(j) -special KG-module then there exists a KN -submodule Uj of Vj such that U = U1 ⊗ · · · ⊗ Un is a P-factorisation of U where each Uj is a πi(j) -special KN -module, 1 ≤ j ≤ n. Since each Xi(j) is a Fitting class and G/ CG (Vj ) ∈ Xi(j) , then the normal subgroup N CG (Vj )/ CG (Vj ) is in Xi(j) . This is to say that N/ CN (Vj ) ∈ Xi(j) . Since Uj is a KN -submodule of Vj , we have that CN (Vj ) is a normal subgroup of CN (Uj ) and then N/ CN (Uj ) is an epimorphic image of N/ CN (Vj ). Since each Xi(j) is also a formation, we have that N/ CN (Uj ) ∈ Xi(j) . Therefore we deduce that U ∈ M(N ). 4. Let G be a group and suppose that M and N are normal subgroups of G such that G = M N . Let V be an irreducible KG-module such that all irreducible KM -submodules of VM are in M(M ) and all irreducible KN submodules of VN are in M(N ). By Lemma 2.5.9 (2), V is P-factorable KGmodule. Suppose that V = V1 ⊗ · · · ⊗ Vn is a P-factorisation of V where Vj is a πi(j) -special KG-module. By Clifford’s theorem [DH92, B, 7.3], (Vj )M and (Vj )N are completely reducible. Suppose that (Vj )M = Zj,1 ⊕ · · · ⊕ Zj,r(j) is a decomposition of (Vj )M in irreducible KM -submodules. By Lemma 2.5.8 every Zj,t is a πi(j) -special KM -module. Therefore VM =

r(1) k(1)=1

Z1,k(1) ⊗ · · · ⊗ Zn,k(n)

r(n)

···

k(n)=1

Any Z1,k(1) ⊗ · · · ⊗ Zn,k(n) is a P-factorisation of an irreducible constituent of VM . Then Z1,k(1) ⊗ · · · ⊗ Zn,k(n) ∈ M(M ). Therefore M/ CM (Zj,l ) ∈ Xi(j) r(j) for any par (j, l). It is clear that CM (Vj ) = i=1 CM (Zj,i ). Since Xi(j) is a formation, the group M/ CM (Vj ) is in Xi(j) . We can argue analogously with VN and deduce that N/ CN (Vj ) ∈ Xi(j) . Moreover since M CN (Vj )/ CM (Vj ) CN (Vj ) ∼ = M/ CM (Vj ) ∈ Xi(j)

2.5 Fitting formations

and

123

N CM (Vj )/ CM (Vj ) CN (Vj ) ∼ = N/ CN (Vj ) ∈ Xi(j) ,

and Xi(j) is a Fitting class, we have that G/ CM (Vj ) CN (Vj ) = [M CN (Vj )/ CM (Vj ) CN (Vj )][N CM (Vj )/ CM (Vj ) CN (Vj )] is in Xi(j) . Finally, notice that CM (Vj ) CN (Vj ) is a normal subgroup of CG (Vj ) and then G/ CG (Vj ) is isomorphic to a quotient group of G/ CM (Vj ) CN (Vj ). Since Xi(j) is a formation, we have that G/ CG (Vj ) ∈ Xi(j) . This implies that V ∈ M(G). Examples and remarks 2.5.11. The Cossey-Kanes construction covers many of the known constructions of Fitting formations. For instance: 1. Let p be a prime and K an algebraically closed field of characteristic p. If P = π1 = {p}, π2 = p and X = {X1 = (1), X2 = S}, then Mp = M(K, P, X ) is a Fitting family of modules in the universe S. The Fitting formation T = T(Mp ) is the one introduced by Hawkes in [Haw70]. 2. The Fitting formations studied by K. L. Haberl and H. Heineken ([HH84] or [DH92, IX, 2.26]) are constructed using a not necessarily algebraically closed field K. Nevertheless they can also be included in the Cossey-Kanes construction thanks to a modification made by Cossey in [Cos89]. According to [HH84, 4.1], every Haberl-Heineken Fitting formation can be seen as a Fitting formation constructed by the Cossey-Kanes method with X1 = S, X2 = (1). 3. The non-saturated Fitting formations introduced by T. R. Berger and J. Cossey in [BC78] are defined in terms of the Cossey-Kanes procedure. The Fitting formations of Berger-Cossey are the first examples of non-saturated Fitting formations composed of soluble groups whose p-length is less or equal to 1 for all primes p. 4. A result due to L. G. Kov´ acs, which appears in [CK87, 4.2], characterises the saturation of the Fitting formations T M(K, P, X ) . This means that some of the Fitting formations constructed by the Cossey-Kanes procedure can be saturated. 5. Let M = G M(G) be a Fitting family. In Theorem 2.5.6, assume that, instead of the class ΓK (G), we work with the class ∆K (G) of all irreducible KG-modules V such that V is a composition factor of the module W K = W ⊗ K, where W is a complemented r-chief factor of G (r is a prime, K is a field with char K = r and G is r-soluble). Then the class C(M) = G : G is r-soluble and ∆K (G) ⊆ M(G) is a Fitting class and a Schunck class (see [CO87]). Moreover, in this paper a criterion to decide which of these classes is a formation is presented.

3 X-local formations

In 1985 P. F¨ orster [För85b] presented a common extension of the Gasch¨ utzLubeseder-Schmid and Baer theorems (see Section 2.2). He introduced the concept of X-local formation, where X is a class of simple groups with a completeness property. If X = J, the class of all simple groups, X-local formations are exactly the local formations and when X = P, the class of all abelian simple groups, the notion of X-local formation coincides with the concept of Baer-local formation. P. F¨ orster also defined a Frattini-like subgroup Φ∗X (G) in each group G, which enables him to introduce the concept of X-saturation. F¨ orster’s definition of X-saturation is not the natural one if our aim is to generalise the concepts of saturation and soluble saturation. Since OJ (G) = G and OP (G) = GS , we would expect the X-Frattini subgroup of a group G subgroup of to be defined as Φ OX (G) , where OX (G) is the largest normal G whose composition factors belong to X. We have that Φ OX (G) is contained in Φ∗X (G), but the equality does not hold in many cases. Nevertheless, F¨ orster proved that X-saturated formations are exactly the X-local ones. If X = J, then we obtain as a special case the Gasch¨ utz-Lubeseder-Schmid theorem. When X = P, Baer’s theorem appears as a corollary of F¨ orster’s result. Since Φ OX (G) is contained in Φ∗X (G) for every group G, we can deduce from F¨ orster’s theorem that every X-local formation fulfils the following property: A group G belongs to F if and only if G/Φ OX (G) belongs to F. (3.1) Therefore from the very beginning the following question naturally arises: Open question 3.0.1. Let F be a formation with the property (3.1). Is F X-local? After studying general properties of X-local formations in Section 3.1, we draw near the solution of Question 3.0.1 in Section 3.2. Products of X-local formations are the theme of Section 3.3, whereas some partially saturated formations are studied in Section 3.4. Throughout this chapter, X denotes a fixed class of simple groups satisfying π(X) = char X. 12 5

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3.1 X-local formations This section is devoted to study some basic facts on X-local formations. We investigate the behaviour of X-local formations as classes of groups, focussing our attention on some distinguished X-local formation functions defining them. We begin with the concept of X-local formation due to F¨ orster [För85b]. Denote by J the class of all simple groups. For any subclass Y of J, we write Y = J \ Y. Let E Y be the class of groups whose composition factors belong to Y. It is clear that E Y is a Fitting class, and so each group G has a largest normal E Y-subgroup, the E Y-radical OY (G). A chief factor of G which belongs to E Y is called a Y-chief factor , and if, moreover, p divides the order of a Y-chief factor H/K of G, we shall say that H/K is a Yp -chief factor of G. Sometimes it will be convenient to identify the prime p with the cyclic group Cp of order p. Definition 3.1.1 (P. F¨ orster). An X-formation function f associates with each X ∈ (char X) ∪ X a formation f (X) (possibly empty). If f is an Xformation function, then the X-local formation LFX (f ) defined by f is the class of all groups G satisfying the following two conditions: 1. if H/K is an Xp -chief factor of G, then G CG (H/K) ∈ f (p), and 2. G/K ∈ f (E), whenever G/K is a monolithic quotient of G such that the composition factor of its socle Soc(G/K) is isomorphic to E, if E ∈ X . Remarks 3.1.2. 1. It is obvious from the definition that LF X (f ) is Q-closed. 2. Applying Theorem 1.2.34, it is only necessary to consider the Xp -chief factors of a given chief series of a group G in order to check whether or not G satisfies Condition 1. 3. If, for some prime p ∈ char X, f (p) = ∅, then every X-chief factor of a group G ∈ LFX (f ) is a p -group. 4. If, for some S ∈ X , f (S) = ∅, then a group G ∈ LFX (f ) cannothave a monolithic quotient whose socle is in E(S). Consequently LFX (f ) ⊆ E (S) . 5. If f (S) = ∅ for some S ∈ X , then LFX (f ) ⊆ E (S) ◦ f (S). Remark 3.1.2 (5) is a consequence of the following lemma: Lemma 3.1.3. Let G be a group and let {Mi : 1 ≤ i ≤ s} be the set of all minimal normal subgroups of G. Then, for each 1 ≤ i ≤ s, G has a normal subgroup Ni such that G/Ni is monolithic and Soc(G/Ni ) is G-isomorphic to Mi . Moreover G ∈ R0 ({G/Ni : 1 ≤ i ≤ s}). Proof. For each 1 ≤ i ≤ s, we consider an element Ni of maximal order of the family {Ti G : Ti ∩ Mi = 1}. Then G/Ni is monolithic, Soc(G/Ni ) is G-isomorphic to Mi and G ∈ R0 ({G/Ni : 1 ≤ i ≤ s}).

3.1 X-local formations

127

Proof (of Remark 3.1.2 (5)). Assume that G ∈ LFX (f ) and f (S) = ∅ for some S ∈ X . Then every minimal normal subgroup of G/N , for N = O(S) (G), is in E(S). Therefore G/N ∈ f (S) by the above lemma. In particular, G ∈ E (S) ◦ f (S). Remark 3.1.2 (5) is proved. We can now deduce the following result. Theorem 3.1.4. Let f be an X-formation function. Then LFX (f ) is a formation. Proof. We prove that LFX (f ) is R0 -closed. Let N1 and N2 be two different minimal normal subgroups of a group G such that G/Ni ∈ LFX (f ) (i = 1, 2). We see that G satisfies Condition 1 of Definition 3.1.1. If N1 ∈ E(X ), then clearly G ∈ LFX (f ). Hence we may assume that N1 ∈ E X. Let p be a prime dividing |N1 |. Then N1 N2 /N1 is an Xp -chief factor of G/N2 and AutG (N1 ) ∼ = AutG/N2 (N1 N2 /N2 ) and G/N2 ∈ LFX (f ). Therefore G/ CG (N1 ) ∈ f (p). Since older theorem G/N1 ∈ LFX (f ), by appealing to the generalised Jordan-H¨ (1.2.34), we infer that G satisfies Condition 1. Consider now a monolithic quotient G/K of G such that Soc(G/K) ∈ E(S) by = ∅, then LF (f ) ⊆ E (S) for some simple group S ∈ X . If f (S) X Remark 3.1.2 (4). Therefore G/N (S) for i ∈ {1, 2}. This implies G ∈ ∈ E i E (S) , contrary to supposition. Hence f (S) = ∅ and so G/Ni ∈ E (S) ◦f (S) by Remark 3.1.2 (5). In particular, G/K ∈ E (S) ◦ f (S) because the latter class is a formation. Since O(S) (G/K) = 1, it follows that G/K ∈ f (S). Hence G satisfies Condition 2 of Definition 3.1.1. Consequently G ∈ LFX (f ). Applying Remark 3.1.2 (1) and [DH92, II, 2.6], LFX (f ) is a formation. Definition 3.1.5. A formation F is said to be X-local if F = LFX (f ) for some X-formation function f . In this case we say that f is an X-local definition of F or f defines F. Examples 3.1.6. 1. Each formation F is X-local for X = ∅ because F = LFX (f ), where f (S) = F for all S ∈ J. 2. If X = J, then an X-formation function is simply a formation function and the X-local formations are exactly the local formations. 3. If X = P, then an X-formation function is a Baer function and the X-local formations are exactly the Baer-local ones. for all i ∈ I. Remarks 3.1.7. Let f and fi be X-formation functions 1. i∈I LFX (fi ) = LFX (g), where g(S) = i∈I fi (S) for all S ∈ (char X)∪ X . 2. Let N G and G/N ∈ LFX (f ). If N ∈ E X and G/ CG (N ) ∈ f (p) for all p | |N |, then G ∈ LFX (f ).

128


Proof. 1. This follows immediately from the definition ofX-local formation. 2. If H/K is an Xp -chief factor of G above N , then G CG (H/K) ∈ f (p) because G/N ∈ LFX (f ). Let H/K be an Xp -chief factor of G below N . Then CG (N ) ≤ CG (H/K) and so G CG (H/K) ∈ Q f (p) = f (p). By the generalised Jordan-H¨ older theorem (1.2.34), we have that G satisfies Condition 1 of Definition 3.1.1. Let K be a normal subgroup of G such that G/K is a monolithic group with Soc(G/K) ∈ E(S), S ∈ X . Then, since N ∈ E X, we have that N ≤ K. Therefore G/K ∈ f (S) because G/N ∈ LFX (f ). Consequently G ∈ LFX (f ). .-

Definition 3.1.8. Let p ∈ char X. Then the subgroup CXp (G) is defined to be the intersection of the centralisers of all Xp -chief factors of G, with CXp (G) = G if G has no Xp -chief factors. Remark 3.1.9. Let LFX (f ) be an X-local formation. Then G satisfies Condition 1 of Definition 3.1.1 if and only if G/ CXp (G) ∈ f (p) for all p ∈ char X such that f (p) = ∅. θ Note that CXp (G) is contained in CXp (Gθ ) for every epimorphism θ of G. Therefore, by [DH92, IV, 1.10], the class Q G/ CXp (G) : G ∈ F is a formation, for each formation F. Let N be a normal subgroup of G and let H/K be a chief factor of G below N . Then, by [DH92, A, 4.13 (c)], H/K is a direct product of chief factors of N . Therefore we have Proposition 3.1.10. CXp (G) ∩ N = CXp (N ) for all normal subgroups N of G. Let f1 and f2 be two X-formation functions. We write f1 ≤ f2 if f1 (S) ⊆ f2 (S) for all S ∈ (char X) ∪ X . Note that in this case LFX (f1 ) ⊆ LFX (f2 ). By Remark 3.1.7 (1), each X-local formation F has a unique X-formation function f defining F such that f ≤ f for each X-formation function f such that F = LFX (f ). We say that f is the minimal X-local definition of F. This X-local formation function will always be denoted by the use of a “lower bar.” Moreover if Y is a class of groups, the intersection of all X-local formations containing Y is the smallest X-local formation containing Y. Such Xlocal formation is denoted by formX (Y). If X = J, we also write lform(Y) = formJ (Y), and if X = P, formP (Y) is usually denoted by bform(Y). Theorem 3.1.11. Let Y be a class of groups. Then F = formX (Y) = LFX (f ), where f (p) = Q R0 G CG (H/K) : G ∈ Y and H/K is an Xp -chief factor of G , if p ∈ char X, and f (S) = Q R0 G/L : G ∈ Y, G/L is monolithic, and Soc(G/L) ∈ E(S) , if S ∈ X . Moreover f (p) = Q G/ CXp (G) : G ∈ F for all p ∈ char X such that f (p) = ∅.


129

Proof. Let g be an X-formation function such that F = LFX (g). Since LFX (f ) is an X-local formation containing Y, we have F ⊆ LFX (f ). Assume that LFX (f ) = F. Then LFX (f ) \ F contains a group G of minimal order. Such a G has a unique minimal normal subgroup N by [DH92, II, 2.5] and G/N ∈ F. If N is an X -chief factor of G, then G ∈ f (S) for some S ∈ X . This implies that G ∈ Q R0 Y ⊆ F, a contradiction. Therefore N ∈ E X. Let p be a prime divisor of |N |. Then G/ CG (N ) ∈ f (p). Now if X is a group in Y and H/K is an Xp -chief factor of X, then X CX (H/K) ∈ g(p) because Y ⊆ F. Therefore f (p) ⊆ g(p), and so G/ CG (N ) ∈ g(p). Applying Remark 3.1.7 (2), G ∈ F, contrary to hypothesis. Consequently F = LFX (f ). Let p ∈ char X and t(p) = Q G/ CXp (G) : G ∈ F . We know that t(p) is a formation. Moreover, if G ∈ F and f (p) = ∅, then G/ CXp (G) ∈ f (p). Therefore t(p) ⊆ f (p). On the other hand, if X ∈ Y, then X/ CXp (X) ∈ t(p). Hence X CX (H/K) ∈ t(p) for every Xp -chief factor H/K of X. This means that f (p) ⊆ t(p) and the equality holds. This completes the proof of the theorem. Remark 3.1.12. If F is a local formation and f is the smallest local definition of F, then f (p) = Q G/ Op ,p (G) : G ∈ F for each p ∈ char F (cf. [DH92, IV, 3.10]). The equality f (p) = Q G/ Op ,p (G) : G ∈ F does not hold for X-local formations in general: Let X = (C2 ) and consider F = LFX (f ), where f (2) = S and f (S) = E for all S ∈ X . Then Alt(5) ∈ F and so Alt(5) ∈ / f (2). Q G/ O2 ,2 (G) : G ∈ F . Since f (2) ⊆ S, it follows that Alt(5) ∈ Consequently f (2) = Q G/ O2 ,2 (G) : G ∈ F . ¯ be classes of simple groups such that X ¯ ⊆ X. Corollary 3.1.13. Let X and X ¯ Then every X-local formation is X-local. ¯ ⊆ char X, we Proof. Let F = LFX (f ) be an X-local formation. Since char X ¯ function g defined by can consider the X-formation ¯ if p ∈ char X, ¯ . if E ∈ X

g(p) = f (p) g(E) = F

LFX¯ (g), and choose a group It is clear that F ⊆ LFX¯ (g). Suppose that F = Y of minimal order in LFX¯ (g) \ F. Then Y has a unique minimal normal ¯ ), then G ∈ F, which contradicts subgroup N , and G/N ∈ F. If N ∈ E(X ¯ and G/ CG (N ) ∈ f (p) for each prime p the choice of G. Therefore N ∈ E X dividing |N |. Applying Remark 3.1.7 (2), we conclude that G ∈ F, contrary ¯ to supposition. Hence F = LF ¯ (g) and F is X-local. X

By [DH92, IV, 3.8], each local formation F = LF(f ) can be defined by a formation function g given by g(p) = F ∩ Sp f (p) for all primes p. The corresponding result for X-local formations is the following:

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Theorem 3.1.14. Let F = LFX (f ) be an X-local formation defined by the X-formation function f . Set f ∗ (p) = F ∩ Sp f (p) f ∗ (S) = F ∩ f (S)

for all p ∈ char X, for all S ∈ X .

Then: 1. f ∗ is an X-formation function such that F = LFX (f ∗ ). 2. Sp f ∗ (p) = f ∗ (p) for all p ∈ char X. Proof. 1. It is clear that f ∗ is an X-formation function. Let F∗ = LFX (f ∗ ) and let G ∈ F∗ . If H/K is an Xp -chief factor of G, then G CG (H/K) ∈ F ∩ Spf (p). Since, by [DH92, A, 13.6], Op G CG (H/K) = 1, it follows that G CG (H/K) ∈ f (p). Now if G/L is a monolithic quotient of G with Soc(G/L) ∈ E(S) for some S ∈ X , it follows that G/L ∈ f (S). Therefore G ∈ F. Now if H/K is an Xp -chief factor of a group A ∈ F, then A CA (H/K) ∈ ∗ Q F∩f (p) ⊆ f (p). If A/L is a monolithic quotient of A with Soc(A/L) ∈ E(S), S ∈ X , then A/L ∈ Q F ∩ f (S) ⊆ f ∗ (S). This implies that A ∈ F∗ and therefore F = F∗ . 2. Let G ∈ Sp f∗ (p), p ∈ char X. Then G/ Op (G) ∈ f ∗ (p) and so G ∈ Sp f (p) because Op G/ Op (G) = 1. Moreover G/ Op (G) ∈ F. If H/K is an Xp -chief factor of G below Op (G), then Op (G) ≤ CG (H/K) by [DH92, B, 3.12 (b)] and so G CG (H/K) ∈ Q f (p) = f (p). If G/L is a monolithic quotient of G such that Soc(G/L) ∈ E(S), S ∈ X , it follows that Op (G) ≤ L. Therefore G/L ∈ Q f ∗ (p) = f ∗ (p) ⊆ F and so G/L ∈ f (S). This proves that G ∈ F. Consequently G ∈ f ∗ (p) and Sp f ∗ (p) = f ∗ (p). Definition 3.1.15. Let f be an X-formation function defining an X-local formation F. Then f is called: 1. integrated if f (S) ⊆ F for all S ∈ (char X) ∪ X , 2. full if Sp f (p) = f (p) for all p ∈ char X. Let F = LFX (f ) be an X-local formation. Then the X-formation function g given by g(S) = F ∩ f (S) for all S ∈ (char X) ∪ X is an integrated X-local definition of F. Moreover f ∗ is, according to the above theorem, an integrated and full X-local definition of F. It is known (cf. [DH92, IV, 3.7]) that if X = J, then every X-local formation has a unique integrated and full X-local definition, the canonical one. This is not true in general. In fact, if ∅ = X = J, we can find an X-local formation with several integrated and full X-local definitions. Example 3.1.16. Let ∅ = X = J. Then we can consider X ∈ J \ X and a prime p ∈ char X. The formation F = Sp is an X-local formation which can be X-locally defined by the following integrated and full X-formation functions:

Sp ∅

∼ Cp , if S = ∼ if S = Cp ,

⎧ ⎪ ⎨Sp f2 (S) = Sp ⎪ ⎩ ∅

if S ∼ = Cp , if S ∼ = X, otherwise

f1 (S) = and


131

for all S ∈ (char X) ∪ X . We say that an X-formation function f defining an X-local formation F is a maximal integrated X-formation function if g ≤ f for each integrated X-formation function g such that F = LFX (g). The next result shows that every X-local formation can be X-locally defined by a maximal integrated X-formation function F . Moreover F is full. Theorem 3.1.17. Let F = LFX (f ) be an X-local formation. Then: 1. F is X-locally defined by the integrated and full X-formation function F given by F (p) = Sp f (p) for all p ∈ char X and F (S) = F for all S ∈ X . 2. For each p ∈ char X, F (p) = (G : Cp G ∈ F). 3. If F = LFX (g), then F (p) = F ∩ Sp g(p) for all p ∈ char X. Proof. 1. Since f ≤ F , it follows that F ⊆ LFX (F ). Suppose, by way of contradiction, that the equality does not hold and let G be a group of minimal order in LFX (F )\F. Then the group G has a unique minimal normal subgroup, N say, and G/N ∈ F. Furthermore N ∈ E X because otherwise G ∈ F (S) for some S ∈ X and then G ∈ F, contrary to supposition. Let p be a prime dividing |N |. Then G/ CG (N ) ∈ Sp f (p) and so G/ CG (N ) ∈ f (p) because Op G/ CG (N ) = 1 by [DH92, A, 13.6 (b)]. Then Remark 3.1.7 (2) implies that G ∈ F. This contradiction yields LFX (F ) ⊆ F and then F = LFX (F ). It is clear that F is full. Let p ∈ char X. If possible, choose a group G of minimal order in F (p) \ F. We know that G has a unique minimal normal subgroup N and, since f (p) ⊆ F, Op (G) = 1. Hence N is a p-group. Moreover G/N ∈ F and G/ CG (N ) ∈ f (p) because Op (G) centralises N . But then G ∈ F. This contradicts the choice of G, and so we conclude that F (p) ⊆ F. 2. Let p ∈ char X and let F¯ (p) denote the class (G : Cp G ∈ F). If G ∈ F (p), then Cp G ∈ Sp F (p) = F (p) ⊆ F by Statement 1. Hence G ∈ F¯ (p) and so F (p) ⊆ F¯ (p). Now consider a group G ∈F¯ (p) and set W = Cp G. Denote B = Cp the base group of W and A = {CW (H/K) : H ≤ B and H/K is a chief factor of W }. Since W ∈ F, it follows that W/A ∈ F (p). On the other hand, A acts as a group of operators for B by conjugation and A stabilises a chain of subgroups of B. Applying [DH92, A, 12.4], we have that A/ CA (B) is a p-group. Then A is itself a p-group because CA (B) = B by [DH92, A, 18.8]. Consequently W ∈ F (p) and G ∈ Q F (p) = F (p). This proves that F¯ (p) = F (p).

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3. Let g be an X-formation function such that F = LFX (g). Since f ≤ g, it follows that F (p) = Sp f (p) ⊆ F ∩ Sp g(p) = g ∗ (p) for all p ∈ char X. Let X be a group in g ∗ (p) and set W = Cp X. As above, denote by B = Cp the base group of W . Then W/B ∈ g ∗ (p). Moreover W/B ∈ F = LFX (g ∗ ) by Theorem 3.1.14. Applying Remark 3.1.7 (2), we conclude that W ∈ F. Hence X ∈ F (p) and F (p) = g ∗ (p). Let g be an integrated X-formation function defining an X-local formation F. Then g(p) ⊆ F ∩ Sp g(p) = F (p) for all p ∈ char X by Theorem 3.1.17 (3). Therefore g ≤ F . We shall say that F is the canonical X-local definition of F = LFX (F ). As in the case of local formations, the canonical X-local definition will be identified by the use of an uppercase Roman letter. Hence if we write F = LFX (F ), we are assuming that F is the canonical X-local definition of F. Corollary 3.1.18. Let F be an X-local formation and Y ⊆ X. Let F1 and F2 be the canonical Y-local and X-local definitions of F, respectively. Then F1 (p) = F2 (p) for all p ∈ char Y. Proof. Applying Corollary 3.1.13, we know that F is Y-local. Let p be a prime in char Y. Then p ∈ char X and by Theorem 3.1.17 (2) we have that F1 (p) = (G : Cp G ∈ F) = F2 (p). Taking Y = (Cp ), p ∈ char X in Corollary 3.1.18 and, applying Theorem 3.1.11 and Theorem 3.1.17, we have: Corollary 3.1.19. Let F be an X-local formation. If p ∈ char X, then F (p) = Sp Q R0 G CG (H/K) : G ∈ F, H/K is an abelian

p-chief factor of G .

Corollary 3.1.20. Let F = LFX (f ) = LFX (F ) and G = LFX (g) = LFX (G) be X-local formations. Then any two of the following statements are equivalent: 1. F ⊆ G 2. F ≤ G 3. f ≤ g Corollary 3.1.21 ([BBCER05, Lemma 4.5]). Let F be a formation and let {Xi : i ∈ I} be a familyof classes of simple groups such that π(Xi ) = char Xi for all i ∈ I. Put X = i∈I Xi . If F is Xi -local for all i ∈ I, then F is X-local. Proof. First of all, note that π(X) = char X. Applying Theorem 3.1.17, F = LFXi (Fi ), where (G : Cp G ∈ F) if S ∼ = Cp , p ∈ char Xi , Fi (S) = F if S ∈ Xi ,


133

for all i ∈ I. Let f be the X-formation function defined by (G : Cp G ∈ F) if S ∼ = Cp , p ∈ char X, f (S) = F if S ∈ X . It is clear that F ⊆ LFX (f ). Assume that the inclusion is proper and derive a contradiction. Let G ∈ LFX (f ) \ F of minimal order. Then G has a unique minimal normal subgroup N such that G/N ∈ F. It is clear that N ∈ E X because otherwise G ∈ F. Hence N ∈ E Xi for some i ∈ I and G/ CG (N ) ∈ f (p) = Fi (p) for all p ∈ π(N ). Therefore G ∈ LFXi (Fi ) = F. This is a contradiction. Consequently F = LFX (f ) and F is an X-local formation. When X is the class of all abelian simple groups, we have X = p∈P (Cp ). Therefore Corollary 3.1.22 ([BBCER05, Corollary 4.6]). A formation F is Baerlocal if and only if F is (Cp )-local for every prime p. A natural question arising from the above discussion is whether an X-local formation has a unique upper bound for all its X-local definitions, that is, if F can be X-locally defined by an X-formation function F 0 such that f ≤ F 0 for each X-local definition f of F. If such F 0 exists, we will refer to it as the maximal X-local definition of F. In [Doe73], K. Doerk presented a beautiful result showing that in the soluble universe each local formation has a maximal local definition (see also [DH92, V, 3.18]). The same author, P. Schmid [Sch74], and L. A. Shemetkov [She78] posed the problem of whether every local formation of finite groups has a maximal local definition. The answer is “no” as the following example shows: Example 3.1.23 ([Sal85]). Let F = S be the local formation of all soluble groups. Then F = LF(f1 ) = LF(f2 ), where f1 and f2 are the formation functions defined by f1 (2) = D0 S, Alt(5) , f1 (p) = S for each prime p = 2, f2 (3) = f2 (5) = D0 S, Alt(5) , f2 (p) = S for each prime p = 3, 5. Assume that F has a maximal local definition, F 0 say. Then fi ≤ F 0 for i = 1, 2. This implies that Alt(5) ∈ LF(F 0 ) = F, a contradiction. Therefore F does not have a maximal local definition. Perhaps the most simple example of a local formation with a maximal local (J-local) definition is given by the class Eπ of all π-groups for a set of primes π. It is rather clear that

134


F(p) =

E if p ∈ π, ∅ if p ∈ / π,

defines the maximal local definition of Eπ . In the following we shall give a description of X-local formations with a maximal X-local definition. The main source for this description is P. F¨ orster and E. Salomon [FS85]. The following concept, introduced for local formations in [Sal85], turns out to be crucial. Definition 3.1.24 ([FS85]). Let F = LFX (F ) be an X-local formation. Denote by bX (F) the class of all groups G ∈ b(F) such that Soc(G) ∈ E X. A group G ∈ bX (F) is called X-dense with respect to F if G ∈ b F (p) for each prime p dividing |Soc(G)|. Further, b(F) is said to be X-wide if there does not exist an X-dense group G ∈ bX (F). Note that a group G ∈ bX (F) with abelian socle cannot be X-dense because F is full. Remark 3.1.25. Let F = LFX (F ) and G ∈ bX (F). G is X-dense with respect to F if and only if there exists an X-formation function f such that F = LFX (f ) and G ∈ b f (p) for all primes p dividing |Soc(G)|. Proof. If G is X-dense with respect to F, then we take f = F . Conversely, assume that G ∈ b f (p) for all p ∈ π Soc(G) for some X-formation function f such that F = LFX (f ). Then G/ Soc(G) ∈ F ∩ Sp f (p) = F (p) for all p ∈ π Soc(G) by Theorem 3.1.17 (3). Since G ∈ / F, it follows that G ∈ b F (p) for every prime p dividing |Soc(G)|. This is to say that G is X-dense with respect to F. Examples 3.1.26. 1. Suppose that X contains a non-abelian group S. Then S is X-dense with respect to any X-local formation F satisfying S ∈ / F and Cp ∈ F for all p ∈ π(S). For example, F = N or S. 2. Let F = NF0 for some formation F0 . Let RX denote the class of all X-groups without abelian chief factors; it is clear that RX = R2X is a Fitting formation. It follows that F = LFX (F ) where F (p) = Sp F0 for all p ∈ char X, and F (S) = F for all S ∈ X . Then b(F) is X-wide if and only if RX F0 = F0 . Proof. 1. It is obvious. 2. It is rather clear that F = LFX (F ). Suppose that b(F) is X-wide and RX F0 = F0 . Let G ∈ RX F0 \ F0 be a group of minimal order. Then G has a / F0 , then unique minimal normal subgroup N such that G/N ∈ F0 . Since G ∈ N is a non-abelian X-group. If G ∈ F, then G ∈ F0 because F(G) = 1, contrary to supposition. Hence / Sp F0 for all p ∈ π(N ). This G∈ b(F). Moreover G ∈ means that G ∈ b F (p) for all p ∈ π(N ) and so G is X-dense with respect to F. This is a contradiction. Hence RX F0 ⊆ F0 and the equality holds.


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Conversely, assume that RXF0 = F0 and suppose that there exists G ∈ bX (F) such that G ∈ b F (p) = b(Sp F0 ) for each p ∈ π Soc(G) . Let p and q be two different primes dividing |Soc(G)|. Then G/N ∈ Sp F0 ∩ Sq F0 . Therefore G ∈ RX F0 = F0 . This contradicts the fact that G ∈ b(F). Consequently b(F) is X-wide. For each prime p, denote E(p) the class of all groups such that p divides 2 the order of every chief factor of G. Then it is clear that E(p) = E(p) is a Fitting formation and E(p) ∩ S = Sp . Note that if p ∈ char X, then E(p) ∩ E X = E(Xp ). Remark 3.1.27. Let F = LFX (f ) = LFX (F ) be an X-local formation. Then F (p) = F ∩ E(p)f (p) for each p ∈ char X. Proof. Let p ∈ char X. By Theorem 3.1.17 (3), F (p) = F ∩ Sp f (p). Therefore F (p). Assume that the equality does not hold and let G ∈ (p) ⊆ F ∩ E(p)f F ∩ E(p)f (p) \ F (p) of minimal order. Then G has a unique minimal normal subgroup N such that N ∈ E(p) and G/N ∈ F (p). Since F is full, we have that N is not a p-group. Hence CG (N ) = 1 and so G ∈ F (p) because G ∈ F. This contradiction yields F (p) = F ∩ E(p)f (p). Let F = LFX (f ) be an X-local formation. Denote f¯ the following Xformation function: E(p)f (p) if p ∈ char X, f¯(p) = f (S) if S ∈ X . In general, F = LFX (f¯); take F = N, X = J, and f (p) = (1) for all p ∈ P. However: Theorem 3.1.28. Let F = LFX (f ) = LFX (F ) be an X-local formation with f integrated. The following statements are pairwise equivalent: 1. F = LFX (f¯); 2. F = LFX (F¯ ); 3. b(F) is X-wide. Proof. 1 implies 2. Let p ∈ char X. Then, by Theorem 3.1.17 (3) F (p) = F ∩ Sp f (p) ⊆ E(p)f (p). Consequently E(p)F (p) ⊆ E(p)f (p). It is then clear that F = LFX (F¯ ). 2 implies 3. Let G ∈ bX (F) be an X-dense group with respect to F. Then Soc(G) ∈ E X and so Soc(G) ∈ E(p) for all primes p dividing |Soc(G)|. Therefore G ∈ E(p)F (p). Applying Remark 3.1.7 (2), we have that G ∈ LFX (F¯ ) = F, contrary to the choice of G. Hence bX (F) is wide. 3 implies 1. Suppose that bX (F) is X-wide. Since f ≤ f¯, it follows that F ⊆ LFX (f¯). Hence the burden of the proof is to show that LFX (f¯) ⊆ F. Assume that this is not true, and let G be a group of minimal order in LFX (f¯) \ F.

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It follows easily that G has a unique minimal normal subgroup, N say, and G/N ∈ F. If N ∈ E(X ), then G ∈ f¯(S) = F for some simple group S ∈ X , contrary to supposition. Hence N ∈ E X and so G/ CG (N ) ∈ E(p)f (p) for each prime p dividing |N |. If N is abelian, then G/ CG (N ) ∈ F∩E(p)f (p) = F (p) by Remark 3.1.27. Now applying Remark 3.1.7 (2), G ∈ F, which is not the case. Hence N is non-abelian and then CG (N ) = 1. Then G/N ∈ F ∩ E(p)f (p) = F (p) for all primes p dividing |N |. Since G ∈ / F (p), we have that G is Xdense with respect to F, and we have reached a final contradiction. Therefore LFX (f¯) ⊆ F and the equality holds. The next result shows that the X-local formations of X-wide boundary are precisely those for which a partial converse of Theorem 3.1.17 (3) holds. Theorem 3.1.29. Let F = LFX (F ) be an X-local formation. Then the following statements are equivalent: 1. b(F) is X-wide. 2. If f is an X-formation function such that F ∩ Sp f (p) = F (p) for all p ∈ char X, and f (S) = F for all S ∈ X , then F = LFX (f ). Proof. 1 implies 2. Let f be an X-formation function such that F ∩ Sp f (p) = F (p) for all p ∈ char X and f (S) = F for all S ∈ X . Denote F1 = LFX (f ). It is clear that F ⊆ F1 because F (p) ⊆ Sp f (p) for all p ∈ char X. Suppose that F1 is not contained in F and let G ∈ F1 \ F of minimal order. As usual, G has a unique minimal normal subgroup N such that G/N ∈ F. Moreover N ∈ E X and G/ CG (N ) ∈ f (p) for all p ∈ π(N ). If N were abelian, then G/ CG (N ) ∈ F ∩ f (p) ⊆ F (p) and we would have G ∈ F by Remark 3.1.7 (2). This would contradict the choice of G. Hence N should be non-abelian and so G ∈ f (p) for all p ∈ π(N ). This implies that G/N ∈ F ∩ f (p) ⊆ F (p). Since G ∈ b(F), we have that G ∈ / F (p). Hence G is X-dense with respect to F and b(F) is not X-wide. This is a contradiction. Consequently F1 ⊆ F and the equality holds. 2 implies 1. Let f be the X-formation function given by f (p) = E(p)F (p) for all p ∈ char X and f (S) = F (S) = F for all S ∈ X . Then, by Remark 3.1.27, we have F ∩ Sp f (p) = F ∩ E(p)F (p) = F (p) for all p ∈ char X. Consequently F = LFX (f ) by Statement 2. Applying Theorem 3.1.28, we conclude that b(F) is X-wide. Theorem 3.1.30. Let F = LFX (F ) be an X-local formation with a maximal X-local definition. Then b(F) is X-wide. Proof. Let p ∈ char X and define the following X-formation function: Fp (p) = Cp . E(p)F (p) and Fp (S) = F (S) for every S ∈ (char X) ∪ X such that S ∼ = Then F ≤ Fp . Hence F ⊆ LFX (Fp ). We suppose that F = LFX (Fp ) and derive a contradiction. Let G ∈ LFX (Fp ) \ F be a group of minimal order. Then G has a unique minimal normal subgroup N and G/N ∈ F. If N ∈ E(X ), then G ∈ F (S) for some S ∈ X and so G ∈ F, which is a contradiction. Hence


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N ∈ E X. Suppose that N is abelian. Since G ∈ / F, we conclude that N is a p-group. But in this case G/ CG (N ) ∈ E(p)F (p)∩F = F (p) by Remark 3.1.27. Hence G ∈ F by Remark 3.1.7 (2). Consequently N should be non-abelian. Let q be a prime different from p such that q divides the order of N . Then G ∈ Fp (q) = F (q) ⊆ F. This is the desired contradiction. Therefore Fp = LFX (Fp ) = F for all p ∈ char X. Let g be the maximal Xlocal definition of F. Then E(p)F (p) ⊆ g(p) for all p ∈ char X. Consequently F = LF(F¯ ). Applying Theorem 3.1.28, b(F) is X-wide. Let F = LFX (F ) be an X-local formation. Define h b F (p) ∩ F if S = p ∈ char X, F (S) = h bS (F) if S ∈ X ⎧ ⎪ G : if S = p ∈ char X, Q R0 F (p) ∪ {G} ⊆ F (p) ⎪ ⎨ F(S) = h bS (F) if S ∈ X \ P, ⎪ ⎪ ⎩ G : Q R F (q) ∪ {G} ⊆ F (q) if S ∈ X ∩ P. 0 Note that h bS (F) is a saturated formation for all S ∈ X \ P by Example 2.3.21. Moreover Q F(p) = F(p) for each prime p. Lemma 3.1.31. F (p) ∩ F = F(p) ∩ F = F (p) for each prime p. Proof. F (p) ⊆ F(p) ∩ F ⊆ F (p) ∩ F. Now if p ∈ char X, then F (p) ∩ F ⊆ F (p) by using familiar arguments. If p ∈ X , then F (p) = F. Therefore in both cases F (p) ∩ F ⊆ F (p) and F (p) = F (p) ∩ F. Lemma 3.1.32. Let p be a prime. If L is a formation contained in F (p), then F (p) ∪ L is contained in F (p). Proof. It is enough to prove R0 F (p) ∪ L ⊆ F (p) since F (p) is a homomorph. Suppose that R0 F(p) ∪ L is not contained in F (p) and take 1 = GL and G ∈ R0 F (p) ∪ L \ F (p) of minimal order. Then GF (p) = there exists a normal subgroup K of G such that G ∈ / F (p). Furthermore, G/K ∈ b F (p) ∩ F or G/K ∈ bp (F) according whether p ∈ char X or p ∈ X . 1 and let N be a minimal normal subgroup of G such Suppose that K∩GF (p) = that N is contained in K ∩ GF (p) . By the choice of G, we have G/N ∈ F (p). Hence G/K ∈ F (p). This is impossible. Consequently K ∩ GF (p) = 1 and, analogously, K ∩ GL = 1. Assume that p ∈ char X. Then G/K ∈ F. Thus G ∈ R0 F = F. This implies that G/GL ∈ L ∩ F ⊆ F (p) ∩ F = F (p) by Lemma 3.1.31 and so GF (p) ≤ GL . Since GF (p) ∩ GL = 1, it follows that G ∈ F (p). This contradicts the choice of G. Now suppose that p ∈ X . In this case G/K ∈ bp (F). Let L/K = Soc(G/K). Then L = GF K = GF × K and so GF is a minimal normal subgroup of G. Let B be a minimal normal subgroup contained in GL . Then G/B ∈ h bp (F) by the choice of G. Suppose Q R0

138


that G/B ∈ / F. Then G has a factor group, G/T say, such that B ≤ T and G/T ∈ b(F). Set M/T = Soc(G/T ). Then M = GF T because GF is a minimal normal subgroup of G. Therefore M/T and G/T ∈ bp (F). This is a p-group is a contradiction. Consequently Q R0 F (p) ∪ L is contained in F (p). Theorem 3.1.33 ([FS85]). Let F = LFX (F ) be an X-local formation. Then F possesses a maximal X-local definition if and only if b(F) is X-wide and, for each prime p, there exists a unique maximal formation contained in F (p). In this case, F is an X-formation function and F is the maximal X-local definition of F. Proof. First, suppose that F possesses a maximal X-local definition, g say. Then b(F) is X-wide by Theorem 3.1.30. Let p be a prime in char X. Then g(p)∩ F is contained in F (p) by Theorem 3.1.17 (3). Hence g(p) is contained in h b(F (p) ∩ F = F (p). Assume now that p ∈ X ∩ P and g(p) is not contained in F (p). Let G be a group of least order in g(p) \ F (p). Then G ∈ bp (F), and from F = LFX (g) we readily get that G ∈ F, the desired contradiction. Consequently g(p) ⊆ F (p). Let L be a formation contained in F (p). By Lemma 3.1.32, Q R0 F (p) ∪ L ⊆ F (p). Consider the X-formation function defined by setting if p = q, Q R0 F (p) ∪ L g1 (q) = F (q) if p = q and g1 (S) = g(S) for every S ∈ X \P. Since g1 (p)∩F ⊆ F (p) by Lemma 3.1.31 and Lemma 3.1.32, we immediately have that F = LFX (g1 ). The maximal character of g implies that g1 (p) ⊆ g(p). Thus L ⊆ g(p). Consequently, g(p) is the unique maximal formation contained in F (p). Conversely, suppose that b(F) is X-wide and for each prime p, there exists a unique maximal formation, g(p), contained in F (p). Consider the X-formation function g1 defined by g1 (p) = g(p) for every prime p and g1 (S) = h bS (F) for every S ∈ X \ P. Clearly F ⊆ LFX (g1 ) because F (S) ⊆ g(p) for all p and F ⊆ g1 (S) for all S ∈ X \ P. If F = LFX (g1 ), then a group G ∈ LFX (g1 ) \ F of minimal order would be an X-dense group. Since b(F) is X-wide, we conclude that F = LFX (g1 ). On the other hand, let j be an X-formation function such that F = LFX (j). Then, for all p, we have j(p) ∩ F ⊆ F (p). Consequently, j(p) ⊆ F (p) and then j(p) ⊆ g(p). Furthermore, it is clear that j(S) ⊆ g1 (S) for every S ∈ X \ P. Consequently, g1 is the maximal X-local definition of F. Note that in this case g(p) = F (p) and g(S) = F(S) for all S ∈ X \ P. Therefore F is an X-formation function and it is actually the maximal X-local definition of F. Proposition 3.1.34. Let Y ⊆ X be classes of simple groups. If F = LFX (F ) has a maximal X-local definition, then F has a unique maximal Y-local definition. If, in addition, char X = char Y, then the converse is valid if, and only if, b(F) is X-wide.


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Proof. Note that F = LFY (F1 ), where F1 (p) = F (p) for all p ∈ char Y and F1 (S) = F for all S ∈ Y (see Corollary 3.1.13). Therefore if F has a maximal X-local definition, then b(F) is X-wide (and so b(F) is Y-wide) and F(p) = F1 (p) for all p ∈ char Y is a formation. We are left to show that F1 (p) is a formation for all p ∈ (char X) ∩ Y . To see this, we prove that F 1 (p) = G = h bq (F) contains a unique maximal formation. Set H = f (G) = (G : H/K is G-central in G for every chief factor of G). Applying Theorem 2.3.20, H is a formation. Suppose that H is not contained in G and let G ∈ H \ G be a group of minimal order. Then G ∈ b(G) = bq (F) and / F, so G is a monolithic group. Moreover X = [N ] G/ CG (N ) ∈ G. If X ∈ then X ∈ bq (F), because G/ CG (N ) ∈ F. Hence X ∈ G ∩ bq (F) = ∅. This is a contradiction. Therefore X ∈ F and G/ CG (N ) ∈ F (p). Applying Remark 3.1.7 (2), we conclude that G ∈ LFX (F ) = F. We have obtained a contradiction. Consequently H ⊆ G. Let now L be a formation contained in G. Then by Theorem 2.3.20 (2), L ⊆ H. This means that F1 (p) is a formation. By Theorem 3.1.33, it follows that F has a maximal Y-local definition. Now if char X = char Y, then F (p) = F1 (p) for all p ∈ char X. Consequently if F has a maximal Y-local definition, then F(p) is a formation for all p ∈ char X. By Theorem 3.1.30, F has a maximal X-local definition if, and only if, b(F) is X-wide. Examples 3.1.35. 1. Let F = S be the J-local (local) formation of all soluble groups. Then F = LFJ (F ) where F (p) = F for all p ∈ P. Hence F(p) = E and so F is a J-formation function. However, F does not have a maximal J-local definition (see Example 3.1.23). This example shows that the requirement that b(F) be X-wide cannot be removed from Theorem 3.1.33. 2. Let F0 be the class of all groups whose Frattini chief factors have odd order. Then F0 is a formation and RJ F0 = F0 . Let F = NF0 . Applying Example 3.1.26 (2), we have that F is a J-local formation with J-wide boundary. Assume that F = S2 F0 and let G ∈ F\S2 F0 be a group of minimal order. Then G has a unique minimal normal subgroup N . Moreover G/N ∈ S2 F0 . Since G∈ / F0 , we conclude that N is a p-group for some odd prime p. Hence F(G) is a p-group. This implies that G ∈ F0 because G/ F(G) has no Frattini 2-chief factors. This is a contradiction. Consequently F = S2 F0 and F = LFJ (F ), where F if p = 2, F (p) = F0 if p = 2. Then F (q) = h b F (q) ∩ F = h b(F0 ) ∩ F = h b(F0 ) = F0 for each odd prime q (note b(F0 ) ⊆ F). Consequently E if p = 2, F (p) = F (p) = F0 if p = 2

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and F = F is a J-formation function. Applying Theorem 3.1.33, we have that F is the maximal J-local definition of F. Let F = LFX (F ) be an X-local formation. In contrast to the condition that b(F) is X-wide, the other condition from Theorem 3.1.33 — namely, that F (X) be a formation for all X ∈ P — is not always easy to check when a concrete formation F is given. We give an example of a local formation for which F is not a formation function. Example 3.1.36 ([FS85]). Let R = RJ be the formation composed of all groups whose chief factors are non-abelian. Consider the local formation F = NRN. Then the canonical definition F of F is given by F (p) = Sp RN for all p. Applying Examples 3.1.26 (2), we have that b(F) is J-wide because RJ (RN) = RN. Let S = SL(2, 5). By [DH92, B, 10.9], S has an irreducible module V over GF(p) such that Ker(S on V ) = CS (V ) = Z(S). Let X = [V ]S be the corresponding semidirect product, and let Y = S Z(X) X be the wreath product of S with respect to the permutation representation of S with X with respect to the permutation representation of X on the set of cosets of Z(X) = Z(S) in X. As usual, for any subgroup U of S, U = U × · · · × U (|X/ Z(X)| copies of U ) shall denote the canonical subgroup of S , the base group of Y , isomorphic to a direct product of |X/ Z(X)| copies of S. Note that Z(X) ≤ Z(Y ) and Z(S) X/ Z(X) is X-isomorphic to the regular wreath product C2 reg X/ Z(X) and this is isomorphic to the semidirect product of the regular X/ Z(X)module over GF(2) with X/ Z(X). Therefore there exists a normal subgroup Z of Y such that Z ≤ Z(S) and Z(S) /Z is a cyclic group of order 2. We consider now G = Y /Z. It is clear that S is isomorphic to a quotient of G. Let A = Z(X)Z/Z and B = Z(S) /Z. It is clear that A and B are subgroups of order 2 contained in Z(G) such that A ∩ B = 1. Hence there exists D ≤ Z(G) of order 2 such that D ∩ A = D ∩ B = 1. In particular G ∈ R0 (G/A, G/D). Assume that p is a prime and p > 5. Then F¨ orster and Salomon [FS85, Example 4.1] proved that G/A, G/D ∈ F (p). However since F (p) is Q-closed, S is isomorphic to a quotient of G and S ∈ b F (p) ∩ F, it follows that G ∈ / F (p). This shows that F(p) is not a formation and hence F = NRN does not have a maximal J-local definition as J-local formation. The above example can be modified to show that h bq (F) , F an X-local formation and q ∈ X ∩ P, does not always contain a unique largest formation. Example 3.1.37. Let F = Sp RN as in the above example. Suppose that X = ∅. Put q = 2 and take G, A, D as in Example 3.1.36. Then Q R0 (G/A) ∪ Q R0 (G/D) ⊆ h b2 (F) , but G ∈ R0 (G/A, G/D) does not belong to h b2 (F) . Consequently F is an ∅-local formation without a maximal ∅-local definition.


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Proof. First of all, that S = SL(2, 5) is a quotient of G and S ∈ b2 (F). we know Therefore G ∈ / h b2 (F) . Moreover, G ∈ R0 (G/A, G/D). Now let B1 = b2 (F)∩ NRN and B2 = b2 (F)\NRN. Thus b2 (F) = B1 ∪B2 and h b2 (F) = h(B1 )∩ orster and Salomon [FS85, Example 4.1] proved that Q R0 (G/A) ∪ h(B2 ). F¨ Q R0 (G/D) ⊆ h(B1 ). Moreover B2 is a class composed by primitive groups. Hence h(B2 ) is a Schunck class by Corollary 2.3.11. Note that [H/K]∗(G/A) ∈ h(B2 ) for each chief factor H/K of G/A (and the same applies to G/D). This implies that G/A and G/B belong to f h(B2 ) , which is the largest formation contained in h(B2 ) by Theorem 2.3.20 (3). Hence Q R0 (G/A) ∪ Q R0 (G/D) ⊆ h(B2 ). In [DH92, pages 364 and 365], the authors study the effect of some closure operations on a local formation. More precisely, they prove: Let F = LF(f ) be a local formation and let C be one of the closure operations S, Sn , or N0 . 1. If f (p) = C f (p) for all p ∈ P, then F = C F, and 2. if F = C F, and F is the canonical local definition of F, then F (p) = C F (p) for all p ∈ P. The natural question is: can the above results be extended to X-local formaor85b, Lemma 3.13]). tions? If C = S, 1 is not always true (compare with [F¨ Example 3.1.38. Let X = (C2 ) and F = LFX (f ), where f (2) = (1) and f (S) = E if S ∼ = C2 . It is clear that S f (S) = f (S) for all S ∈ (char X) ∪ X , but F is / F. not S-closed because Alt(5) ∈ F but Alt(4) ∈ Our next result shows that 1 is true for

C

= Sn or

N0 .

Proposition 3.1.39. Let F = LFX (f ) be an X-local formation and let C be one of the closure operations Sn or N0 . If f (S) = C f (S) for all S ∈ (char X) ∪ X , then F = C F. Proof. Let C = Sn . Let G ∈ F, and let N be a normal subgroup of G. We prove that N ∈ F by induction on |G|. Let A be a minimal normal subgroup of G. Then N A/A ∈ F. If B were another minimal normal subgroup of G, then N B/B ∈ F. This would imply N ∈ F. Consequently we may assume that A = Soc(G) minimal normal subgroup of G. Let p ∈ char X. is the unique Then N/ N ∩ CXp (G) ∼ = N CXp (G)/ CXp (G) and N CXp (G)/ CXp (G) is a normal subgroup of G/ CXp (G) ∈ f (p). Since N ∩ CXp (G) = CXp (N ) by Proposition 3.1.10, it follows that N/ CXp (N ) ∈ f (p). Assume now that N/L is a monolithic quotient of N such that T /L = Soc(N/L) ∈ E(S) for some simple group S ∈ X . If A is not contained in L, then T /L is contained in AL/L = 1 and so A ∈ E(S). Since G is a monolithic F-group, it follows that G ∈ f (S). Hence N ∈ Sn f (S) = f (S) and N/L ∈ Q f (S) = f (S). Suppose that A is contained in L. We have that N/A ∈ F by induction. Therefore N/L ∈ f (S) because N/L is isomorphic to a monolithic quotient of N/A whose socle belongs to E(S). Therefore N ∈ F and F = Sn F.

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Now suppose C = N0 . Applying [DH92, II, 2.11], it is enough to show that G ∈ F provided that G = N1 N2 , where Ni is a normal subgroup of G and Ni ∈ Fi , i ∈ {1, 2}. We argue by induction on |G|. It is rather clear that we may assume that G has a unique minimal normal subgroup, A say, and G/A ∈ F. Let p ∈ char X. Then G/ CXp (G) = N1 CXp (G)/ CXp (G) N2 CXp (G)/ CXp (G) . Moreover Ni CXp (G)/ CXp (G) ∼ = Ni / Ni ∩ CXp (G) = Ni / CXp (Ni ) ∈ f (p). Hence G/ CXp (G) ∈ N0 f (p) = f (p). Suppose that G/L is a monolithic quotient of G such that Soc(G/L) ∈ E(S) for some simple group S ∈ X . If L = 1, then G/L ∈ F by induction. This implies G/L ∈ f (S). Thus we may assume that L = 1. In this case A ∈ E(S). It is clear that Soc(Ni ) ∈ E(S) for i ∈ {1, 2}. Therefore, applying Remark 3.1.2 (5), Ni ∈ f (S) because Ni ∈ F, i ∈ {1, 2}. Consequently G ∈ N0 f (S) = f (S) and G ∈ F. We conclude that F is N0 -closed. The next proposition shows that Statement 2 holds for X-local formations. Proposition 3.1.40. Let F = LFX (F ) be an X-local formation. If C is one of the closure operations S, Sn , or N0 and F = C F, then F (S) = C F (S) for all S ∈ (char X) ∪ X . Proof. If S ∈ X , then F (S) = F. Hence we have to prove that F (p) = C F (p) for all p ∈ char X. Assume C = S and p ∈ char X. Let G ∈ F (p), and let H be a subgroup of G. Then if W = Cp G, we know that W ∈ F. Hence BH ∈ F, where B is the base group of W . Therefore BH/ CXp (BH) ∈ F (p). Now CXp (BH) centralises every chief factor of BH below B. Since B ≤ CXp (BH) and CW (B) = B, we have that CXp (BH)/B is a p-group by [DH92, A, 12.4]. Thus H ∈ F (p) and F (p) is subgroup-closed. The case C = Sn is analogous. Now assume that C = N0 . By [DH92, II, 2.11], it will suffice to show that if G = N1 N2 with Ni a normal subgroup of G and Ni ∈ F (p), i = 1, 2, then G ∈ F (p). Let W = Cp G with B as the base group of W . Note that W = (BN1 )(BN2 ), BNi W , and BNi ∈ Sp F (p) = F (p) ⊆ F for i = 1, 2. Therefore W ∈ N0 F = F. By Theorem 3.1.17 (3), G ∈ F (p). Given a group G, denote by SX (G) the set of all subgroups H of G such that H ∈ E X. If L is a class of groups, write L(X) = G : SX (G) ⊆ L . It is clear that L(X) is the unique largest subgroup-closed class such that L(X) ∩ E X ⊆ L. If F is a formation, then F(X) is clearly a formation, but if F is an Xlocal formation, then F(X) is not an X-local formation in general as the next example shows. Example 3.1.41. Consider X = J, the class of all simple groups, let G = Alt(5), and let F = N2 D0 (1, G). In this case, F(X) is the class of all groups U such that every subgroup of U belongs to F. Hence G belongs to F(X). If F(X) were


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an X-local formation, then [V ]G would be an F(X)-group for every irreducible and faithful GF(2)G-module V . In particular, if D is the dihedral group of order 10, then V D ∈ F. This would be a contradiction. Hence F(X) is not an X-local formation. The next result provides precise conditions to ensure that F(X) is again an X-local formation. Theorem 3.1.42 ([BB91]). Let F be an X-local formation. The following statements are pairwise equivalent: 1. For each primitive group G of type 2 in F(X) such that Soc(G) ∈ E X, and for every irreducible and faithful G-module V over GF(p), p ∈ π Soc(G) , the corresponding semidirect product [V ]G is an F(X)-group. 2. For each primitive group G of type 2 in F(X) such that Soc(G) ∈ E X and for every irreducible and faithful G-module V over GF(p), p ∈ π Soc(G) , and for every X ∈ SX (G) such that G = X Soc(G), the semidirect product [V ]X is an F-group. 3. F(X) is an X-local formation. Proof. 2 implies 3. Suppose F = LFX (F ). Define F ∗ (p) = F (p)(X), for each prime p ∈ char X and F ∗ (E) = F (E)(X), for every E ∈ X . Then F ∗ is an X-formation function. We see that F(X) = LFX (F ∗ ). Assume that F(X) is not contained in LFX (F ∗ ) and derive a contradiction. We choose a group G ∈ F(X)\LFX (F ∗ ) of minimal order. Using familiar arguments, we have that G is a monolithic group. Denote N = Soc(G). If N ∈ E(X ), then G ∈ F (E)(X) for some E ∈ X and so G ∈ LFX (F ∗ ), which is a contradiction. Hence N ∈ E X. Suppose that N is abelian. Then N is a p-group for some prime p ∈ char X. Let X be a subgroup of G such that X ∈ E X. Without loss of generality, we may assume that N is contained in X. Certainly X ∈ F(X) as F(X) is subgroup-closed. If X is a proper subgroup of G, then X ∈ LFX (F ∗ ) by the choice of G. This implies that X/ ChX (N ) ∈ F ∗ (p), where ChX (N ) is the intersection of the centralisers in X of all chief factors of X below N . Applying [DH92, A, 2.11], ChX (N )/ CX (N ) is a p-group. Hence X/ CX (N ) ∈ F ∗ (p) and so X/ CX (N ) ∈ F (p). If X = G, then G/ CG (N ) ∈ F (p) because F is X-local. Consequently G/ CG (N ) ∈ F ∗ (p). Applying Remark 3.1.7 (2), we have that G ∈ LFX (F ∗ ) and we have the desired contradiction. Therefore N is a non-abelian group. Let p be a prime dividing the order of N and let X ∈ E X. Assume that T = XN is a proper subgroup of G. Arguing as above, T = XN ∈ LFX (F ∗ ) and ChT (N ) ∼ = ChT (N )/ CT (N ) is a p-group h (note CT(N ) = 1). Hence T / CT (N ) ∈ F ∗ (p). Since X ChT (N )/ ChT (N ) ∈ that h h h SX T / CT (N ) , it follows that X CT (N )/ CT (N ) is in F (p) and so X ∈ F (p). Suppose that T = G and consider an irreducible and faithful G-module V over GF(p) (such V exists by [DH92, B, 10.9]). By Statement 2, the semidirect product [V ]X is an F-group. It implies that X ∈ F (p). Therefore G ∈ F ∗ (p) and G ∈ LFX (F ∗ ) and we have the desired contradiction.

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On the other hand, taking into account that LFX (F ∗ ) is subgroup-closed, it is easy to see that LFX (F ∗ ) is contained in F(X). Consequently F(X) is an X-local formation. 3 implies 1. Taking into account that the F(X) can be locally defined by an X-formation function, it is clear that if G is a primitive group of type 2 in F(X) and Soc(G) ∈ E X, then the semidirect product [V ]G is an F(X)-group for every irreducible and faithful G-module V over GF(p), p ∈ π Soc(G) . Hence Statement 1 holds. Finally, it is clear that 1 implies 2. The circle of implications is now complete. Example 3.1.43. Assume that X is the class of all simple groups and consider the class F = G : Alt(5) ∈ / Q(G) . Then b(F) = Alt(5) . Hence F is a saturated formation by Example 2.3.21. If G is a primitive group of type 2 in F(X), then every subgroup of [V ]X is an F-group, for every subgroup X of G such that G = X Soc(G) and for every irreducible and faithful G-module V over GF(p), p ∈ π Soc(G) . Consequently F(X) is a saturated formation. It is clear that F(X) is the largest subgroup-closed formation contained in F.

3.2 A generalisation of Gasch¨ utz-Lubeseder-Schmid-Baer theorem In this section we study two different Frattini-like subgroups associated with a class of simple groups which lead to the corresponding notion of saturation. We then present an extension of Gasch¨ utz-Lubeseder-Schmid and Baer theorems. We begin with the following definition due to P. F¨ orster. Definition 3.2.1 ([F¨ or85b]). Let G be a group. For a prime p, we define ΦpX (G) as follows: • If Op (G) = 1, ΦpX (G) =

Φ(G) Φ OX (G)

if Soc G/Φ(G) and Φ(G) belong to otherwise.

E X,

• In general, ΦpX(G) is the subgroup of G such that ΦpX (G)/ Op (G) = ΦpX G/ Op (G) . • Finally put Φ∗X (G) = OX (G) ∩ p∈char X ΦpX (G). If q is a prime such that q ∈ / char X, then Φ∗X (G) is a q -group because q ∗ ∗ π(X) = char X. pHence ΦX (G) ≤ Oq (G) ≤ ΦX (G). Consequently ΦX (G) = OX (G) ∩ p∈P ΦX (G). The basic properties of Φ∗X (G) are displayed in the next result.

3.2 A generalisation of Gasch¨ utz-Lubeseder-Schmid-Baer theorem

145

Proposition 3.2.2. Let G be a group. p 1. Φ∗X (G) and p a prime, are characteristic subgroups of G. ΦX (G), 2. Φ OX (G) ≤ Φ∗X (G) ≤ OX (G) ∩ Φ(G). 3. Let p be a prime. If Op (G) = 1, then Φ∗X (G) = ΦpX (G). 4. Let p be a prime. If N is a normal subgroup of G contained in ΦpX (G), then Op (G/N ) = Op (G)N/N . 5. If N is a normal subgroup of G contained in Φ∗X (G), then Φ∗X (G/N ) = Φ∗X (G)/N . Proof. 1. It is clear. ∗ is isomorphic to a sub2. Let p be a prime. Then ΦX (G) Op (G)/ Op (G) group of Φ G/ Op (G) , which is a p-group. Hence Φ∗X (G)∩Op (G) is a normal Hall p -subgroup of Φ∗X (G) and so Φ∗X (G) is p-nilpotent. Therefore Φ∗X (G) is nilpotent. Assume, arguing by contradiction, that Φ∗X (G) is not contained in Φ(G). Then there exists a maximal subgroup M of G such that G = M Φ∗X (G). Since Φ∗X (G) is nilpotent, we can find a prime p and a Sylow p-subgroup P of Φ∗X (G) such that G = M P . In particular, Op (G) is contained in M . Hence ΦpX (G)/ Op (G) is a subgroup of M/ Op (G) and so Φ∗X (G) ≤ M . This contradiction leads to Φ∗X (G) ≤ Φ(G). Now Φ OX (G) Op (G)/ Op (G) ≤ Φ OX (G) Op (G) Op (G)/ Op (G) ≤ Φ OX (G) Op (G)/ Op (G) ≤ Φ OX G/ Op (G) ≤ ΦpX (G)/ Op (G) for each prime p. Consequently Φ OX (G) ≤ Φ∗X (G). 3. Suppose that Op (G) = 1 for some prime p. Since ΦpX (G) is contained in Φ(G), it follows that ΦpX (G) is a p-group. Hence if q is a prime, q = p, we have that ΦpX (G) ≤ Oq (G) ≤ ΦqX (G). Therefore ΦpX (G) ≤ Φ∗X (G) and so Φ∗X (G) = ΦpX (G). 4. Let N be a normal subgroup of G such that N ≤ ΦpX (G) for some prime p. Put Q/N = Op (G/N ) and M = N ∩ Op (G). Then N Op (G)/ Op (G) ≤ ΦpX (G)/ Op (G) ≤ Φ G/ O p (G) , which is a p-group. Therefore N/M is a p-group. Since (Q/M ) (N/M ) is a p -group, it follows that Q/M = (N/M )(H/M ) for some Hall p -subgroup H/M of Q/M . It is clear that H is a Hall p -subgroup of Q G. Moreover the Hall p -subgroups of Q are conjugate. Therefore G = NG (H)N by the Frattini argument. Since N Op (G)/ Op (G) is contained in Φ G/ Op (G) , it follows that G = NG (H) and H ≤ Op (G). Consequently Q/N = Op (G)N/N . 5. Let N be a normal subgroup of G contained in Φ∗X (G). Let p be a prime. in N 4. Suppose that Op (G/N ) = 1. Then Op (G) is contained by Statement Moreover ΦpX (G/N ) is Φ(G/N ) = Φ(G)/N or Φ OX (G/N ) = Φ OX (G)/N . Suppose that ΦpX (G/N ) = Φ OX (G)/N . Then Soc (G/N ) Φ(G)/N and Op (G) and Φ(G)/N belongs to E X and for Soc G/ Op (G) Φ(G)/ Φ G/ Op (G) the same is true. Hence we have that Φ pX G/ Op (G) = p p Φ(G)/ Op (G) and ΦX (G/N ) = ΦX (G)/N .

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ΦpX (G/N ) = Φ OX (G)/N . Then ΦpX (G)/ Op (G) = Assume now that it follows that ΦpX (G) is nilpotent. Φ OX (G)/ Op (G) . By [DH92, A, 9.3 (c)], p Hence ΦX (G) is contained in Φ OX (G) Op (G)by [DH92, A, 9.11]. Therefore ΦpX (G)/N is contained in Φ O (G) N/N ≤ Φ OX (G/N ) . Since N/ Op (G) X is contained in Φ OX (G)/ Op (G) , it follows that ΦpX (G/N ) is isomorphic to ΦpX (G)/N . It leads to ΦpX (G)/N = ΦpX (G/N ). Assume now that Op (G/N ) = Op (G)N/N = 1. Denote with bars ¯ N ¯ ) = 1 and N ¯ ≤ Φp (G), ¯ ¯ = G/ Op (G). Since Op (G/ the images in G X p ¯ ¯ p ¯ p ¯ it follows that ΦX (G/N ) = ΦX (G)/N . By definition of ΦX (G), we have ¯ N ¯ ) under the nat¯ = Φp (G). Therefore the image of Φp (G/ that ΦpX (G) X X ¯ N ¯ and G/N Op (G) is Φp (G)/N Op (G). This ural isomorphism between G/ X implies that ΦpX G/N Op (G) = ΦpX (G)/N Op (G). On the other hand, by definition we have ΦpX (G/N )/ Op (G/N ) = ΦpX (G/N )/N Op (G)/N = ΦpX (G/N ) N Op (G)/N . Now the image of ΦpX (G/N ) N Op (G)/N under the natural isomorphism between the groups (G/N ) N Op (G)/N and G/N Op (G) is the subgroup ΦpX G/N Op (G) . Therefore we have that ΦpX (G)/N = ΦpX (G/N ). Consequently ΦpX (G)/N = ΦpX (G/N ) for all primes p and so Φ∗X (G)/N = ∗ ΦX (G/N ). Remark 3.2.3. If N is a normal subgroup of a group G, then Φ(G)N/N ≤ Φ(G/N ) and Φ(N ) ≤ Φ(G) ([DH92, A, 9.2]). This is not true for Φ∗X (G) in general, as the next examples show. Examples 3.2.4. 1. Let H = SL(2, 5). Then H has an irreducible module V over GF(2) such that Ker(H on V ) = Z(H) (cf. [DH92, B, 10.9]). Let G = [V ]H be the corresponding semidirect product. Put X = (C2 ). Then Φ∗X (G) = Φ(G) = Φ(H) and Φ∗X (G/V ) = 1. 2. If G1 = G × Alt(5), where G and X are as in 1, it follows that Φ(H) = Φ∗X (G) ≤ Φ∗X (G1 ) = 1. If X = J, then Φ∗X (G) = Φ(G) for every group G by Proposition 3.2.2 (2). However, if ∅ = X = J, then we can find a group G such that Φ OX (G) is a proper subgroup of Φ∗X (G) as the next example shows. Example 3.2.5 ([BBCER05]). Assume that ∅ = X = J. Then there exist a non-abelian simple group S ∈ X and a prime p ∈ π(S) such that p ∈ char X. It is certainly true that char X is the set of all prime numbers. Suppose that char X = P and take p ∈ char X and q ∈ / char X. If S is the alternating group of degree p + q, then S ∈ X and p ∈ char X ∩ π(S). Let T be the group algebra GF(p)S and consider G = [T ]S, the corresponding semidirect product. It is rather Op (G) = 1 and Φ(G) clear that Φ(G) = Rad T . Since ∗ and Soc G/Φ(G) belong to E X, we have that ΦX (G) = ΦpX (G) = Φ(G) by Proposition 3.2.2 (3). It is certainly true that Φ(G) = 1 because Rad T = 1. However, OX (G) = T and Φ(T ) = 1.


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This example shows, in particular, that Φ∗X (G) is not always the Frattini subgroup of the soluble radical when X is the class of all abelian simple groups. In [BBCER05] another Frattini-like subgroup associated with a class of simple groups is introduced and analysed. It is smaller than F¨ orster’s one and coincides with the Frattini subgroup of the E X-radical except in a very few number of cases. We present here a slight variation of this subgroup as it appears in [BBCER05]. Definition 3.2.6. Let p be a prime. A group G belongs to AXp (P2 ) provided that G is monolithic and there exists an elementary abelian normal p-subgroup N of G such that 1. N ≤ Φ(G) and G/N is a primitive group of type 2, 2. Soc(G/N ) ∈ E X \ Ep , and 3. ChG (N ) ≤ N , where ChG (N ) := {CG (H/K) : H/K is a chief factor of G below N }. The next result shows that AXp (P2 ) = ∅ if X contains non-abelian simple groups. Proposition 3.2.7. Let G be a primitive group of type 2 such that Soc(G) ∈ Then, for each prime p ∈ π Soc(G) , there exists a group Ep ∈ AXp (P2 ) and a minimal normal p-subgroup Tp of Ep contained in Φ(Ep ) such that Ep / CEp (Tp ) is isomorphic to G.

E X.

Proof. Note that p ∈ char X because π(X) = char X. Let Ep be the maximal Frattini extension of G with p-elementary abelian kernel Ap (G).Then [GS78]). Moreover, by [GS78, Ep / Ap (G) ∼ = G and Ap (G) = Φ(Ep ) (see Theorem 1], we have that Ker G on Soc Ap (G) = Op ,p (G) = 1. Hence there exists a minimal normal subgroup Tp of Ep such that Tp ≤ Ap (G) and CEp (Tp ) = Ap (G). If Ep is monolithic, then clearly Ep ∈ AXp (P2 ) and the proposition is proved. Suppose that Ep is not monolithic. By Lemma 3.1.3, there exists a normal subgroup N of Ep such that N ∩ Tp = 1, Ep /N is monolithic, and Soc(Ep /N ) = Tp N/N . Now N ≤ CEp (Tp ) = Ap (G) = Φ(Ep ) and CEp /N (Tp N/N ) = CEp (Tp )/N = Φ(Ep )/N = Φ(Ep /N ). Therefore Ep /N ∈ AXp(P2 ) and Tp N/N is a minimal normal subgroup of Ep /N such that (Ep /N ) CEp /N (Tp /N ) ∼ = Ep / CEp (T ) ∼ = G. Definition 3.2.8. The X-Frattini subgroup of a group G is the subgroup ΦX (G) defined as follows: Φ OX (G) if G ∈ / AXp (P2 ) for all p ∈ char X, ΦX (G) := Φ(G) otherwise.

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It is clear that ΦX (G) is a characteristic subgroup of G. Moreover if X = J, then obviously ΦX (G) = Φ(G) and if X = P, then AXp (P2 ) = ∅ for all p ∈ char X. Hence ΦX (G) = Φ(GS ) for every group G. Moreover, Proposition 3.2.9. Let G be a group. Then ΦX (G) is contained in Φ∗X (G). Proof. We know, by Proposition 3.2.2 (2), that Φ OX (G) is contained in Φ∗X (G). Suppose now that G ∈ AXp (P2 ) for some prime p ∈ char X. Then Op (G) = 1 and Φ(G) is a p-group. Since Φ(G) and Soc G/Φ(G) belong to p p ∗ E X, ΦX (G) = Φ(G). In addition, ΦX (G) = ΦX (G) by Proposition 3.2.2 (3). ∗ Therefore Φ(G) = ΦX (G) = ΦX (G). Remarks 3.2.10. 1. Example 3.2.5 shows that the equality ΦX (G) = Φ∗X (G) does not hold in general. 2. If X1 ⊆ X2 , then ΦX1 (G) ≤ ΦX2 (G) for all groups G. By definition, if G ∈ / AXp (P2 ) for p ∈ char X, then ΦX (G) = Φ OX (G) . We do not know whether in groups belonging to AXp (P2 ) for some p ∈ char X the above equality holds. This raises the following question: Open question 3.2.11. Let X be a class of simple groups such that char X= π(X) and let p ∈ char X. If G ∈ AXp (P2 ), is it true that Φ(G) = Φ OX (G) ? Moreover, the compatibility of ΦX (G) with quotients of G is not visible and doubtful. In fact, we do not know whether ΦX (G/N ) = ΦX (G)/N for N G such that N ≤ ΦX (G). In the sequel, using the ideas contained in the paper [BBCER05], we shall prove that the X-local formations are exactly those formations which are closed under extensions by the Frattini-like subgroups studied above. It leads to extensions of the Gasch¨ utz-Lubeseder-Schmid and Baer theorems. We begin with the following definitions. Definitions 3.2.12. Let F be a formation. We say that: 1. F is X-saturated (N) if F contains a group G whenever it contains G/Φ OX (G) . 2. F is X-saturated (F) if G ∈ F provided that G/Φ∗X (G) ∈ F. 3. F is X-saturated if G ∈ F provided that G/ΦX (G) ∈ F. 4. G has property X∗ if F contains every group G ∈ AXp (P2 ), p ∈ char X, whenever it contains G/Φ(G). Remarks 3.2.13. Let F be a formation. 1. F is X-saturated if and only if F is X-saturated (N) and F has property X∗ . 2. If F is X-saturated (F), then F is X-saturated. 3. If X = J, then F is X-saturated if and only if F is saturated. 4. If X ⊆ P, then F is X-saturated if and only if F is X-saturated (N). The main result in this section is the following.


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Theorem 3.2.14. Let F be a formation. The following statements are pairwise equivalent: 1. F 2. F 3. F 4. F

is is is is

X-local. X-saturated (F). X-saturated. X-saturated (N) and F has property X∗ .

We begin with some preliminary results. Lemma 3.2.15. Let p be a prime in char X, let G be a group, and let N be a normal subgroup of G such that N ≤ OX (G). Then CXp G/Φ(N ) = CXp (G)/Φ(N ). Proof. Put A/Φ(N ) = CXp G/Φ(N ) . It is clear that A is a normal subgroup of G such that Φ(N ) ≤ Op ,p (G) ≤ CXp (G) ≤ A. We prove that A ≤ CXp (G); we consider A acting on G and N by conjugation, and define the following formation function: (1) for q = p, f (q) = E for q = p. Next we see that A acts f -hypercentrally on N (cf. [DH92, IV, 6.2]). Let H/K be an A-composition factor of G between A∩N and N . Since [A, N ] ≤ A∩N , it is true that CA (H/K) = A. Let H/K be a chief factor of G between Φ(N ) and A∩N such that p divides |H/K|. Then H/K is an Xp -chief factor of G because N ≤ OX (G). Hence CA (H/K) = A and so A centralises every A-composition factor of N between K and H. It yields that A acts f -hypercentrally on N/Φ(N ). By [DH92, IV, 6.7], A acts f -hypercentrally on N . Let H/K be an Xp -chief factor of G below Φ(N ). Since H/K is a minimal normal subgroup of G/K and H/K ≤ A/K, we can apply [DH92, A, 4.13] to conclude that H/K = L1 /K × · · · × Lr /K, where Li /K is a minimal normal subgroup of A/K for all 1 ≤ i ≤ r. Since Li /K is an A-composition factor of N and p divides |Li /K|, it follows that A ≤ CG (Li /K). Hence CA (H/K) = A. Consequently A centralises all Xp -chief factors of G below Φ(N ) and so A ≤ CXp (G). Theorem 3.2.16. If F is an X-local formation, then F is X-saturated (F). Proof. Let G be a group such that G/Φ∗X (G) ∈ F. We prove that G ∈ F p by induction on |G|. Let p be a prime in char X. Then G/ΦX (G) ∈ F and p ∗ ΦX (G)/ Op (G) = ΦX G/ Op (G) by Proposition 3.2.2 (3). Consequently, if Op (G) = 1, we have G/ Op (G) ∈ F. This implies that every Xp -chief factor H/K of G is G-isomorphic to an Xp -chief factor of G/ Op (G). Hence G CG (H/K) ∈ F (p), where F is the canonical X-local definition of F. We may assume that Op (G) = 1 for some prime p ∈ char X. In this ∗ Suppose that Φ (G) = Φ OX (G) . case ΦpX (G) = Φ∗X (G) is a p-group. X Then p divides OX (G) Φ OX (G) and so G has an Xp -chief factor above

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Φ OX (G) . In particular, F (p) = ∅. Since G/Φ OX (G) ∈ F, we have that G/Φ OX (G) CXp G/Φ OX (G) ∈ F (p). By Lemma 3.2.15, it follows G/ CXp (G) ∈ F (p). We conclude then that G satisfies Condition 1 in Definition 3.1.1. Assume now that Φ∗X (G) = Φ OX (G) , then Φ(G) and Soc G/Φ(G) = F (G)/Φ(G) belong to E X and Φ∗X (G) = Φ(G). Note that in this case p divides the order of every X-chief factor of G below F (G). Let T be the intersection of the centralisers in G of the Xp -chief factors of G between Φ(G) and F (G). Then G/T ∈ F (p) because G/Φ(G) ∈ F. Moreover, T /Φ(G) centralises F (G)/Φ(G) because F (G)/Φ(G) is a direct product of Xp -chief factors of G. By [För85b, Satz 1.2], T /Φ(G) is a p-group. This yields T is a p-group and so G ∈ F (p). Consequently, in both cases, G satisfies Condition 1 in Definition 3.1.1. Let L be a normal subgroup of G such that G/L is monolithic and Soc(G/L) belongs to E(S) for some S ∈ X . Then Φ∗X (G) ≤ OX (G) ≤ L and so G/L ∈ F = F (S). Hence G satisfies Condition 2 in Definition 3.1.1 and therefore G ∈ F. This is to say that F is X-saturated (F). Lemma 3.2.17. Let p be a prime and let F be a (Cp )-saturated formation. 1. Let X be a group, and let M , N be GF(p)X-modules with N irreducible and X acting faithfully on M . If [M ]X ∈ F, then [N ]X ∈ F. 2. Let N be an elementary abelian normal p-subgroup of a group G. Assume that [N ](G/N ) ∈ F and that Cp ∈ F. Then G ∈ F. Proof. 1 and 2 follow from the proofs of [DH92, IV, 4.1] and [DH92, IV, 4.15], respectively, taking into account that the Hartley group used there plays the role of the normal p-subgroup. Lemma 3.2.18. Let F be a (Cp )-saturated formation, p a prime. If X ∈ G CG (H/K) : G ∈ F and H/K is an abelian p-chief factor of G , then [N ]X ∈ F for every irreducible GF(p)X-module. R0

Proof. The group X has a set {N1 , . . . , Nn } of normal subgroups satisfying: 1. X/Ni is isomorphic to Gi CGi (Hi /Ki ), where Gi ∈ F and Hi /Ki is an abelian p-chief factor of Gi , n 2. i=1 Ni = 1. By Corollary 2.2.5, [Hi /Ki ](X/Ni ) ∈ F, 1 ≤ i ≤ n. Note that Hi /Ki can be regarded as X-modules over GF(p) and Ker(X on Hi /Ki ) = Ni , 1 ≤ i ≤ n. Moreover, the semidirect product [Hi /Ki ]X has normal subgroups Hi /Ki and Ni satisfying [Hi /Ki ]X (Hi /Ki ), [Hi /Ki ]X/Ni ∈ F. Therefore [Hi /Ki ]X ∈ R0 F = F, 1 ≤ i ≤ n. Put M = H1 /K1 × · · · × Hn /Kn . Then M is an n X-module and Ker(X on M ) = i=1 Ni = 1. Hence X acts faithfully on M . Consider the set {M1 , . . . , Mn } of normal subgroups of [M ]X: M1 = H2 /K2 × · · · × Hn /Kn , . . . , Mn = H1 /K1 × · · · × Hn−1 /Kn−1 and Mi =


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H1 /K1 × · · · × Hi−1 /Ki−1 × Hi+1 /Ki+1 × · · · × Hn /Kn , 2 ≤ i ≤ n − 1. Then n ∼ j=1 Mj = 1 and [M ]X/Mj = [Hj /Kj ]X ∈ F. Therefore [M ]X ∈ R0 F = F. By Lemma 3.2.17, [N ]X ∈ F for every irreducible GF(p)X-module. Theorem 3.2.19. If F is an X-saturated formation, then F is X-local. Proof. By Remark 3.2.10 (3.2), F is a (Cp )-saturated formation for all p ∈ char X. Bearing in mind Theorem 3.1.17, the natural candidate f for an X-local definition of F is given by f (p) = Sp Q R0 G CG (H/K) : G ∈ F and H/K is an abelian p-chief factor of G for p ∈ char X, f (S) = F for S ∈ X . It is clear that f is an X-formation function. Put H = LFX (f ). Suppose that F is not contained in H and let G ∈ F \ H of minimal order. We shall show that this supposition leads to a contradiction. Since H is a formation, it follows that G has a unique minimal normal subgroup, N say, and that G/N ∈ H. If N has composition type S ∈ X , then G ∈ f (S) = F. This is impossible. Therefore N is an X-chief factor of G. If N is non-abelian, then G is a primitive group of type 2. Let p be a prime divisor of |N |. Then p ∈ char X and, by Proposition 3.2.7, there exists E ∈ AXp (P2 ) such that E/ CE (T ) ∼ = G for some minimal normal subgroup T of G. Moreover T is a p-group. Since Φ(E) = CE (T ) = ΦX (E) and F is X-saturated, it follows that E ∈ F. This means that G ∈ f (p). Then we conclude that G ∈ F because / F by supposition, and so we must have that N is a Op (G) = 1. But G ∈ p-group for some prime p ∈ char X. In this case, G/ CG (N ) ∈ f (p) and so G ∈ H by Remark 3.1.7 (2), and we reach a contradiction. Therefore F ⊆ H. Suppose that H is not contained in F, and let G be a group of minimal order in H \ F. Then, as usual, G has a unique minimal normal subgroup N and G/N ∈ F. Moreover neither N ∈ E(X ) nor N is a non-abelian E Xgroup because G ∈ / F. Consequently, N is an abelian p-group for some prime p ∈ char X. In particular, f (p) = ∅ and therefore H contains the cyclic group of order p. By Corollary 2.2.5, A = [N ](G/N ) ∈ H. Assume that N < CG (N ). Then M = (G/N ) ∩ CA (N ) is a non-trivial normal subgroup of A. Since |A/M | < |G|, we have that A/M ∈ F by minimality of G. Hence A ∼ = A/(N ∩ M ) ∈ R0 F = F. We can apply Lemma 3.2.17 (2) and deduce that G ∈ F. This is a contradiction. Hence we must have CG (N ) = N and so G/N ∈ ∈ f (p). Since Op (G/N ) = 1 by [DH92, B, 3.12 (b)], it follows that G/N Q R0 B CB (H/K) : B ∈ F and H/K is an abelian p-chief factor of B . This yields that G/N ∼ = X/T for some normal subgroup T of X ∈ R0 B CB (H/K) : B ∈ F and H/K is an abelian p-chief factor of B . Now N can be regarded as an irreducible X-module over GF(p) such that T = Ker(X on N ). By Lemma 3.2.18, we have [N ]X ∈ F. Consequently G ∼ =

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[N ](G/N ) ∼ = [N ](X/T ) belongs to F. We have reached a contradiction. Hence H ⊆ F and the equality holds. Return for the moment to Theorem 3.2.14. It can be deduced at once from Theorem 3.2.16, Theorem 3.2.19, and Remarks 3.2.13. Note that the Gasch¨ utz-Lubeseder-Schmid theorem is a special case of Theorem 3.2.14 when X = J, the class of all simple groups. Another generalisation of Gasch¨ utz’s concept of local formation in the general finite universe is due to L. A. Shemetkov, who introduced in 1973 the notion of composition formation. The most general version of these kind of formations was presented in [She01]. Let us describe Shemetkov’s approach. Let Y = ∅ be a class of simple groups. A function which associates with every group A ∈ Y a formation f (A) and with every group B ∈ Y a formation ∅ = f (Y ) is called a CY -satellite. If f is a CY -satellite, then the class CFY (f ) of all groups G satisfying: 1. if H/Kis a Y-chief factor of G and S is the composition factor of H/K, then G CG (H/K) ∈ f (S), and 2. G/ OY (G) ∈ f (Y ) is a formation. We say that a formation F is a Y-composition formation if F = CFY (f ) for some CY -satellite f . Remark 3.2.20. Let ∅ = Y be a class of simple groups. Denote X = char Y = {Cp : p ∈ char Y}. Then the Y-composition formations are exactly the X-saturated ones. Proof. Let F = CFY (f ) be a Y-composition formation. Then it is clear that F = LFX (f0 ), where f0 is the X-formation function defined by f (p) if S ∼ = Cp ∈ X, f0 (S) = F if S ∈ X . By Theorem 3.2.14, F is X-saturated. Conversely, suppose that F is an X-saturated formation. Then, by Theorem 3.2.14, F = LFX (F ), where F is the canonical X-local definition of F. We define a CY -satellite f by the following formula: F (p) if S ∼ = Cp ∈ X, f (S) = F if S ∈ X . Then F = CFY (f ).

Assume that X ⊆ P, then F is X-saturated if and only if F is (Cp )-saturated for all p ∈ char X by Theorem 3.2.14, Corollary 3.1.13 and Corollary 3.1.21. Therefore we have:

3.3 Products of X-local formations

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Corollary 3.2.21 ([She97, Theorem 3.2], [She01, Lemma 7]). Let F be a formation, ∅ = Y a non-empty class of simple groups and π = char Y. The following statements are pairwise equivalent: 1. F is closed under extensions by the Frattini subgroup of a normal soluble π-subgroup. 2. F contains each group G provided that F contains G/Φ F(G)π , where F(G)π is the Hall π-subgroup of the Fitting subgroup of G. 3. A group G belongs to F if and only if G/Φ Op (G) belongs to F for all p ∈ π. 4. F is a Y-composition formation. When Y = P, the class of all abelian simple groups, we have: Corollary 3.2.22 ([F¨ or84a, Korollar 3.11]). Let F be a formation. The following statements are pairwise equivalent: 1. F is solubly saturated. 2. A group G belongs to F if and only if G/Φ F(G) ∈ F. 3. F contains a group G provided that F contains G/Φ Op (G) for every prime p. Final remark 3.2.23. In the sequel we make use of the fact that the concepts of “X-saturated formation” and “X-local formation” are equivalent without appealing to Theorem 3.2.14.

3.3 Products of X-local formations As a point of departure, consider the following observations: if F and G are saturated formations, then the formation product F ◦ G is again saturated ([DH92, IV, 3.13 and 4.8]). However, the formation product of two solubly saturated formations is not solubly saturated in general as the following example shows. Example 3.3.1 ([Sal85]). Let F = D0 1, Alt(5) and G = S2 . Then it is clear that F and G are solubly saturated. Assume that H = F ◦ G is solubly saturated. Then H = LFP (H), where H is the canonical P-local definition of H. Since G ⊆ H, it follows that H(2) = ∅. Consider G = SL(2, 5). Then G/ Z(G) ∈ H and G/ CG Z(G) ∈ H(2). Applying Remark 3.1.7 (2), we have that G ∈ H. This is not true. Hence H is not solubly saturated. Taking the above example into account, the following question arises: Which are the precise conditions on two X-local formations F and G to ensure that F ◦ G is an X-local formation?

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The problem of the existence of solubly saturated factorisations of solubly saturated formations was taken up by Salomon [Sal85]. A complete answer to the general question was obtained in [BBCER06]. In the first part of the section we are concerned with the above question. We stay close to the treatment presented in [BBCER06]. In the following F and G are formations and H = F ◦ G If p ∈ char X, denote GX (p) = Sp Q R0 G CG (H/K) : G ∈ G and

H/K is an Xp -chief factor of G .

By Theorem 3.1.11, the smallest X-local formation formX (G) containing G is X-locally defined by the X-formation function G given by G(p) = GX (p), p ∈ char X, and G(S) = F for every S ∈ X . The next theorem provides an X-local definition of formX (H). Theorem 3.3.2. Assume that F is an X-local formation defined by an integrated X-formation function f . Then the smallest X-local formation formX (H) containing H is X-locally defined by the X-formation function h given by f (p) ◦ G if Sp ⊆ F h(p) = p ∈ char X if Sp ⊆ F GX (p) h(S) = H

S ∈ X

¯ = LFX (h) and Proof. It is clear that h is an X-formation function. We set H ¯ ¯ first prove that H ⊆ H. Assume that H \ H contains a group G of minimal ¯ Let order. Then G has a unique minimal normal subgroup N and G/N ∈ H. G ¯ A = G G. If A = 1, then G ∈ G ⊆ H, contrary to supposition. Therefore ¯ G would N is contained in A. If N were an X -chief factor of G, since G/N ∈ H, ¯ Since G ∈ H, the second condition satisfy the first condition to belong to H. would also be satisfied, bearing in mind that h(S) = H for every simple group ¯ Hence N ∈ E X. Applying [DH92, S ∈ X . This would imply that G ∈ H. A, 4.13], N = N1 × · · · × Nn , where Ni is a minimal normal subgroup of A, 1 ≤ i ≤ n. Since A ∈ F, it follows that f (p) = ∅ for each prime p dividing |N |. Moreover A/ CN (Ni ) ∈ f (p), for all i ∈ {1, . . . , n}, and p | |N |. Consequently G G/ CG (N ) ∼ = A/ CA (N ) ∈ R0 f (p) = f (p) and so G/ CG (N ) ∈ f (p) ◦ G = ¯ h(p) for all p | |N |. Hence, applying Remark 3.1.7 (2), we have that G ∈ H. ¯ ¯ This contradiction proves that H ⊆ H. Since H is X-local, it follows that ¯ formX (H) ⊆ H. On the other hand, we know by Theorem 3.1.17 that formX (H) = LFX (H), where H is the X-formation function defined by H(p) = HX (p) if p ∈ char(X) H(E) = H if E ∈ X


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¯ is not contained in formX (H) and choose a group Z ∈ Suppose that H ¯ \ formX (H) of minimal order. Then Z has a unique minimal normal subH group N and Z/N ∈ formX (H). Moreover it is clear that N ∈ E X. Let p be ¯ we have a prime dividing |N |. If Sp ⊆ F, then h(p) = GX (p). Since Z ∈ H, we are that Z/ CZ (N ) ∈ GX (p) ⊆ H(p). Assume case Sp ⊆ F. Then in the Z/ CZ (N ) ∈ h(p) = f (p)◦G and Cp Z/ CZ (N ) ∈ Sp f (p)◦G ⊆ Sp f (p)◦G. By Theorem 3.1.17, we know that Sp f (p) ⊆ F and, hence, Cp Z/ CZ (N ) ∈ F ◦ G ⊆ formX (H). This implies that Z/ CZ (N ) ∈ HX (p) = H(p) by Theorem 3.1.17. Applying Remark 3.1.7 (2), we can conclude that Z ∈ formX (H). ¯ = formX (H). ¯ ⊆ formX (H) and, hence, H This contradiction shows that H The following definition was introduced in [Sal85] for Baer-local formations. Definition 3.3.3. We say that the boundary b(H) is XG-free if every group G ∈ b(H) such that Soc(G) is a p-group for some prime p ∈ char X satisfies / GX (p). that G/ CG Soc(G) ∈ Remark 3.3.4. Note that in Example 3.3.1, b(H) is not PG-free. The next result provides a test for X-locality of H in terms of its boundary. Theorem 3.3.5. Assume that F is X-local. Then H is an X-local formation if and only if b(H) is XG-free. Proof. Suppose that H is X-local. Then H = LFX (H), where H is the canonical X-local definition of H. Let Gbe a group in b(H) such that Soc(G) is a p-group Soc(G) were in GX (p), then we would have that for some p ∈ char X. If G/ C G G/ CG Soc(G) ∈ HX (p) = H(p), since G ⊆ H. By Remark 3.1.7 (2), it would imply that G ∈ H. This would be a contradiction. Therefore G/ CG Soc(G) ∈ / GX (p) and b(H) is XG-free. Conversely, suppose that b(H) is XG-free. Consider an integrated X-local definition f of F. By Theorem 3.3.2, formX (H) = LFX (h), where f (p) ◦ G if Sp ⊆ F h(p) = p ∈ char X if Sp ⊆ F GX (p) h(S) = H

S ∈ X

We shall prove that H = formX (H). Assume that this is not the case and choose a group G of minimal order in formX (H) \ H. Then G ∈ b(H) and so G has a unique minimal normal subgroup, N say, and G/N ∈ H. If N were an X -group, we would have that G ∈ h(S) for some S ∈ X . This would imply that G ∈ H, contrary to supposition. Hence N is an X-chief factor of G. Let p be a prime dividing |N |. Since p ∈ char X, it follows that G/ CG (N ) ∈ h(p). Since h(p) ⊆ Sp H and Op G/ CG (N ) = 1, we have that G/ CG (N ) ∈ H. Therefore CG (N ) = 1 and so N is an abelian p-group.

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Assume that Sp is not contained in F. Then h(p) = GX (p). We conclude that b(H) is not XG-free. This contradiction shows that Sp is contained in F. Then G/ CG (N ) ∈ f (p) ◦ G. It follows that GG / CGG (N ) ∈ f (p). Since GG /N ∈ F, we can apply Remark 3.1.7 (2) to conclude that GG ∈ F, that is, G ∈ H. This contradiction shows that formX (H) is contained in H and, therefore, H is X-local. Example 3.3.6. Let S be a non-abelian simple group with trivial Schur multiplier. Consider F = D0 (1, S), the formation of all groups which are a direct product of copies of S together with the trivial group. Let X be a class of simple groups such that S ∈ / X. Notice that F is X-local. Let G be any formation. Suppose that G ∈ b(H), N = Soc(G) is the minimal normal subgroup of G, and N is a p-group for some p ∈ char X. If G/ CG (N ) ∈ GX (p), then N ≤ Z(GG ) because 1 = GG ≤ CG (N ). Since G/N ∈ H, it follows that GG /N ∈ F. Assume that GG /N = 1. This implies that GG /N , a direct product of copies of S, has non-trivial Schur multiplier, contrary to [Suz82, Exercise 4 (c), page 265]. Thus GG = N and then G ∈ formX (H) by Remark 3.1.7 (2). Therefore if formX (G) ⊆ Np G for all primes p ∈ char(X), it follows that G ∈ G, and this contradicts our choice if G. Hence b(H) is XG-free and H is X-local by Theorem 3.3.5. Consequently, H is X-local for all formations G satisfying formX (G) ⊆ Np G for all primes p ∈ char(X). As an application of Theorem 3.3.5 we have: Theorem 3.3.7. Assume that F is X-local and G is a formation satisfying one of the following conditions: 1. G is X-local, or 2. Sp G = G for all p ∈ char X \ char F. Then H is X-local if F and G satisfy the following condition: If p ∈ char X ∩ π(F) and Sp ⊆ G, then Sp ⊆ F.

(3.2)

Proof. Consider the canonical X-local definition F of F. We will obtain a contradiction by assuming that H is not X-local. Then, by Theorem 3.3.5, there exists a group G ∈ b(H) such that N = Soc(G) is the unique minimal normal subgroup of G, N is a p-group for some prime p ∈ char X and G/ CG (N ) ∈ GX (p). Since GX (p) ⊆ Sp G and Op G/ CG (N ) = 1, it follows that G/ CG (N ) ∈ G. Then GG ≤ CG (N ). Since GG = 1, it follows that N ≤ GG . Hence N ≤ Z(GG ). Moreover GG /N ∈ F because G/N ∈ H. Suppose that N is not contained in Φ(GG ). Then there exists a maximal subgroup M of GG such that GG = M N . Notice that M is normal in GG . Then Op (GG ) is contained in M and is a normal subgroup of G. If Op (GG ) = 1, it follows that N ≤ Op (GG ) ≤ M . This contradiction proves that GG is a p-group. Assume that p ∈ / char F. In this case, since GG /N ∈ F, it follows that N = GG . This means that G/N ∈ G. If G is X-local, we conclude that G ∈ G by Re/ char F. mark 3.1.7 (2). If G is not X-local, we have G ∈ Sp G = G because p ∈


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In both cases, we reach a contradiction. Hence we have that p ∈ char F. In this case F (p) = ∅. In particular, Sp ⊆ F as F is X-local. Therefore GG ∈ F. This contradiction proves that N is contained in Φ(GG ). This implies that p divides |GG /N | and so p ∈ π(F). If p ∈ char F, then F (p) = ∅ and GG ∈ F as F is X-local and Remark 3.1.7 (2) can be applied. Suppose that p ∈ / char F. If G is X-local, we have that Sp ⊆ G because GX (p) = ∅. The same holds / char F, we have that Sp is contained in G. By if Sp G = G. Hence if p ∈ Condition (3.2), Sp ⊆ F. This contradiction completes the proof. Since local formations are X-local for every class of simple groups X (see Corollary 3.1.13), we obtain as a special case of Theorem 3.3.7 the following results: Corollary 3.3.8. Suppose that either of the following conditions is fulfilled: 1. F is local and G is X-local. 2. F is local and Sp G = G for all p ∈ char X \char F. Then H is an X-local formation. Proof. If F is local, then condition (3.2) in Theorem 3.3.7 is satisfied, since Sp ⊆ F for every p ∈ π(F). Corollary 3.3.9 ([DH92, IV, 3.13 and 4.8]). H is a local formation if either of the following conditions is satisfied: 1. F and G are both local. / char F. 2. F is local and Sp G = G for all p ∈ Example 3.3.6 shows that there are cases in which a product of an X-local formation and a non X-local formation is X-local. This observation leads to the following question: Are there X-local products of non X-local formations? The local version of the above question is the one appearing in The Kourovka Notebook ([MK90]) as Question 9.58. It was posed by L. A. Shemetkov and A. N. Skiba and answered affirmatively in several papers (see [BBPR98, Ved88, Vor93]). The next example gives a positive answer to the above question when |char X| ≥ 2. Example 3.3.10 ([BBPR98]). Assume that p and q are different primes in char X. Consider the formations F = Sp Aq ∩ Aq Sp and G = Sq Ap , where Ar denotes the formation of all abelian r-groups for a prime r. F is not (Cq )local and G is not (Cp )-local. Therefore, by Corollary 3.1.13, F and G are not X-local. However H = F ◦ G is local and so it is X-local.

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Note that if the formation of all p-groups, p a prime, were a product of two proper subformations, Question 9.58 in [MK90] would be solved automatically. Perhaps it was the reason to put forward the following question in The Kourovka Notebook [MK90]: Question 10.72 (Shemetkov). To prove indecomposability of Sp , p a prime, into a product of two non-trivial subformations. This question was solved positively by L. A. Shemetkov and A. N. Skiba in [SS89]. We present a general version of this conjecture as a corollary of a more general result at the end of the section. On the other hand, bearing in mind Example 3.3.10, the following question naturally arises: Which are the precise conditions on two formations F and G to ensure that H = F ◦ G is X-local? Our next results answer this question. Notation 3.3.11. If Y is a class of groups, denote YG = (Y G : Y ∈ Y). Lemma 3.3.12. If T is a group such that T ∈ / G and Sp (T ) ⊆ H for some prime p, then Sp (T G ) ⊆ F. Proof. Let Z be a group in Sp (T G ). Then Z has a normal subgroup P such that P is a p-group and Z/P is isomorphic to T G = 1. Assume that ps is the exponent of the abelian p-group P/P . Consider Q = P nat H, where H = (1, 2, . . . , ps ) is a cyclic group of order ps regarded as a subgroup of the symmetric group of degree ps . Here the wreath product is taken with respect to the natural permutation representation of H of degree ps . Set D = {(a, . . . , a) : s a ∈ P } the diagonal subgroup of P , the base group of Q. Since ap ∈ P , we have that D is contained in [P , H] by [DH92, A, 18.4]. In particular D is contained in Q . Let Y = Q T be the regular wreath product of Q with T . Since Q ∈ Sp (T ) ⊆ H, it follows that Q ∈ H. Therefore Y G ∈ F. Now, by Proposition 2.2.8, we know that Y G = (B ∩Y G )T G , where B = Q is the base group of Y . Now, by [DH92, A, 18.8], BT G is isomorphic to (Qn ) T G , where n = |T : T G | and C ≤ [C, T G ], for C = (Qn ) , by virtue of [DH92, A, 18.4]. This implies that B = [B, T G ] ≤ [B, Y G ] ≤ B ∩Y G . Hence B T Gis contained in Y G . Applying Theorem 2.2.6, B T G ∈ F. Therefore Q )n T G ∈ F. Since P is isomorphic to a subgroup of Q , it follows that P n T G ∈ F by Theorem 2.2.6. Since P can be regarded as a subgroup of P n , we have that P T G is a subgroup of P n T G supplementing the Fitting subgroup of P n T G . Applying again Theorem 2.2.6, we have that P T G ∈ F. By [DH92, A, 18.9], Z is isomorphic to a subgroup of P T G supplementing the Fitting subgroup of P T G . Therefore Z ∈ F by virtue of Theorem 2.2.6. This completes the proof of the lemma.


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Theorem 3.3.13. H is an X-local formation if and only if the following two conditions hold: 1. If p ∈ (char X) ∩ char formX (H) and HX (p) is not contained in G, then Sp HX (p)G ⊆ F. 2. If p ∈ (char X) ∩ char formX (H) , G ∈ b(H), and N = Soc(G) is a / H. p-group, then [N ](G/N ) ∈ Proof. Assume that H is an X-local formation, that is, H = formX (H). We know that H = LFX (H), where H is the X-formation function defined in Theorem 3.1.17. Consider a prime p ∈ char(X) and assume there exists a group T ∈ HX (p) \ G. We have that Sp (T ) ⊆ Sp HX (p) = HX (p) ⊆ H. Hence, by Lemma 3.3.12, we have that Sp (T G ) ⊆ F. Now consider a group G in Sp HX (p)G . Then G has a normal p-subgroup N such that G/N ∼ = T¯G , where G G ¯ ¯ ¯ T ∈ HX (p). If T = 1, we have just proved that Sp (T ) ⊆ F and, therefore, G ∈ F. If T¯G = 1, then G ∈ Sp . Consider the group A := G × T G . We have that A ∈ Sp (T G ) ⊆ F and, therefore, G ∈ Q(F) = F. We conclude that Sp HX (p)G ⊆ F and Statement 1 holds. Let G be such that N = Soc(G) is a p-group for a prime p ∈ a group in b(H) (char X)∩ char formX (H) . Note that N is a minimal normal subgroup of G. If H := [N ](G/N ) ∈ H, we would have that H/ CH (N ) ∈ HX (p) and, therefore, G/ CG (N ) ∈ HX (p). Since G/N ∈ H, this would imply by Remark 3.1.7 (2) that G ∈ LFX (H) = H. This contradiction proves Condition 2. To prove the sufficiency, assume that H is the product of F and G and H satisfies Conditions 1 and 2. We will obtain a contradiction by supposing that formX (H) \ H contains a group G of minimal order. Such a G has a unique minimal normal subgroup, N , and G/N ∈ H. This is to say that G ∈ b(H). If N ∈ E(X ), then there exists S ∈ X such that G ∈ H(S) = H, contrary to supposition. Therefore N ∈ E X. Let p be a prime dividing |N |. Then G/ CG (N ) ∈ HX (p). In particular p ∈ (char X) ∩ char formX (H) . If N were non-abelian, then CG (N ) = 1 and G ∈ HX (p). This would imply that G ∈ H because Op (G) = 1. It would contradict the choice of G. Therefore N is an abelian p-group. Applying Corollary 2.2.5, A = [N ](G/N ) ∈ formX (H). Suppose that the intersection B of CA (N ) with G/N is non-trivial. Then B A and A/B ∈ H by the choice of G. Since G/N ∈ H, we have that A ∈ R0 H = H. This contradicts Statement 2. Hence B = 1 and N = CG (N ). In particular G ∈ HX (p)\G. Applying Statement 1, we have that Sp HX (p)G ⊆ F. We deduce then that GG ∈ F and so G ∈ H. We have reached a final contradiction. Therefore formX (H) ⊆ H and H is X-local. Remark 3.3.14. If X = J, then Condition 1 implies Condition 2 in the above theorem. Proof. Assume that H satisfies Condition 1. Let G ∈ b(H) such that N = Soc(G) is the unique minimal normal subgroup of G. Suppose that N is a p-group for some p ∈ (char X) ∩ char formX (H) .

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Suppose that Φ(G) = 1. Then G is a primitive group, CG (N ) = N and G is isomorphic to [N ](G/N ). Therefore, [N ](G/N ) ∈ / H and the remark follows. Now assume that Φ(G) = 1. Consider T /N := Op (G/N ). Since T /N is p-nilpotent and N ≤ Φ(G), we have by [Hup67, VI, 6.3] that T is p-nilpotent. This implies that T = N because otherwise we would find a non-trivial normal p -subgroup of G. Hence, Op (G/N ) = 1. Consequently, G ∈ HX (p) by [DH92, IV, 3.7]. By Condition 1, Sp (GG ) ⊆ F. In particular, GG ∈ F. We conclude that G ∈ H, which contradicts our supposition. Corollary 3.3.15 ([BBPR98, Theorem A]). A formation product H of two formations F and G is local if and only if H satisfies the following condition: If p ∈ char lform(H) and HJ (p) is not contained in G, then Sp HJ (p)G ⊆ F. When a product is X-local, the formation G has a very nice property. Corollary 3.3.16. If H = F ◦ G is X-local, then formX (G) ⊆ Np G for all primes p ∈ char(X) \ π(F). Proof. Let p ∈ char(X) \ π(F). By Theorem 3.3.13, we have that HX (p) ⊆ G. Consider the canonical X-formation function G defining formX (G). Suppose that formX (G) is not contained in Np G, and let G ∈ formX (G) \ Np G be a group of minimal order. Then G ∈ H and G has a unique minimal normal subgroup, N say. In addition, N ≤ GG and G/N ∈ Np G. If N ∈ E X , it follows that G ∈ G(S) for some S ∈ X . But, in this case, G ∈ G. This is a contradiction. Hence N is an E X-group. Consider q ∈ π(N ). If N were non-abelian, then G would belong to G(q) ⊆ Sq G. Hence G ∈ G because Oq (G) = 1. This would contradict our assumption. Therefore N is an elementary abelian q-group for some prime q ∈ char X. Assume that Φ(G) = 1. Then G is a primitive group and N = CG (N ). Therefore G ∈ G(q). If p = q, then G ∈ Np G because G(q) ⊆ Sq G and if p = q, then G ∈ Sp HX (p) = HX (p) ⊆ G. In both cases, we reach a contradiction. Hence N is contained in Φ(G). If p = q, then F(G) is a p -group and G/ F(G) ∼ = (G/N ) F(G/N ) ∈ G. Hence, G ∈ Np G, contrary to supposition. Assume that p = q. Then, since G/N ∈ Np G, it follows that (G/N )G = GG /N is a p -group. Thus GG /N is contained in Op (G/N ) which is trivial by [Hup67, VI, 6.3]. Hence N = GG . Since G ∈ H, we have that GG = N ∈ F and p ∈ π(F). This final contradiction proves that formX (G) ⊆ Np G. If X = J, we have: Corollary 3.3.17 ([She84]). If H = F◦G is local, then lform(G) is contained in Np G for all primes p ∈ / π(F). This result motivates the following definition.


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Definition 3.3.18. Let ω be a non-empty set of primes, and let F be a formation. 1. (see [She84]) F is said to be ω-local if lform(F) is contained in Nω F. 2. (see [SS00a]) F is called ω-saturated if the condition G/ Φ(G)∩Oω (G) ∈ F always implies G ∈ F. When ω = {p}, we shall say p-local (respectively, p-saturated) instead of {p}-local (respectively, {p}-saturated). Remarks 3.3.19. Let ∅ = ω be a set of primes and let F be a formation. 1. F is ω-local if and only if F is p-local for all p ∈ ω. Hence F is local if and only if F is p-local for all primes p. 2. If F is an ω-local formation, then F is ω-saturated. 3. If F is ω-saturated, then Nω F is local. Therefore every ω-saturated formation is ω-local (see [SS95]). 4. Every formation composed of ω -groups is ω-saturated. 5. Every ω-saturated formation is Xω -saturated, where Xω is the class of all simple ω-groups. Proof. 1, 2, and 4 are clear. To prove Statement 3, suppose that F is ωsaturated. If q is a prime such that q ∈ ω , then H = Nω F is q-saturated. Assume that p is a prime in ω such that H is not p-saturated. Then there / H. Let us choose exists a group G such that G/ Φ(G) ∩ Op (G) ∈ H but G ∈ G of least order. Then G has a unique minimal normal subgroup N , N is contained in Φ(G) ∩ Op (G), and G/N ∈ H. Since F is contained in H, it follows that GF = 1 and so N is also contained in GF . Now Op (G/N ) = 1 and GF /N is a p -group because G/N ∈ H. This implies that GF = N . But then G/N ∈ F and so G ∈ F because F is p-saturated. This contradiction shows that H is p-saturated for all p ∈ ω. Therefore H is saturated. In particular, lform(F) ⊆ Nω F and F is ω-local. 5 follows directly from the fact that ΦXω (G) ⊆ Φ(G) ∩ Oω (G) for every group G. The family of Xω -saturated formations does not coincide with the one of ω-saturated formations in general. This follows from the fact that there exist Baer-local formations which are not ω-saturated for any non-empty set of primes ω. Example 3.3.20 ([BBCER03]). Let Y = {Alt(n) : n ≥ 5} and F = E Y. It is clear that F is a Baer-local formation. In particular, F is X-saturated for every X ⊆ P by Corollary 3.1.13. Assume that F is p-saturated for a prime p. If p ≥ 5, set k := p; otherwise, set k := 5. As p divides |Alt(k)|, there exists a group E with a normal ∼ elementary abelian p-subgroup A ≤ Φ(E) and E/A = Alt(k) A = 1 such that ∼ ([DH92, B, 11.8]). Then E/ Φ(E) ∩ Op (E) = E/A ∈ F. Therefore E ∈ F, and we have a contradiction.

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This implies that F is not ω-saturated for any non-empty set of primes ω. In particular, F is (C2 )-saturated but not 2-saturated. Suppose that G is a p-saturated formation, p a prime. Then lform(G) ⊆ Np G. Therefore G(p) ⊆ Np G and so G(p) = GJ (p) ⊆ G. The converse is also true as the following lemma shows. Lemma 3.3.21. G is p-saturated if and only if G(p) ⊆ G. Proof. Only the sufficiency is in doubt. Suppose that G is not p-saturated and GJ (p) ⊆ G. Let G be a group of minimal order satisfying G/ Φ(G) ∩ Op (G) ∈ G and G ∈ / G. G is a monolithic group and N := Soc(G) ≤ Φ(G) ∩ Op (G). We have that Op ,p (G/N ) = Op ,p (G)/N , since N ≤ Φ(G). Moreover, G/N ∈ G and, therefore, G/ Op ,p (G) ∈ GJ (p), bearing in mind that p ∈ π(G/N ). Since Op ,p (G) = Op (G), G ∈ GJ (p) ⊆ G. This is not possible. Theorem 3.3.22. Let F and G be formations. Let M be a p-saturated formation contained in F ◦ G, where p is a prime. If MJ (p) is not contained in G, then Sp MJ (p)G ⊆ F. Proof. Assume that M is p-saturated. Then MJ (p) is contained in M by Lemma 3.3.21. There exists a group T ∈ MJ (p) \ G. We have that Sp (T ) ⊆ MJ (p) ⊆ M ⊆ F ◦ G. Hence Sp (T G ) ⊆ F by Lemma 3.3.12. Now consider a group G in Sp MJ (p)G . Then G has a normal p-subgroup N such that G/N ∼ = T¯G , where T¯ ∈ MJ (p). If T¯G = 1, we have just proved G ¯ that Sp (T ) ⊆ F and, therefore, G ∈ F. If T¯G = 1, then G ∈ Sp . Consider the group A := G × T G . We have that A ∈ Sp (T G ) ⊆ F and, therefore, G ∈ Q(F) = F. We conclude that Sp MJ (p)G ⊆ F. Corollary 3.3.23. Let F and G be formations and let p be a prime. Then the following statements are equivalent: 1. H = F ◦ G is a p-saturated formation. 2. If HJ (p) is not contained in G, then Sp HJ (p)G ⊆ F. Proof. 1 implies 2 by virtue of Theorem 3.3.22. Let us prove that 2 implies 1. We shall derive a contradiction by supposing that HJ (p) \ H contains a group G of minimal order. Then G has a unique minimal normal subgroup N , and G/N ∈ H. Since HJ (p) is contained in Sp H, it follows that N is a p-group. It is clear that HJ (p) is not contained in G. Hence Sp HJ (p)G ⊆ F. Note that N ≤ GG and GG /N ∈ HJ (p)G . Therefore GG ∈ Sp HJ (p)G ⊆ F. This contradiction shows that HJ (p) ⊆ H and that H is p-saturated by Lemma 3.3.21. Theorem 3.3.22 also confirms a more general version of the abovementioned conjecture of L. A. Shemetkov concerning the non-decomposability of the formation of all p-groups (p a prime) as formation product of two non-trivial subformations.

3.4 ω-local formations

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Corollary 3.3.24. Let F, G, and M be formations such that M is contained in F ◦ G and M is p-saturated. If F ⊆ Sp and F = Sp , then M ⊆ G. Proof. If MJ (p) = ∅, it follows that M ⊆ Ep . In this case, we have that M ⊆ Ep ∩ (F ◦ G) ⊆ Ep ∩ (Sp ◦ G). Therefore, M ⊆ G. If MJ (p) = ∅, we have that M ⊆ Ep MJ (p). If MJ (p) is contained in G, then M ⊆ Ep MJ (p) ∩ (Sp G) ⊆ (Ep G) ∩ (Sp G) = G and the result holds. Suppose that MJ (p) is not contained in G. Then Sp MJ (p)G is contained in F by Theorem 3.3.22. In particular, Sp ⊆ F, and we have a contradiction.

3.4 ω-local formations The family of ω-local formations, ω a set of primes, emerges naturally in the study of local formations that are products of two formations as it was observed in Section 3.3. There it is also proved that the ω-local formations are exactly the ones which are closed under extensions by the Hall ω-subgroup of the Frattini subgroup. In this section ω-saturated formations are studied by using a functional approach. This method was initially proposed by L. A. Shemetkov in [She84] for p-local formations, and further developed in [SS00a, SS00b, BBS97]. The second part of the section is devoted to study the relation between ω-saturated formations and X-local formations, where X is a class of simple groups which is naturally associated with ω. Definition 3.4.1 ([SS00a]). Let ω be a non-empty set of primes. Every function of the form f : ω ∪ {ω } −→ {formations} is called an ω-local satellite. If f is an ω-local satellite, define the class LFω (f ) = G : G/Gωd ∈ f (ω ) and G/ Op ,p (G) ∈ f (p) for all p ∈ ω ∩ π(G) , where Gωd is the product of all normal subgroups N of G such that every composition factor of N is divisible by at least one prime in ω (Gωd = 1 if π Soc(G) ∩ ω = ∅).

If f is an ω-local satellite, we write Supp(f ) = p ∈ ω ∪ {ω } : f (p) = ∅ . Denote π1 = Supp(f ) ∩ ω, π2 = ω \ π1 . Then LFω (f ) = ∩ p∈π2 Ep E S ◦ f (p) ∩ E ◦ f (w ). Here E is the class of all groups G such p ωd ωd p∈π1 p that every composition factor of G is divisible by at least one prime in ω. Since the intersection and the formation product of two formations are again formations, the above formula implies that LFω (f ) is a formation. Theorem 3.4.2 ([SS00a]). Let ω be a non-empty set of primes and let F be a formation. The following statements are equivalent:

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1. F is ω-saturated. 2. F = LFω (f ), where f (p) = FJ (p), p ∈ ω, and f (ω ) = F. Proof. 1 implies 2. It is clear that F ⊆ LFω (f ). Suppose that the equality does not hold and derive a contradiction. Choose a group G ∈ LFω (f ) \ F of minimal order. Then, as usual, G has a unique minimal normal subgroup N and G/N ∈ F. Since G/Gωd ∈ f (ω ) = F, it follows that Gωd = 1. This implies that π(N ) ∩ ω = ∅. Let p ∈ ω be a prime dividing |N |. If N were non-abelian, then G ∈ FJ (p). Since, by Lemma 3.3.21, FJ (p) ⊆ F, we would have G ∈ F. This would be a contradiction. Therefore N is an abelian p-group. Moreover N ∩ Φ(G) = 1 because F is ω-saturated. Hence N = CG (N ) and G/N ∈ FJ (p). This implies that G ∈ Sp FJ (p) = FJ (p), and we have a contradiction. Consequently F = LFω (f ). 2 implies 1. Suppose that F is not ω-saturated. Then there exists a prime / F. Denote p ∈ ω and a group G such that G/ Φ(G) ∩ Op (G) ∈ F but G ∈ L = Φ(G)∩Op (G). Then (G/L)ωd = Gωd /L and Oq ,q (G/L) = Oq ,q (G)/L for all primes q. Hence G/Gωd ∈ f (ω ) and G/ Oq ,q (G) ∈ f (q) for all q ∈ ω∩π(G) because G/L ∈ F. Consequently G ∈ F. This contradiction completes the proof of the theorem. Remark 3.4.3. An ω-saturated formation can be ω-locally defined by two distinguished ω-local satellites: the minimal ω-local satellite and the canonical one. Moreover, if Y is a class of groups, the intersection of all ω-local formations containing Y is the smallest ω-local formation containing Y. Such ωlocal formation is denoted by lformω (Y). It is clear that lformω (Y) = LFω (f ), where f is given by: f (p) = Q R0 G/ Op ,p (G) : G ∈ Y if p ∈ π(Y) ∩ ω, f (p) = ∅, p ∈ ω \ π(Y), f (ω ) = Q R0 (G/Gωd : G ∈ Y) (see [SS00a] for details). Let ω be a non-empty set of primes. One can ask the following question: Is it possible to ensure the existence of a class X(ω) of simple groups such that char X(ω) = π X(ω) satisfying that a formation is ωsaturated if and only if it is X(ω)-saturated? The following example shows that the answer is negative. Example 3.4.4 ([BBCER03]). Consider the formation F := (G : all abelian composition factors of G are isomorphic to C2 ). Suppose that F is X-saturated for a class X containing a non-abelian simple group E and π(X) = char X. Let p = 2 be a prime dividing |E|. Then p ∈ char X. Since E ∈ F, it follows that if F = LFX (f ), then f (p) = ∅. This means


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that Cp ∈ F. This is a contradiction. Hence X should be composed of abelian simple groups. Since F is solubly saturated, we have that F is X-saturated exactly for the classes of simple groups X contained in P by Corollary 3.1.13. Since F is clearly 2-saturated, if we assume the existence of a class X(2) fulfilling the property, it follows that X(2) ⊆ P. This is not possible because the formation in Example 3.3.20 is X(2)-saturated but not 2-saturated. The following theorem shows that an X-local formation always contains a largest ω-local formation for ω = char X. Theorem 3.4.5 ([BBCS05]). Let X be a class of simple groups such that ω = char X = π(X). Let F = LFX (F ) be an X-local formation. Then the ω-local formation Fω = LFω (f ), where f (p) = F (p) for every p ∈ ω and f (ω ) = F, is the largest ω-local formation contained in F. Proof. Suppose, for a contradiction, that Fω is not contained in F. Let G be a group of minimal order in Fω \ F. Then, as usual, G has a unique minimal normal subgroup N , and G/N ∈ F. If Gωd = 1, we would have that G ∈ f (ω ) = F, contradicting the choice of G. Assume that Gωd = 1. Then N is contained in Gωd . This means that there exists a prime p ∈ ω dividing |N |. Hence G/ CG (N ) ∈ f (p) = F (p). If N is a p-group, it follows that N is an X-chief factor of G. By Remark 3.1.7 (2), we conclude that G ∈ LFX (F ) = F, against the choice of G. Hence N is non-abelian and so CG (N ) = 1 and G ∈ F (p). Since F (p) = Sp f (p) and Op (G) = 1, it follows that G ∈ f (p) ⊆ F. This contradiction proves that Fω ⊆ F. Now let G = LFω (g) be an ω-local formation contained in F. Suppose, if possible, that G is not contained in Fω and let A be a group of minimal order in the supposed non-empty class G\Fω . Then A has a unique minimal normal subgroup B, and A/B ∈ Fω . Since A ∈ G ⊆ F, we have that A/Aωd ∈ F = f (ω ). Suppose that p ∈ ω ∩π(B). If B is an X-chief factor of A, it follows that A/ CA (B) ∈ F (p) = f (p). If B is an X -chief factor of A, then B is non-abelian and A ∼ = A/ CA (B) ∈ g(p). Then Op (A) = 1 and so, by [DH92, B, 10.9], A has a faithful irreducible representation over GF(p). Let M be the corresponding module and G = [M ]A the corresponding semidirect product. Let us see that G ∈ G. Since M is contained in Gωd , it follows that G/Gωd ∈ g(ω ) because A/Aωd ∈ g(ω ). Moreover, we have that G/ CG (M ) ∼ = A ∈ g(p). We can conclude that G ∈ G and, consequently, G = [M ]A ∈ F. This implies that A ∼ = G/ CG (M ) ∈ f (p). Now we can state that A ∈ Fω , contradicting the choice of A. Therefore G is contained in Fω . An immediate application of Theorem 3.4.5 is the following corollary: Corollary 3.4.6 ([BBCER03]). Let ω be a set of primes and let Xω be the class of all simple ω-groups. If F is an Xω -local formation composed of ω-separable groups, then F is ω-local.

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Proof. Suppose that F is an Xω -local formation. According to Theorem 3.4.5, F = LFXω (F ) contains a largest ω-local formation Fω , where f (p) = F (p) for every p ∈ ω and f (ω ) = F. Suppose that the inclusion is proper, and let G be a group of minimal order in F \ Fω . Then G has a unique minimal normal subgroup N , and G/N ∈ Fω . It is clear that G/Gωd ∈ f (ω ) = F. If p ∈ π(N ) ∩ ω, it follows that N is an ω-group, since G is ω-separable. Hence, N is an Xω -chief factor of G and, therefore, G/ CG (N ) ∈ F (p) = f (p). Taking into account that G/N ∈ Fω , we conclude that G ∈ Fω . This contradiction proves that F = Fω is ω-local. Corollary 3.4.7 ([BBCER03]). Let F be a formation composed of ω-separable groups. Then F is ω-saturated if and only if F is Xω -saturated, where Xω is the class of all simple ω-groups. The following consequence of Theorem 3.4.5 is of interest. Corollary 3.4.8 ([Sal85]). Every solubly saturated formation contains a maximal saturated formation with respect to inclusion. Remarks 3.4.9. 1. The converse of Corollary 3.4.8 does not hold. It is enough to consider F = D0 S2 , Alt(5) . By Lemma 2.2.3, F is a formation. The group SL(2, 5) shows that F is not solubly saturated. However S2 is the maximal saturated formation contained in F. 2. There exist formations not containing a maximal saturated formation as the Example 5.5 in [Sal85] shows: Let F be the class of all soluble groups G such that Sylow subgroups corresponding to different primes permute. By [Hup67, VI, 3.2], F is a formation. Let q be a prime and consider the formation function fq given by: fq (p) = S{p,q} for all p ∈ P. Then the saturated formation Fq = LF(fq ) is contained in F by [Hup67, VI, 3.1]. Let q1 and q2 be two different primes and let Fq1 ,q2 be the smallest saturated formation containing Fq1 and Fq2 . Then Cq1 × Cq2 ∈ F (p) for all p ∈ P, where F is the canonical local definition of Fq1 ,q2 . This is due to the fact that Cq1 ∈ Fq1 (p) and Cq2 ∈ Fq2 (p), where Fq1 and Fq2 are the canonical local definitions of Fq1 and Fq2 , respectively. Let q3 be a prime, q3 = q1 , q2 . By [DH92, B, 10.9], Cq1 × Cq2 has an irreducible and faithful module M over GF(q3 ). Let G = [M ](Cq1 × Cq2 ) / F. This be the corresponding semidirect product. Then G ∈ Fq1 ,q2 , but G ∈ shows that F does not contain a maximal saturated formation with respect to the inclusion. A natural question arising from the above results is the following: What are the precise conditions to ensure that an X-local formation is ω-local for ω = char X? The next result gives the answer. Theorem 3.4.10. Let F = LFX (f ) = LF(F ) be an X-local formation and ω = char X. The following conditions are pairwise equivalent:


167

1. F is ω-local. 2. f (S) ⊆ f (p) for every S ∈ X and p ∈ π(S) ∩ ω. 3. Sp f (S) ⊆ F for every S ∈ X and p ∈ π(S) ∩ ω. Proof. 1 implies 2. Assume that F is ω-local. Then, by Theorem 3.4.5, F = LFω (f ), where f (p) = F (p) = Sp f (p)

if p ∈ ω,

f (ω ) = F. Let S ∈ X and p ∈ π(S) ∩ ω. Then S is non-abelian. By Theorem 3.1.11, f (S) = Q R0 G/L : G ∈ F, G/L is monolithic, and Soc(G/L) ∈ E(S) . Let G be a group in F and let L be a normal subgroup of G such that G/L is monolithic and Soc(G/L) ∈ E(S). Since G/L is a primitive group of type 2, L = CG Soc(G/L) . Moreover G/L ∈ F. This implies that G/L ∈ F (p) = Sp f (p). Hence G/L ∈ f (p) because Op (G/L) = 1. Consequently f (S) ⊆ f (p) for all p ∈ π(S) ∩ ω. 2 implies 3. Let S ∈ X and p ∈ π(S) ∩ ω. Then Sp f (S) ⊆ Sp f (p) = F (p) ⊆ F. 3 implies 2. Applying Theorem 3.4.5, it is known that Fω = LFω (f ), where f (p) = F (p) f (ω ) = F,

if p ∈ ω, and

is the largest ω-local formation contained in F. Suppose, by way of contradiction, that F is not ω-local. Then Fω = F. Let G be a group of minimal order in F \ Fω . By a familiar argument, G has a unique minimal normal subgroup N , and G/N ∈ Fω . If π(N ) ∩ ω = ∅, then Gωd = 1 and so G ∈ Fω , which contradicts the fact that G ∈ / Fω . Therefore π(N ) ∩ ω = ∅. Let p be a prime in π(N ) ∩ ω. If N is an Xp -chief factor of G, G/ CG (N ) ∈ F (p) = f (p). Assume that N is an X -chief factor of G and N ∈ E(S). Then S is nonabelian and so Op (G) = 1. By [DH92, B, 10.9], G has an irreducible and faithful module M over GF(p). Let Z = [M ]G be the corresponding semidirect product. Since G ∈ f (S), it follows that Z ∈ Sp f (S) ⊆ F. This implies that G ∼ = Z/ CZ (M ) ∈ F (p) = f (p). Consequently G/ CG (N ) ∈ f (p) for all p ∈ π(N ) ∩ ω and G ∈ Fω . This contradicts our initial supposition. Therefore F = Fω and F is ω-local.

4 Normalisers and prefrattini subgroups

The aim of this chapter is to obtain information about the structure of a finite group through the study of H-normalisers and subgroups of prefrattini type. In the soluble universe, after the introduction of saturated formations and covering subgroups by W. Gasch¨ utz, R. W. Carter, and T. O. Hawkes introduced in [CH67] a conjugacy class of subgroups associated to saturated formations F of full characteristic, the F-normalisers, defined in terms of a local definition of F, which generalised Hall’s system normalisers. The CarterHawkes’s F-normalisers keep all essential properties of system normalisers and, in the case of the saturated formation N of the nilpotent groups, the N-normalisers of a group are exactly Hall’s system normalisers. In this context, and having in mind the known characterisation of Fnormalisers by means of F-critical subgroups, it is natural to think about H-normalisers associated with Schunck classes H for which the existence of H-critical subgroups is assured in each soluble group not in H. A. Mann [Man70] chose this characterisaton as his starting point and was able to extend introduced the normaliser concept to certain Schunck classes following this arithmetic-free way. Concerning the prefrattini subgroups, we said in Sections 1.3 and 1.4 that the classical prefrattini subgroups of soluble groups were introduced by W. Gasch¨ utz ([Gas62]). A prefrattini subgroup is defined by W. Gasch¨ utz as an intersection of complements of the crowns of the group. They form a characteristic conjugacy class of subgroups which cover the Frattini chief factors and avoid the complemented ones. Gasch¨ utz’s original prefrattini subgroups have been widely investigated and variously generalised. The first extension is due to T. O. Hawkes ([Haw67]). He introduced the idea of obtaining analogues to Gasch¨ utz’s prefrattini subgroups, associated with a saturated formation F, by taking intersections of certain maximal subgroups defined in terms of F into which a Hall system of the group reduces. Note that Hawkes restricts the set of maximal subgroups considered to the set of F-abnormal maximal subgroups. He observed that all of the relevant properties of the original idea were kept and, furthermore, he presented an original new theorem of 16 9

170


factorisation of the F-normaliser and the new prefrattini subgroup associated to the same Hall system. The extension of this theory to Schunck classes, still in the soluble realm, was done by P. F¨ orster in [F¨ or83]. Another generalisation of the Gasch¨ utz work in the soluble universe is due to H. Kurzweil [Kur89]. He introduced the H-prefrattini subgroups of a soluble group G, where H is a subgroup of G. The H-prefrattini subgroups are conjugate in G and they have the cover-avoidance property; if H = 1 they coincide with the classical prefrattini subgroups of Gasch¨ utz and if F is a saturated formation and H is an F-normaliser of G the H-prefrattini subgroups are those described by Hawkes. The first attempts to develop a theory of prefrattini subgroups outside the soluble universe appeared in the papers of A. A. Klimowicz in [Kli77] and A. Brandis in [Bra88]. Both defined some types of prefrattini subgroups in π-soluble groups. They manage to adapt the arithmetical methods of soluble groups to the complements of crowns of p-chief factors, for p ∈ π, of π-soluble groups. Also the extension of prefrattini subgroups to a class of non finite groups with a suitable Sylow structure, made by M. J. Tomkinson in [Tom75], has to be mentioned. All these types of prefrattini subgroups keep the original properties of Gasch¨ utz: they form a conjugacy class of subgroups, they are preserved by epimorphic images and they avoid some chief factors, exactly those associated to the crowns whose complements are used in their definition, and cover the rest. Moreover, some other papers (see [Cha72, Mak70, Mak73]) analysed their excellent permutability properties, following the example of the theorem of factorisation of Hawkes. At the beginning of the decade of the eighties of the past twentieth century, when the classification of simple groups was almost accomplished, H. Wielandt proposed, as a main aim after the classification, to progress in the universe of non-necessarily soluble groups trying to extend the magnificent results obtained in the soluble realm. As we have mentioned in Section 2.3, R. P. Erickson, P. F¨ orster and P. Schmid answered this Wielandt’s challenge analysing the projective classes in the non-soluble universe. It seems natural to progress in that direction and think about normalisers and prefrattini subgroups in the general finite universe. This was the starting point A. Ballester-Bolinches’ Ph. Doctoral Thesis at the Universitat de València in 1989 [BB89b]. This chapter has two main themes which are organised in three sections. The first two sections are devoted to study the theory of normalisers of finite, non-necessarily soluble, groups. The second subject under investigation is the theory of prefrattini subgroups outside the soluble universe. This is presented in Section 4.3.

4.1 H-normalisers

171

4.1 H-normalisers Obviously the definition of H-normalisers in the general universe has to be motivated by the characterisation of H-normalisers of soluble groups by chains of H-critical subgroups. In this section, H will be a Schunck class of the form H = EΦ F, for some formation F. Thus, by Theorem 2.3.24, the existence of H-critical subgroups is assured in every group which does not belong to H. Here we present the extension of the theory of H-normalisers to general non-necessarily soluble groups done by A. Ballester-Bolinches in his Ph. Doctoral Thesis [BB89b] and published in [BB89a]. Previous ways of extending the soluble theory had been looked at. J. Beidleman and B. Brewster [BB74] studied normalisers associated to saturated formations in the π-soluble universe, π a set of primes, and L. A. Shemetkov [She76] introduced normalisers associated to saturated formations in the general universe of all finite groups by means of critical supplements of the residual. The definition of H-normaliser presented here is obviously motivated by the most abstract characterisation of the classical H-normalisers. Definition 4.1.1. Let G be a group. A subgroup D of G is said to be an H-normaliser of G if either D = G or there exists a chain of subgroups D = Hn ≤ Hn−1 ≤ · · · ≤ H1 ≤ H0 = G

(4.1)

such that Hi is H-critical subgroup of Hi−1 , for each i ∈ {1, . . . , n}, and Hn contains no H-critical subgroup. The condition on Hn is equivalent to say that D ∈ H. Moreover D = G if and only if G ∈ H. The non-empty set of all H-normalisers of G will be denoted by NorH (G). If we restrict ourselves to the universe of soluble groups, this definition is equivalent to the classical ones of R. W. Carter and T. O. Hawkes in [CH67] and A. Mann in [Man70] (see [DH92, V, 3.8]). In this section, we analyse the main properties of H-normalisers, primarily motivated by their behaviour in the soluble universe. In particular, we study their relationship with systems of maximal subgroups and projectors. Each H-normaliser of a soluble group is associated with a particular Hall system of the group ([Man70]). Obviously this is no longer true in the general case. But bearing in mind the relationship between systems of maximal subgroups and Hall systems (see Theorem 1.4.17 and Corollary 1.4.18), it seems natural to wonder about the relationship between H-normalisers and systems of maximal subgroups. Assume that D is an H-normaliser of a group G constructed by the chain D = Hn ≤ Hn−1 ≤ · · · ≤ H1 ≤ H0 = G

(4.2)

172


such that Hi is H-critical subgroup of Hi−1 , for each i ∈ {1, . . . , n}, and Hn contains no H-critical subgroup. Let X(D) be a system of maximal subgroups of D. Applying several times Theorem 1.4.14, we can obtain a system of maximal subgroups X of G such that there exist systems of maximal subgroups Xi of Hi , for i = 0, 1, . . . , n, with X0 = X, Xn = X(D) and for each i, Hi ∈ Xi−1 and (Xi−1 )Hi = {Hi ∩ S : S ∈ Xi−1 , S = Hi } ⊆ Xi . This motivates the following definition. Definition 4.1.2. Let D be an H-normaliser of a group G constructed by a chain (4.2) and let X be a system of maximal subgroups of G such that there exist systems of maximal subgroups Xi of Hi , i = 0, 1, . . . , n, with X0 = X, Xn = X(D) and for each i, Hi ∈ Xi−1 and (Xi−1 )Hi = {Hi ∩ S : S ∈ Xi−1 , S = Hi } ⊆ Xi . We will say that D is an H-normaliser of G associated with X. By the previous paragraph, every H-normaliser is associated with some system of maximal subgroups. Next we see that every system of maximal subgroups has an associated H-normaliser. Proposition 4.1.3. Given a system of maximal subgroups X of a group G, there exists an H-normaliser of G associated with X. Proof. We argue by induction on the order of G. We can assume that G ∈ / H. Let M be an H-critical maximal subgroup of G such that M ∈ X. By Corollary 1.4.16, there exists a system of maximal subgroups Y of M , such that XM ⊆ Y. By induction, there exists an H-normaliser D of M associated with Y. Then D is an H-normaliser of G associated with X. Remarks 4.1.4. 1. An H-normaliser can be associated with some different systems of maximal subgroups. Consider the symmetric group of order 5, G = Sym(5), and H = N the class of nilpotent groups. Write D = (12), (45). The subgroups M1 = D(123) and M2 = D(345) are N-critical maximal subgroups of G and X1 = {M1 , Alt(5)} and X2 = {M2 , Alt(5)} are systems of maximal subgroups of G. Observe that D is an N-normaliser of G associated with X1 and X2 . 2. Given a system of maximal subgroups X of a group G, there is not a unique H-normaliser of G associated with X. In the soluble group G = a, b : a9 = b2 = 1, ab = a−1 , the Hall system Σ = {G, a, b} reduces into the N-critical subgroup M = a3 , b and then the N-normalisers D1 = b and D2 = a3 b are associated with the system of maximal subgroups defined by Σ: X(Σ) = {a, a3 , b}. For a non-soluble example, consider the Example of 1 and observe that D1 = (12), (45), D2 = (13), (45) and D3 = (23), (45) are N-normalisers associated with X1 .

4.1 H-normalisers

173

One of the basic properties of H-normalisers of soluble groups is that they are preserved by epimorphic images (see [DH92, V, 3.2]). This is also true in the general case. Proposition 4.1.5. Let G be a group. Let N be a normal subgroup of G. If D is an H-normaliser of G associated with a system of maximal subgroups X, then DN/N is an H-normaliser of G/N associated with X/N . In particular, the H-normalisers of a group are preserved under epimorphic images. Proof. We argue by induction on the order of G. Suppose first that N is a minimal normal subgroup of G. If G ∈ H, D = G and there is nothing to prove. If G ∈ / H, then G has an H-critical subgroup M ∈ X such that D is an H-normaliser of M associated with a system of maximal subgroups Y of M and XM ⊆ Y. If N is contained in M , then DN/N is, applying induction, an H-normaliser of M/N associated with the system of maximal subgroups Y/N of M/N . Since X/NM/N = XM /N is contained in Y/N and M/N is H-critical in G/N by Lemma 2.3.23, it follows that DN/N is an H-normaliser of G/N associated with X/N . Suppose that G = M N . By induction, D(M ∩ N )/(M ∩ N ) is an H-normaliser of M/(M ∩ N ) associated with Y/(M ∩ N ). Therefore, by virtue of the canonical isomorphism between G/N and M/(M ∩ N ), it follows that DN/N is an H-normaliser of G/N associated with X/N (note that the image of X/N = {Y N/N : Y ∈ XM } under the above isomorphism is just Y/(M ∩ N )). Assume now that N is not a minimal normal subgroup of G and let A be a minimal normal subgroup of G contained in N . Then, by induction, DA/A is an H-normaliser of G/A associated with X/A and (DN/A) (N/A) is H-normaliser of (G/A) (N/A) associated with (X/A) (N/A). Consequently, DN/N is an H-normaliser of G/N associated with X/N . The proof of the proposition is now complete. It is well-known that H-normalisers of soluble groups cover the H-central chief factors and avoid the H-eccentric ones (see [DH92, V, 3.3]). The coveravoidance property is a typical property of the soluble universe that we cannot expect to be satisfied in the general one. We present here some results to show partial aspects of the cover-avoidance property of H-normalisers in the general universe. Lemma 4.1.6. Let M be an H-critical subgroup of a group G. If H/K is an H-central chief factor of G, then M covers H/K and [H/K] ∗ G ∼ = [(H ∩ M )/(K ∩ M )] ∗ M . In particular (H ∩ M )/(K ∩ M ) is an H-central chief factor of M . Proof. If M does not cover H/K, then K = H ∩ CoreG (M ) and M supplements H/K. Moreover H CoreG (M )/ CoreG (M ) is the socle of the monolithic primitive group G/ CoreG (M ). Since H CoreG (M )/ CoreG (M ) ∼ =G H/K, then

174


∼ [H/K] ∗ G ∈ H, contrary to the H-abnormality of M in G. G/ CoreG (M ) = Hence M covers H/K. Since H/K is H-central in G, then CG (H/K) is not contained in CoreG (M ) and therefore G = M CG (H/K). Now the result follows from [DH92, A, 13.9]. Corollary 4.1.7. Let D be an H-normaliser of a group G. If H/K is an Hcentral chief factor of G, then D covers H/K and (H ∩ D)/(K ∩ D) is an H central chief factor of D. Moreover, AutG (H/K) ∼ = AutD (H ∩ D)/(K ∩ D) . Proposition 4.1.8. Let D be an H-normaliser of a group G. If H/K is a supplemented chief factor of G covered by D, then [H/K]∗G ∼ = [(H ∩D)/(K ∩ D)] ∗ D ∈ H. Proof. If D = G the result is clear. Suppose that D is an H-critical subgroup of G. Since H/K is avoided by Φ(G) and covered by D, then(H ∩ D)/(K ∩ D) is a chief factor of D, AutG (H/K) ∼ = = AutD (H ∩D)/(K ∩D) and [H/K]∗G ∼ [(H ∩ D)/(K ∩ D)] ∗ D, by Statements (1), (2), and (3) of Proposition 1.4.11. Thus, if H/K is non-abelian, then [H/K] ∗ G is isomorphic to a quotient group of D and therefore [H/K] ∗ G ∈ H. If H/K is abelian, then H/K it is complemented by a maximal subgroup M of G. By Proposition 1.4.11 (4), we have that M ∩ D is a maximal subgroup of D, and (H ∩ D)/(K ∩ D) is a chief factor of D complemented by M ∩ D. Since D ∈ H, the primitive group associated with (H ∩ D)/(K ∩ D) is isomorphic to a quotient group of D and therefore [(H ∩ D)/(K ∩ D)] ∗ D ∈ H. In the general case, we consider the chain (4.2) of subgroups of G. If H/K is a supplemented chief factor of G covered by D, then H/K is covered by H1 and avoided by Φ(G). By Proposition 1.4.11, (H ∩ H1 )/(K ∩ H1 ) is a supplemented chief factor of H1 . Now, since D is an H-normaliser of H1 , then [(H ∩ H1 )/(K ∩ H1 )] ∗ H1 ∼ = [(H ∩ D)/(K ∩ D)] ∗ D by induction. Since clearly [(H ∩ H1 )/(K ∩ H1 )] ∗ H1 ∼ = [H/K] ∗ G, we deduce that [H/K] ∗ G ∼ = [(H ∩ D/(K ∩ D)] ∗ D ∈ H. Corollary 4.1.9. Let D be an H-normaliser of a group G. Then, among all supplemented chief factors of G, D covers exactly the H-central ones. We show next that nothing can be said about the H-eccentric chief factors of G. Example 4.1.10. Let S be the alternating group of degree 5. Consider the class F = G : S ∈ / Q(G) . Then b(F) = S . Hence F is a saturated formation by Example 2.3.21. Let E be the maximal Frattini extension of S with 3elementary abelian kernel (see [DH92, Appendix β] for details). The group E possesses a 3-elementary abelian normal subgroup N such that N ≤ Φ(E), and E/N ∼ = S. Let M be a maximal subgroup of E, such that M/N ∼ = Alt(4). Then M is F-critical in E and, since M is soluble, and then M ∈ F, we have that M is an F-normaliser of E. Observe also that if a minimal normal subgroup K of E in N is F-central in E, then K ≤ Z(E). Recall that N ∼ = A3 (S), the

4.1 H-normalisers

175

3-Frattini module, and we can think of N as an GF(3)[S]-module. If we denote S(N ) the socle of such module, we have that Ker S on S(N ) = O3 ,3 (S) = 1, by a theorem of R. Griess and P. Schmid [GS78]. Therefore there exists an F-eccentric minimal normal subgroup K of E, such that K ≤ N . It is clear that M covers K. Note that the group E has at least three conjugacy classes of F-normalisers. Moreover, none of these F-normalisers has the cover-avoidance property in E. Lemma 4.1.11. Let G be a group. Consider a system of maximal subgroups X of G and an H-normaliser D of G associated with X. Then, for any monolithic H-abnormal maximal subgroup H ∈ X, we have that D is contained in H. Proof. We prove the assertion by induction on |G|. Let H be a monolithic H-abnormal maximal subgroup in X. Assume that G has an H-central minimal normal subgroup, N say. By Corollary 4.1.7, N is contained in D ∩ H. Moreover, applying Proposition 4.1.5, D/N is an H-normaliser of G associated with X/N . By induction, D/N ≤ H/N and then D ≤ H. Thus, we can assume that every minimal normal subgroup of G is H-eccentric in G. If N is contained in H, then, again by Proposition 4.1.5 and induction, we have that D ≤ DN ≤ H. Therefore we assume that CoreG (H) = 1 and G is a monolithic primitive group. There exists a unique minimal normal subgroup N of G. Observe that F (G) = N and so H is H-critical in G. Since H ∈ X, we have that D is contained in H by construction of D. Lemma 4.1.12. If a maximal subgroup M of a group G contains an Hnormaliser of G, then M is H-abnormal in G. Proof. Suppose that D is an H-normaliser of the group G and D is contained in the maximal subgroup M of G. If H/K is a chief factor supplemented by M and H/K is H-central in G, then D covers H/K, by Corollary 4.1.9, and so does M , a contradiction. Hence H/K is H-eccentric in G and M is H-abnormal in G. The previous lemmas allow us to discover the relationship between Hnormalisers and monolithic maximal subgroups. The corresponding result in the soluble universe is in [DH92, V, 3.4]. Corollary 4.1.13. Let M be a monolithic maximal subgroup of a group G. Then M is H-abnormal in G if and only if M contains an H-normaliser of G. It is not true in general that an H-abnormal maximal subgroup M of a group G contains an H-normaliser of G. Example 4.1.14. Consider the saturated formation F composed of all S-perfect groups, for S ∼ = Alt(5), the alternating group of degree 5 as in Example 4.1.10. Let G be the direct product G = S1 × S2 of two copies S1 , S2 of S. Clearly each core-free maximal subgroup is F-abnormal in G. Suppose, arguing by

176


contradiction, that U is a core-free maximal subgroup of G and there exists E ∈ NorF (G) such that E is contained in U . Let M be an F-critical maximal subgroup of G such that E is contained in M and E is an F-normaliser of M . Since M is monolithic, we can assume that S1 = CoreG (M ). Therefore M = S1 × (M ∩ S2 ). It is clear that M ∩ S2 = 1. Let N be a minimal normal subgroup of M contained in M ∩ S2 . Since N is a supplemented F-central chief factor of M , then N is covered by E by virtue of Corollary 4.1.9. Consequently, N ≤ M ∩ S2 ∩ U = 1. This contradiction yields that no core-free maximal subgroup of G contains an F-normaliser of G. The fundamental connection between H-normalisers and H-projectors of a soluble group is that every H-projector contains an H-normaliser (see [Man70, Theorem 9] and [DH92, V, 4.1]). This is no longer true in the general case: any Sylow 5-subgroup of G = Alt(5), the alternating group of degree 5, is an N-projector of G and contains no N-normaliser of G. However we can prove some interesting results that confirm the close relation between H-normalisers and H-projectors, especially when saturated formations H are considered. Definitions 4.1.15. Let G be a group. 1. A maximal subgroup M of G is said to be H-crucial in G if M is Habnormal and M/ CoreG (M ) ∈ H. 2. If G ∈ / H, an H-normaliser D of G is said to be H-crucial in G if there exists a chain of subgroups D = Hn ≤ Hn−1 ≤ · · · ≤ H1 ≤ H0 = G

(4.3)

such that Hi is H-crucial H-critical subgroup of Hi−1 , for each i ∈ {1, . . . , n}, and Hn contains no H-critical subgroup. Proposition 4.1.16. If D is an H-crucial H-normaliser of a group G, then D is an H-projector of G. Proof. Clearly G ∈ / H. Suppose first that D is maximal in G. Then we have that D/ CoreG (D) is an H-maximal subgroup of the group G/ CoreG (D) and G/ CoreG (D) is a primitive group in the boundary of H. Since D/ CoreG (D) is an H-projector of G/ CoreG (D), then D is an H-projector of G by Proposition 2.3.14. Suppose that D is not maximal in G, and let M be an H-crucial H-critical subgroup of G such that D is an H-crucial H-normaliser of M . By induction, D is an H-projector of M . By Proposition 2.3.14, D is an H-projector of G. Lemma 4.1.17. Let G be a group and E an H-maximal subgroup of G such that G = E F(G), then E is an H-normaliser of G.

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Proof. We proceed by induction on |G|. If E = G, there is nothing to prove. We can assume that G ∈ / H and E is then a proper subgroup of G. Let M be a maximal subgroup of G containing E. Since M = E F(M ) and E is H-maximal in M , then E is an H-normaliser of M , by induction. Applying Proposition 2.3.17, E is an H-projector of G and then M is H-critical in G. Therefore E is an H-normaliser of G. Let F be a saturated formation. It is known that in a soluble group in NF, the F-projectors and the F-normalisers coincide (see [DH92, V, 4.2]). The above lemma allows us to extend this result to Schunck classes in the general universe. Theorem 4.1.18. If G is a group in NH, then the H-projectors and the Hnormalisers of G coincide. Proof. We prove by induction on the order of G that the H-normalisers of G are H-crucial in G. If G ∈ H, the result is trivial. Thus, we can assume that G ∈ / H. Let M be an H-critical subgroup of G. Then G = M F(G) and F(G)/Φ(G) is abelian. Hence M ∩ F(G) is contained in CoreG (M ) because M / CoreG (M ) is a quotient group of M/ M ∩ F(G) ∼ = G/ F(G), and then M/ CoreG (M ) ∈ H. Therefore M is H-crucial in G. If D ∈ NorH (G), then there exists an H-critical subgroup M of G such that D ∈ NorH (M ). Since M ∈ NH, we have that D is an H-crucial H-normaliser of M by induction. Therefore D is an H-crucial H-normaliser of G. Therefore we can apply Proposition 4.1.22 to conclude that each Hnormaliser of G is an H-projector of G. Now, let E be an H-projector of G. Since G ∈ NH, it follows that G = E F(G). By Lemma 4.1.17, E is an H-normaliser of G. The previous result can be used to show that, for saturated formations F, the F-normalisers of groups with soluble F-residual can be described in terms of projectors. The corresponding result for soluble groups appears in [DH92, V, 4.3]. Theorem 4.1.19. 1. Let F be a formation and H = EΦ F. Then, for any group G, if D is an NF-normaliser of G, the H-projectors of D are Hnormalisers of G. 2. Let F be a saturated formation and let G be a group such that the Fresidual GF is a soluble group of nilpotent length r. We construct the chain of subgroups Dr ≤ Dr−1 ≤ Dr−2 ≤ · · · ≤ D1 ≤ D0 = G where Di is an Nr−i F-projector of Di−1 , for i ∈ {1, . . . , r}. Then Dr is an F-normaliser of G.

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Proof. 1. By Corollary 3.3.9, NF is a saturated formation. Moreover, H is contained in NF. If G ∈ NF, then G ∈ NH and so ProjH (G) = NorH (G) by Theorem 4.1.18. Thus we can assume that G ∈ / NF. Let D be an NF-normaliser of G. Then there exists a chain of subgroups (4.2), such that Hi−1 is an NF-critical subgroup of Hi , for each index i. Since H ⊆ NF, every H-normaliser of D is an H-normaliser of G. Since D ∈ NF ⊆ NH, we have that ProjH (D) = NorH (D) by Theorem 4.1.18. Hence each H-projector of D is an H-normaliser of G. 2. Let F be a saturated formation and let G be a group whose F-residual, GF , is a soluble group of nilpotent length r. This is to say that G ∈ Nr F. We construct the chain of subgroups Dr−1 ≤ Dr−2 ≤ · · · ≤ D1 ≤ D0 = G where Di is an Nr−i F-projector of Di−1 , for i ∈ {1, . . . , r − 1}. Since G ∈ N(Nr−1 F), then the Nr−1 F-projectors and the Nr−1 F-normalisers of G coincide by Theorem 4.1.18. Therefore D1 is an Nr−1 F-normaliser of G. By Statement 1, the Nr−2 F-projectors of D1 are Nr−2 F-normalisers of G. Thus, D2 is an Nr−2 F-normaliser of G. Repeating this argument, we obtain that Dr−1 is an NF-normaliser of G. Hence, every F-projector of Dr−1 is an Fnormaliser of G by Statement 1. Consequently Dr is an F-normaliser of G. The next result yields a sufficient condition for a subgroup of a group in NH to contain an H-normaliser. Theorem 4.1.20. Let G be a group in NH and E a subgroup of G that covers all H-central chief factors of a given chief series of G. Then E contains an H-normaliser of G. Proof. We argue by induction on the order of G. Clearly we can assume that G∈ / H and that E is a proper subgroup of G. If M is a maximal subgroup of G such that E ≤ M , then M is an H-abnormal subgroup of G and G = M F(G) because E covers the section G/ F(G). This is to say that M is H-critical in G. Moreover M is has the cover-avoidance property and the intersections of M with all normal subgroups of a chief series of G give a chief series of M . If H/K is a chief factor of G in that series covered by M , then (M ∩H)/(M ∩K) is a chief factor of M such that [H/K] ∗ G ∼ = [(M ∩ H)/(M ∩ K)] ∗ M by Proposition 1.4.11. Consequently, E covers all H-central chief factors of a chief series of M . By induction, E contains an H-normaliser of M which is an Hnormaliser of G. We end this section with the analysis of the relation between the Fnormalisers and the F-hypercentre, F a saturated formation. Recall that a normal subgroup N of a group G is said to be F-hypercentral in G if every chief factor of G below N is F-central in G. The product of Fhypercentral normal subgroups of a group is again an F-hypercentral normal

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subgroup of the group (see [DH92, IV, 6.4]). Thus every group G possesses a unique maximal normal F-hypercentral subgroup called the F-hypercentre of G and denoted by ZF (G). Let G be a group. By Corollary 4.1.7, the F-hypercentre of G is contained in every F-normaliser of G. Therefore ZF (G) is contained in CoreG (D), for every H-normaliser D of G. However, the equality does not hold in general. Example 4.1.21. Consider E and F as in Example 4.1.10. By [GS78, Example 1 (b)], ZF (E) = 1. If M is a maximal subgroup of E such that M/N ∼ = Alt(4), then M is an F-normaliser of E and CoreE (M ) = N = 1. In the next section, we shall see that the equality holds in groups with soluble F-residual. Next we describe the F-hypercentre of a group in terms of the F-residual of the group and an F-normaliser. A similar description appears in [DH92, IV, 6.14] for F-maximal subgroups supplementing the F-residual. Note that, in general, the F-normalisers are not F-maximal subgroups. Proposition 4.1.22. Let F be a saturated formation. If D is an F-normaliser of a group G, then ZF (G) = CD (GF ). Proof. Applying [DH92, IV, 6.10]), we have that [GF , ZF (G)] = 1. Therefore ZF (G) is contained in CD (GF ). Next we prove that CD (GF ) is an Fhypercentral normal subgroup of G. Since G = DGF , the CD (GF ) is normal in G. Let H/K be a chief factor of G below CD (GF ). Then GF ≤ CG (H/K). This implies that G = D CG (H/K). Consequently H/K is a chief factor of D by [DH92, A, 13.9]). Since D ∈ F, the chief factor H/K is F-central in D and then in G by [DH92, A, 13.9]). Consequently CD (GF ) is an F-hypercentral normal subgroup of G and hence it is contained in ZF (G).

4.2 Normalisers of groups with soluble residual In this section we assume that F is a saturated formation. Most of the properties of F-normalisers of soluble groups, such as conjugacy, cover-avoidance property, relation with F-projectors, do not hold in the general case (see examples of the previous section). However F-normalisers of groups G in which the F-residual GF is soluble (i.e. groups in the class SF) do really satisfy these classical properties. The purpose of the section is to give a full account of these results. We remark that no use of the corresponding results for soluble groups occurs in our arguments. The following elementary result will be used frequently in the section. Let M be an F-abnormal maximal subgroup of a group G. Then G = M GF . Assume, in addition, that GF is soluble. Then every chief factor of G supplemented by M is abelian. In particular, M is a maximal subgroup of G of type 1.

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Our starting point is a result of P. Schmid which proves that the Fprojectors of a group with soluble F-residual form a conjugacy class of subgroups. Theorem 4.2.1 ([Sch74]). Let F be a saturated formation. Let G be group whose F-residual GF is soluble. Then ProjF (G) is a conjugacy class of subgroups of G. Proof. We argue by induction on |G|. Obviously we can assume that GF = 1. Let N be a minimal normal subgroup of G such that N ≤ GF and suppose that E and D are F-projectors of G. By induction, X = EN = Dg N for some g ∈ G. Since N is abelian, we have that E and Dg are F-projectors of X, by Lemma 4.1.17 and Theorem 4.1.18. If X is a proper subgroup of G, then E and Dg are conjugate in X by induction. Thus we can assume that G = EN , for every minimal normal subgroup N which is contained in GF . Since G/N ∼ = E/(E ∩ N ) ∈ QF = F, we have that N = GF . This is to say F that G is an abelian minimal normal subgroup of G and every F-projector of G is a maximal subgroup of G. Let p be the prime dividing |G|. Let F be the canonical local definition of F = LF(F ), and consider the F (p)-residual T = GF (p) of G. Clearly T contains N . Since G/N ∈ Ep F (p) (see [DH92, IV, 3.2]), it follows that T /N is a p -group. Moreover, since F is full, we have that Op (T ) = T . Hence, for any E ∈ ProjF (G), we have that T = N (T ∩ E) and T ∩ E is a Hall p -subgroup of T . By the Schur-Zassenhaus theorem [Hup67, I, 18.1 and 18.2], the Hall p -subgroups of T are a conjugacy class of subgroups of T . If T ∩E is normal in G, then T ∩E = Op (T ) = T . This is a contradiction. Hence E = NG (T ∩ E) and then ProjF (G) is a conjugacy class of subgroups of G. Assume that G is a group with soluble F-residual, F a saturated formation. Then ProjF (G) = CovF (G). This can be proved by reducing the problem to the case G ∈ b(F) (note that if E is an F-projector of G, then E is an F-projector of EN for every minimal normal subgroup N of G by Proposition 2.3.16). In such case, the equality is obviously true because G is a primitive group of type 1 (see [DH92, III, 3.9]). We show next that in groups with soluble F-residual, the F-normalisers can be joined to the group by means of some special chains. Lemma 4.2.2. Let G be a group whose F-residual GF is soluble. If D is an F-normaliser of G, there exists a chain of subgroups D = Hn ≤ Hn−1 ≤ · · · ≤ H1 ≤ H0 = G

(4.4)

such that Hi is H-critical maximal subgroup of Hi−1 of type 1, for each i ∈ {1, . . . , n}, and Hn contains no F-critical subgroup. Proof. We prove the assertion by induction on |G|. We can assume that G ∈ / F. If M is an F-critical subgroup of G containing D as F-normaliser, then M is


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a maximal subgroup of type 1. Moreover M F ≤ GF by Proposition 2.2.8 (3). Hence M F is soluble. By induction, D can be joined to M by means of a chain of F-critical maximal subgroups of type 1. This completes the proof the lemma. Lemma 4.2.3 (see [Ezq86]). If M is a maximal subgroup of a group G which supplements the Fitting subgroup F(G), then every subgroup with the cover-avoidance property in M is a subgroup with the cover-avoidance property in G. Proof. Let D be a subgroup with the cover-avoidance property in M . Let H/K be a chief factor of G covered by M . Observe that G = M F(G) = M CG (H/K). Then (H ∩ M )/(K ∩ M ) is a chief factor of M . If D covers (H ∩ M )/(K ∩ M ), then H ∩ M = (K ∩ M )(H ∩ D). Since H = K(H ∩ M ), we have that H = K(H ∩ D) and D covers H/K. If D avoids (H ∩ M )/(K ∩ M ), then D ∩H ≤ K and D avoids H/K. Finally D avoids all chief factors avoided by M . Theorem 4.2.4. Let G be a group whose F-residual GF is soluble. If D is an F-normaliser of G, then D covers all the F-central chief factors of G and avoids all the F-eccentric ones. Proof. We use induction on the order of G to prove that F-normalisers are subgroups with the cover-avoidance property in G. Let D = G be an Fnormaliser of G and suppose that D is maximal in G. If H/K is a non-abelian chief factor of G, then D covers H/K since D is of type 1. If H/K is abelian and D does not cover H/K, then G = DH and K ≤ D. In the group G/K, the minimal normal subgroup H/K is abelian and complemented by the maximal subgroup D/K. Then D avoids H/K. If D is not maximal in G, there exists an F-critical maximal subgroup M of G such that D ∈ NorF (M ). By induction, D has the cover-avoidance property in M . Since M supplements F(G), D has the cover-avoidance property in G by Lemma 4.2.3. If H/K is an F-central chief factor of G, then, by Corollary 4.1.7, D covers H/K. Suppose that H/K is an F-eccentric chief factor of G which is covered by D. Suppose that D is defined by a chain (4.4) as in Lemma 4.2.2. Observe that G = H1 F(G) = H1 CG (H/K) and H1 covers H/K. Hence, (H ∩H1 )/(K ∩H1) is a chief factor of H1 such that AutG (H/K) ∼ = AutH1 (H ∩ H1 )/(K ∩ H1 ) . By repeating the argument we obtain that (H ∩ D)/(K ∩ D) is an F-eccentric chief factor of D. Since D ∈ F, all chief factors of D are F-central. This contradiction yields that H/K is avoided by D. Combining Corollary 4.1.7 and Theorem 4.2.4, a chief series of an Fnormaliser D of a group G with soluble F-residual can be obtained by intersecting D with the members of a given chief series of G. Our next result partially extends a result of J. D. Gillam (see [DH92, V, 3.3]) on the cover-avoidance property of F-normalisers. We wonder whether

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the cover-avoidance property characterises the F-normalisers of groups whose F-residual is soluble. The answer in general is negative even in soluble groups (see an example in [DH92, page 401]). Gillam’s result characterises the Fnormaliser of a soluble group associated with a particular Hall system by the cover-avoidance property together with the permutability with the Hall system. Obviously this is not possible in our context. However, Theorem 4.1.20 allows us to show that the characterisation of the F-normalisers by the coveravoidance property, holds in groups whose F-residual is nilpotent. Corollary 4.2.5. If F is a saturated formation and G is a group in NF, then, for a subgroup D of G, the following sentences are equivalent: 1. D is an F-normaliser of G, 2. D covers the F-central chief factors of G and avoids the F-eccentric ones. We have seen in Example 4.1.21 that, in general, the F-hypercentre of a group G is not the core in G of an F-normaliser of G. The equality in groups with soluble F-residual follows from the cover-avoidance property of the F-normalisers. Proposition 4.2.6. Let G be a group such that the F-residual GF is a soluble group. If D is an F-normaliser of G, then ZF (G) = CoreG (D). Proof. If ZF (G) = 1, the core of any F-normaliser is trivial by Theorem 4.2.4. If ZF (G) is non-trivial, the group G/ ZF (G) has trivial F-hypercentre and the quotient D ZF (G)/ ZF (G) is an F-normaliser of G/ ZF (G) by Proposition 4.1.5. Consequently CoreG (D) ≤ ZF (G). Our next major objective is to show that the connections between Fnormalisers and F-projectors of groups with soluble F-residual are similar to the ones of the soluble case. In particular every F-normaliser is contained in an F-projector. Since, by Theorem 4.2.1, the F-projectors of groups in SF form a conjugacy class of subgroups, every F-projector contains an F-normaliser. Theorem 4.2.7. Let F be a saturated formation. If G ∈ NF and H is a subgroup of G such that G = H F(G), then each F-projector of H is of the form H ∩ E, for some F-projector E of G. Proof. Clearly we can assume that F(G) = 1, G = H, and G ∈ / F. Moreover, arguing by induction on the order of G, we can assume that H is a maximal subgroup of G. Since H/ H ∩ F(G) ∈ F, each F-projector D of H satisfies H = D H ∩ F(G) . Then G = D F(G). If E is an F-maximal subgroup of G such that D ≤ E, then E ∈ ProjF (G) by Proposition 2.3.17. It is rather easy to show that D and E ∩ H cover and avoid the same chief factors of a given chief series of G. Consequently D = E ∩ H. Theorem 4.2.8. Let F be a saturated formation. If G is a group whose Fresidual GF is soluble, and H is a subgroup of G such that G = H F(G), then there exist an F-projector A of H and an F-projector E of G such that A = H ∩ E.


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Proof. By Theorem 4.2.7, we can assume that G ∈ / NF. The quotient group ¯ = G/ F(G) has soluble non-trivial F-residual G ¯ F = GF F(G)/ F(G), Since G ¯ F = 1, we can consider a chief factor of G of the form G ¯ F /K. ¯ Since F is satG F ¯ is a complemented abelian chief factor of G. ¯ Let M/ F(G) ¯ /K urated, then G ¯ in G. ¯ Then M is an F-crucial maximal subgroup ¯ F /K be a complement of G of G. If N/ Core G (M ) = Soc G/ CoreG (M ) , then H covers N/ CoreG (M ) and (N ∩ H)/ CoreG (M) ∩ H is an F-eccentric chief factor of H. Moreover, H = (N ∩ H)(M ∩ H) and (N ∩ H)/ CoreG (M ) ∩ H is an abelian chief factor of H. Consequently M ∩ H is an F-crucial maximal subgroup of H. On the other hand, M = (M ∩ H) F(M ) and so M F F(M ) = (M ∩ H)F F(M ) by Proposition 2.2.8 (2). Analogously GF F(G) = H F F(G). This implies that M F is soluble. By induction, there exist A ∈ ProjF (M ∩ H) and E ∈ ProjF (M ) such that A = H ∩ E ∩ M = H ∩ E. By Proposition 2.3.16, the F-projectors of any F-crucial monolithic maximal subgroup of a group are F-projectors of the group. Since M ∩ H is F-crucial in H, we have that A is an F-projector of H, and since M is F-crucial in G, then E is an F-projector of G. Theorem 4.2.9. Let F be a saturated formation. Let G be a group whose F-residual GF is soluble. Then each F-normaliser of G is contained in an F-projector of G and each F-projector contains an F-normaliser. Proof. We argue by induction of the order of G. We can assume that G ∈ / F. Let D be an F-normaliser of G. There exists an F-critical subgroup M of G such that D ∈ NorF (M ). Since M F is soluble, there exists an F-projector A of M such that D is contained in A. Since M is critical in G, we can apply Theorem 4.2.8 to conclude that there exist B ∈ ProjF (M ) and E ∈ ProjF (G) such that B = M ∩ E. By Theorem 4.2.1, the subgroups A and B are conjugate in M . Hence there exists an element x ∈ M such that A = B x . Thus, A = M ∩ E x and D is contained in E x which is an F-projector of G. By Theorem 4.2.1, the F-projectors of G form a conjugacy class of subgroups. Hence, every F-projector contains an F-normaliser. Assume that F is a saturated formation. Let G be a group whose Fresidual GF is soluble. If Σ is a Hall system of GF , then we denote NG (Σ) = {NG (H) : H ∈ Σ}. Sometimes NG (Σ) is said to be the absolute system normaliser in G of Σ. In [Yen70], it is proved that if G is a soluble group, then the F-projectors of T are F-normalisers of G. Our next objective is to show that this result holds not only in soluble groups but also in groups whose F-residual is soluble. As a consequence we will obtain the conjugacy of F-normalisers in such groups. In general, if N is a soluble normal subgroup of a group G and Σ is a Hall system of N , then Σ g is also a Hall system of N , for all g ∈ G. Since Hall systems of a soluble group are conjugate, there exists an element x ∈ N such that Σ g = Σ x . Hence, by the Frattini argument, we have that G = NG (Σ)N . Then NG (Σ) ∩ N is a system normaliser of N . Hence NG (Σ) ∩ N is nilpotent by [DH92, I, 5.4] and NG (Σ)/ NN (Σ) is isomorphic to G/N . If, in addition,

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N contains GF , it follows that G/N ∈ F and so NG (Σ) belongs to NF. In that case, NG (Σ)F is contained in NN (Σ) and so Σ reduces into NG (Σ)F . The next lemma will be used in subsequent proofs. Lemma 4.2.10. Let G be a group whose F-residual GF is soluble. Consider a Hall system Σ of GF and write T = NG (Σ). If N is a normal subgroup of G, then T N/N = NG/N (ΣN/N ). Therefore, if E is an F-projector of T , then EN/N is an F-projector of NG/N (ΣN/N ). Proof. We argue by induction on the order of G. Clearly T N/N is contained in NG/N (ΣN/N ). Assume that N is a minimal normal subgroup of G. Suppose that N ∩ GF = 1. Note that G acts transitively by conjugation on the set of Hall systems of GF N/N . Hence |G/N : NG/N (ΣN/N )| is the number of Hall systems of GF N/N . Moreover, by the same argument, the number of Hall systems of GF is |G : T |. Hence |G/N : NG/N (ΣN/N )| = |G : T |. Now |G : T N | ≤ |G : T | = |G/N : NG/N (ΣN/N )| ≤ |G/N : T N/N |. This implies that T N/N = NG/N (ΣN/N ). Assume now that N ≤ GF = R. Since system normalisers are preserved under epimorphisms by [DH92, I, 5.8], we have that NR/N (ΣN/N ) = NR (Σ)N/N . Hence, since G = RT , we have that |G/N : NG/N (ΣN/N )| = |R/N : NR/N (ΣN/N )| = |R : NR ( Σ)N | = |R : (T ∩ R)N | = |R : R ∩ T N | = |G : T N | = |G/N : T N/N | and then T N/N = NG/N (ΣN/N ). If N is not a minimal normal subgroup of G and A is a minimal normal subgroup of G contained in N , it followsthat T A/A = NG/A (ΣA/A). By induction, (T N/A) (N/A) = N(G/A)/(N/A) (ΣN/A)/(N/A) . Then T N/N = NG/N (ΣN/N ). The following result is also useful. Proposition 4.2.11 ([Hal37]). Let G be a soluble group and N a normal subgroup of G. Let Σ ∗ be a Hall system of N such that Σ ∗ = Σ ∩ N for some Hall system Σ of G. Put M = NG (Σ ∗ ). We have 1. NG (Σ) is contained in M , 2. Σ1 = Σ ∩ M is a Hall system of M , and 3. NM (Σ1 ) = NG (Σ). Proof. 1. For any Hall subgroup H ∗ of N in Σ ∗ , there exists a Hall subgroup H of G in Σ such that H ∗ = H ∩ N . If x ∈ NG (Σ), then H ∗ x = (H ∩ N )x = H x ∩ N = H ∩ N = H ∗ , since N is normal in G. Then x ∈ NG (Σ ∗ ). Hence NG (Σ) ≤ NG (Σ ∗ ) = M . 2. Let p be any prime dividing the order of G, H the Hall p -subgroup of N in Σ ∗ and P the Sylow p-subgroup of G in Σ. There exists a Hall p -subgroup S of G in Σ such that S ∩ N = H. Since S normalises H and G = P S, it follows that T = NG (H) ∩ P is a Sylow p-subgroup of NG (H). Moreover, for any prime q = p, P is contained in the Hall q -subgroup Sq of G in Σ.


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Hence, T ≤ Sq . The subgroup Sq ∩ N is the Hall q -subgroup of N in Σ ∗ and Sq ∩ N is normal in Sq . Therefore T normalises Sq ∩ N . This means that T ≤ NG (Σ ∗ ) = M and T = M ∩ P . For two different primes pi , i = 1, 2, dividing the order of G consider the corresponding Sylow subgroups Pi ∈ Sylpi (G) of G in Σ and Ti = Pi ∩ M , i = 1, 2. Note that P1 P2 is a subgroup of G and T1 , T2 is contained in P1 P2 ∩ M . Hence, T1 , T2 is a {p1 , p2 }-subgroup and so T1 , T2 = T1 T2 . Therefore Σ ∩ M = Σ1 is a Hall system of M . 3. Clearly, Σ ∗ g is a Hall system of N , for all g ∈ G. Therefore, there exists x ∈ N , such that Σ ∗ g = Σ ∗ x . The Frattini argument implies that G = M N . Therefore, if P ∈ Sylp (G) ∩ Σ, then (P ∩ M )N/N = P N/N ∈ Sylp (G/N ). Hence (P ∩ M )(P ∩ N ) = P ∩ (P ∩ M )N = P ∩ P N = P . If x ∈ NG (Σ), then x ∈ M and, for any Sylow subgroup P ∈ Σ, we have that (P ∩ M )x = (P ∩ M ). Hence NG (Σ) ≤ NM (Σ1 ). Conversely, if x ∈ NM (Σ1 ), for any Sylow subgroup P ∈ Σ, we have that x ∈ NG (P ∩ M ) and x ∈ M ≤ NG (P ∩ N ). Hence x ∈ NG (P ). Consequently NM (Σ1 ) ≤ NG (Σ) and the equality holds. Lemma 4.2.12. Let G be a group with a soluble normal subgroup H such that GF ≤ H. Let Σ be a Hall system of H. Denote R = NG (Σ). Then each F-projector of R is contained in an F-projector of NG (Σ ∩ GF ). Proof. Assume that the result is not true and let G be a minimal counterexample. Let H be a normal subgroup of G of minimal index |H : GF | among all normal subgroups for which the assertion does not hold. Let H/K a chief factor of G such that GF ≤ K. Note that Σ ∩ K is a Hall system of K and denote B = NG (Σ ∩ K). Since the lemma is true for G, K, and Σ ∩ K, we have that each F-projector of B is contained in an F-projector of NG (Σ ∩ GF). By Proposition 4.2.11 (2), we have that Σ ∗ = Σ ∩ (H ∩ B) is a Hall system of H ∩ B = NH (Σ ∩ K). On the other hand, since G = NG (Σ ∩ K)K = BH, and then B/(B ∩ H) ∼ = G/H ∈ F, the subgroups B, H ∩ B, and the Hall system Σ ∗ satisfy the hypotheses of the lemma. If B is a proper subgroup of G, each F-projector of Q = NB (Σ ∗ ) is contained in an F-projector of NB (Σ ∗ ∩ B F ). Note that NH∩B (Σ ∗ ) = NH (Σ) by Proposition 4.2.11 (3). Moreover NG (Σ) ≤ Q. Since G = H NG (Σ), we have that B = (H ∩B) NG (Σ). Consequently Q = NG (Σ)(Q ∩ H ∩ B) = NG (Σ) NH∩B (Σ ∗ ) = NG (Σ) = R. This contradiction yields B = G. In other words, every Sylow subgroup of K is normal in G. In particular, G ∈ NF. Suppose that p is the prime divisor of the order of H/K. If P is the Sylow p-subgroup of H in Σ, we have that H = P K and R = NG (Σ) = NG (P ). Let E be an F-projector of R, then G = HR = KR = K(ERF ) = EK = E F(G) because RF is contained in GF . By Theorem 4.2.7, E is contained in an F-projector of G = NG (Σ ∩ GF). This is the final contradiction. Theorem 4.2.13. Let G be a group whose F-residual GF is soluble. Consider a Hall system Σ of GF and denote T = N G (Σ ). Suppose that M is an

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F-abnormal maximal subgroup of G. If Σ reduces into M ∩ GF, then there exists an F-projector of T contained in an F-projector of NM (Σ ∩ M F ). Proof. We split the proof in two steps. 1. There exists an F-projector of T contained in M . We use induction on the order of G. Note that, by [DH92, I, 4.17a] and Lemma 4.2.10, the hypotheses of the lemma hold in G/ CoreG (M ) and M/ CoreG (M ). If CoreG (M ) is non-trivial, then, by induction, there exists an F-projector of T CoreG (M )/ CoreG (M ), D/ CoreG (M ) say, contained in M/ CoreG (M ). We know that the F-residual of T CoreG (M )/ CoreG (M ) is nilpotent and therefore the F-projectors of T CoreG (M )/ CoreG (M ) are conjugate by Theorem 4.2.1. If E is an F-projector of T , then there exists g ∈ T such that D = E g CoreG (M ). Hence E g is an F-projector of T contained in M . Assume now that CoreG (M ) = 1. Since M is F-abnormal in G, the group G is a primitive group of type 1 and G = M N , where N is the minimal normal subgroup of G. Clearly we can assume that G is not an F-group. Then N ≤ GF and, by Proposition 2.2.8 (3), M∩ GF = M F . If M F = 1, then M is an F-group and NM (Σ ∩ M F ) = M . Then M is an F-projector of G and G ∈ NF. In this case G = T and our claim is true. Suppose that M F = 1. We see that in this case T is contained in M . Consider an element am ∈ T , with a = 1, a ∈ N , and m ∈ M . If p is the prime divisor of |N | = |GF : M F | and S p is the Hall p subgroup of GF in Σ, then (S p )am = S p . Moreover, S p ≤ M F ≤ M and then (S p )a ≤ M . If x ∈ S p , then [x, a] ∈ M ∩ N = 1. Consequently a centralises S p and N ≤ Z(GF ) by [DH92, I, 5.5]. Thus GF is contained in CG (N ) which is equal to N by Theorem 1. Hence M F ≤ N ∩ M = 1. This contradiction shows that a = 1 and T is contained in M . 2. Conclusion. Let D be an F-projector of T contained in M . Since T F ≤ GF , we have that G = T GF = DGF . Put R = M ∩ GF ; by hypothesis Σ ∩R is a Hall system of R. Then we have that D ≤ NM (Σ) ≤ NM (Σ ∩ R) = D GF ∩ NM (Σ ∩ R) = D NR (Σ ∩ R). Since system normalisers of soluble groups are nilpotent, it follows that NR (Σ ∩ R) is a nilpotent normal subgroup of NM (Σ ∩ R). Hence NM (Σ ∩ R) ∈ NF and D supplements the Fitting subgroup of NM (Σ ∩ R). By Theorem 4.2.8, D is contained in an F-projector E of NM (Σ ∩ R). Since M F ≤ R and R is soluble, we can apply Lemma 4.2.12 to M , R, and Σ ∩ R and deduce that each F-projector of NM (Σ ∩ R) is contained in an F-projector of NM (Σ ∩ M F ). Therefore E, and then D, is contained in an F-projector of NM (Σ ∩ M F ). Theorem 4.2.14. Let G be a group whose F-residual GF is soluble. Consider a Hall system Σ of GF and denote T = NG (Σ). If D is an F-projector of T , then D covers all F-central chief factors of G and avoids the F-eccentric ones. Proof. By Lemma 4.2.10, it is enough to prove that D covers the F-central minimal normal subgroups of G and avoids the F-eccentric ones. Let N be a F-central minimal normal subgroup of G. Then N ≤ CG (GF ). It implies


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that N is contained in T and G = DGF = D CG (N ). Hence N is a minimal normal subgroup of N D and [N ] ∗ (N D) ∼ = [N ] ∗ G ∈ F. Since F is a saturated formation and N D/N ∈ F, we have that N D ∈ F. Since D is F-maximal in T , we have that N ≤ D. Suppose now that N is F-eccentric in G. Then N ≤ GF and N is abelian. If D does not avoid N , then N ∩ D = 1. By [DH92, I, 5.5], we deduce that N ≤ Z(GF ), and then N is F-central in G, contrary to supposition. Therefore N is avoided by D. Now we can give a characterisation of the F-normalisers of a group G whose F-residual is soluble in terms of the F-projectors of the absolute system normalisers of the Hall systems of GF . Theorem 4.2.15. Let G be a group whose F-residual GF is soluble. For every Hall system Σ of GF , every F-projector of NG (Σ) is an F-normaliser of G. Thus

NorF (G) = E ∈ ProjF NG (Σ) : Σ is a Hall system of GF , and NorF (G) is a conjugacy class of subgroups of G. Proof. We can assume that G is not an F-group. Let Σ be a Hall system of GF and let M be an F-critical subgroup of G such that Σ reduces into M ∩ GF . By Theorem 4.2.13 there exists an F-projector D of NG (Σ) contained in an F-projector D∗ of NM (Σ ∩ M F ). Arguing by induction, D∗ is an F-normaliser of M , and then of G. Applying Theorem 4.2.4, for D∗ , and Theorem 4.2.14, for D, we have that both cover simultaneously all F-central chief factors of G and avoid the F-eccentric ones. Therefore D and D∗ have the same order and D = D∗ . Since NG (Σ) ∈ NF, the F-projectors of NG (Σ) are a conjugacy class of subgroups by Theorem 4.2.1. Therefore, every F-projector of NG (Σ) is an F-normaliser of G. Conversely, if D is an F-normaliser of G and D = G, then D is an Fnormaliser of an F-critical subgroup M of G. By induction, there exists a Hall system Σ ∗ of M F such that D ∈ ProjF NM (Σ ∗ ) . Since, by Proposition 2.2.8 (3), M F is contained in GF , we can find a Hall system Σ of GF which reduces into M ∩ GF and Σ ∩ M F = Σ ∗ by [DH92, I, 4.16]. Applying Theorem 4.2.13, NM (Σ ∗ ) contains an F-projector of NG (Σ ). Since ∗ ProjF NM (Σ ) is a conjugacy class of subgroups of NM (Σ ∗ ), it follows that there exists an F-projector E of NG (Σ g ), for some g ∈ G, contained in D. Thus, D is an F-projector of NG (Σ g ) by Theorem 4.2.4 and Theorem 4.2.14. Consequently,

E ∈ ProjF NG (Σ) : Σ is a Hall system of GF = NorF (G). Corollary 4.2.16. Let G be a group whose F-residual GF is soluble. If H is an F-projector of G complementing GF in G, then H normalises some Sylow p-subgroup of GF , for each prime p dividing the order of GF .

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Proof. By Theorem 4.2.9, H contains an F-normaliser of G. Since in this case both complement GF , then H is an F-normaliser of G. By Theorem 4.2.15, there exists a Hall system Σ of GF such that H ≤ NG (Σ). This means that H normalises every Sylow subgroup of GF in Σ. The following useful splitting theorem is a generalisation of a theorem due to G. Higman on complementation of abelian normal subgroups. The corresponding result for finite soluble groups was obtained by R. W. Carter and T. O. Hawkes (see [CH67] and [DH92, IV, 5.18]). Theorem 4.2.17. Let F be a saturated formation and let G be group whose F-residual GF is abelian. Then GF is complemented in G and two any complements are conjugate in G. The complements are the F-normalisers of G. Proof. First we prove that an F-normaliser of G is a complement of GF . Suppose that this is not true and let G be a minimal counterexample. Put R = GF . Then there exists D ∈ NorF (G) such that D ∩ R = 1. Observe that, since R is abelian and G = RD, the subgroup R ∩ D is normal in G. Assume that there exists an F-eccentric minimal normal subgroup N of G such that N ≤ R. The quotient DN/N is an F-normaliser of G/N and R/N = F (G/N ) . By minimality of G, we have that R ∩ D = N . But then D covers N and N has to be F-central in G by Theorem 4.2.4. This is a contradiction. Hence every minimal normal subgroup of G below R is F-central in G. Then, if N is any minimal normal subgroup of G below R, we have that N ≤ D and, by minimality of G, R ∩ D = N . Consequently, N is the unique minimal normal subgroup of G below R. Let M be an F-critical subgroup of G such that D ∈ NorF (M ). Since M F is contained in R, we have that M F is an abelian normal subgroup of G. If M F = 1, then N is contained in M F and, by minimality of G, we have that M F ∩ D = 1. This is a contradiction. Hence M ∈ F and then M = D. This implies that R/N is chief factor of G complemented by D. Let p be the prime dividing the order of N . Then R is an abelian p-group. Suppose that F is the integrated and full local definition of F. Then F (p) = ∅ and R ≤ GF (p) . Observe that F is contained in Ep F (p) and that GF (p) /R is therefore a p group. Thus R ∈ Sylp (GF (p) ). By the Schur-Zassenhaus Theorem [Hup67, I, 18. 1 and 18.2], there exists a complement Q of R in GF (p) . Observe that R/N is a chief factor avoided by D. Therefore R/N is F-eccentric in G. Consequently / F (p), and GF (p) is not contained in CG (R/N ). Consider the G CG (R/N ) ∈ p -group Q acting on the normal p-subgroup R by conjugation. Then R = [R, Q] × CR (Q) by [DH92, A, 12.5]. Observe that both CR (Q) = CR (QR) = CR (GF (p) ) and [R, Q] = [R, QR] = [R, GF (p) ] are normal subgroups of G. Since N is the unique minimal normal subgroup of G below R, then either CR (Q) = 1 or [R, Q] = 1. Since N is F-central in G, we have that N ≤ CR (GF (p) ) = CR (Q). Consequently, GF (p) = QR ≤ CG (R) ≤ CG (R/N ), contrary to supposition. Therefore each F-normaliser complements GF in G.


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Consider now a subgroup H of G such that G = HGF and H ∩ GF = 1. Since every chief factor of G below GF is F-eccentric, the subgroup H covers all F-central chief factors of a chief series of G through GF . By Theorem 4.1.20, there exists D ∈ NorF (G) such that D ≤ H. Therefore D = H ∈ NorF (G). Finally, by Theorem 4.2.15, NorF (G) is a conjugacy class of subgroups of G. Hence the complements of GF are the F-normalisers of G and they are conjugate. A consequence of Theorem 4.2.17 is the following result due to P. Schmid. Corollary 4.2.18 ([Sch74]). For every group G, we have that GF ∩ ZF (G) ≤ (GF ) ∩ Z(GF ). Proof. Theorem 4.2.17, applied to the group G/(GF ) , leads to ZF (G) ∩ GF ≤ (GF ) . By [DH92, IV, 6.10]), we have that [GF , ZF (G)] = 1. Therefore GF ∩ ZF (G) ≤ (GF ) ∩ Z(GF ). Next, we use Corollary 4.2.18 to give a short proof of a well-known result of L. A. Shemetkov ([She72]). Theorem 4.2.19. Let G be a group such that for some prime p, the Sylow p-subgroups of GF are abelian. Then every chief factor of G below GF whose order is divisible by p is an F-eccentric chief factor of G. Proof. Suppose that the theorem is false and let G be a minimal counterexample. Then GF = 1. Let N be a minimal normal subgroup of G such that N ≤ GF . From minimality of G, every chief factor of G between N and GF whose order is divisible by p is F-eccentric, the prime p divides |N | and N is an F-central chief factor of G. Then N ≤ GF ∩ ZF (G) ≤ (GF ) ∩ Z(GF ) by Corollary 4.2.18. Let P be a Sylow p-subgroup of GF . Since P is abelian, we have that N ≤ (GF ) ∩ Z(GF ) ∩ P = 1 by Taunt’s Theorem (see [Hup67, VI, 14.3]). This contradiction concludes the proof. We round the section off with another interesting splitting theorem. Theorem 4.2.20. Let G be a group such that every chief factor of G below GF is F-eccentric. Assume that GF is p-nilpotent for every prime p in π = π(|G : GF |), Then 1. (P. Schmid, [Sch74]) GF is complemented in G and any two complements are conjugate; 2. (A. Ballester-Bolinches, [BB89a]) the complements of GF in G are the (F ∩ Sπ )-normalisers of G. Proof. First we note that the class L = F ∩ Sπ is a saturated formation and GL = G F . We argue by induction on the order of G. Consider N = Oπ (GF ) and supF pose that N = 1. The quotient group GF /N = (G/N ) is a nilpotent π-group.

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By induction, GF /N is complemented in G/N and any two complements are conjugate. If L/N is a complement of GF /N in G/N , then N is a normal Hall π -subgroup of L. By the Schur-Zassenhaus Theorem [Hup67, I, 18.1 and 18.2], there exists a Hall π-subgroup H of L and two Hall π-subgroups of L are conjugate in L. Observe that H ∩ GF = 1 and then GF is complemented in G. Moreover if A and B are two complements of GF in G, then AN/N and BN/N are conjugate in G/N . Without loss of generality we can assume that AN = BN . Since A and B are Hall π-subgroups of AN and N is a normal Hall π -subgroup of AN , it follows that A and B are conjugate by the Schur-Zassenhaus Theorem. If E is an L-normaliser of G, then EN/N is an L-normaliser of G/N by Proposition 4.1.5. By induction, E ∩ GF ≤ N . Since E is a π-group and N is a π -group, we have that E ∩ GF = 1 and E complements GF in G. Therefore we can assume that N = 1, i.e. GF is a nilpotent π-group, and G is a π-group in NF. Here the L-normalisers and the F-normalisers of G coincide. Since every chief factor of G below GF is F-eccentric in G, if D is an F-normaliser of G, then D ∩ GF = 1, by Corollary 4.2.5, and D is a complement of GF in G. Any complement E of GF is an F-group. By Lemma 4.1.17, E is contained in an F-normaliser. Hence E is an F-normaliser of G. Thus, the complements of GF in G are the F-normalisers of G, and they are conjugate, by Theorem 4.2.15. Postscript K. Doerk (see [DH92, V, 3.18]) used the F-normalisers to show that a saturated formation F has a unique upper bound for all local definitions, that is, a maximal local definition, in the soluble universe. In fact, he proved that the formation function g given by g(p) = G : the F- normalisers of G are in F (p) , for all primes p, is the maximal local definition of F. As we have seen in Chapter 3, the situation in the general finite universe is not so clear cut. However, it is possible to use the F-normalisers of finite, non-necessarily soluble, groups to give necessary and sufficient conditions for a saturated formation F to have a maximal local definition ([BB89a], [BB91]).

4.3 Subgroups of prefrattini type The introduction of systems of maximal subgroups in [BBE91] made possible the extension of prefrattini subgroups to finite, non-necessarily soluble, groups. Later, in [BBE95], we introduced the concept of a weakly solid (or simply w-solid ) set of maximal subgroups following some ideas due to M. J. Tomkinson [Tom75]. Equipped with these new notions, we were able to present

4.3 Subgroups of prefrattini type

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a common generalisation of all prefrattini subgroups of the literature. These new subgroups enjoy most of the properties of the soluble case, for instance they are preserved by epimorphic images and enjoy excellent factorisation properties. Unfortunately, we cannot expect to keep cover-avoidance property and conjugacy. In fact, conjugacy characterises solubility, and conjugacy and cover-avoidance property are equivalent in some sense (see Corollary 4.3.14). In fact we can repeat here the comment said in the introduction of Section 1.4: we lose the arithmetical properties, but we find deep relations between maximal subgroups which are general to all finite groups. We present here a distillation of the preceding concepts. Observe, for instance, that the definition of system of maximal subgroups given in [BBE91] is different, but equivalent, to the one in Section 1.4. In fact this presentation allows us to speak of a particular subgroup of prefrattini type, which is defined by the intersection of all maximal subgroups in a subsystem of maximal subgroups. This point of view is new since all precedents of prefrattini subgroups in the past were families of subgroups of the group. To recover this classical idea of a set of prefrattini subgroups, we include the concept of w-solid set as a union-set of subsystems of maximal subgroups. Definitions 4.3.1. Let X be a (possibly empty) set of monolithic maximal subgroups of a group G. 1. We will say that X is a weakly solid (w-solid) set of maximal subgroups of G if for any U , S ∈ X such that CoreG (U ) = CoreG (S) and both complement the same abelian chief factor H/K of G, then M = (U ∩ S)H ∈ X. (4.5) 2. X is said to be solid if it satisfies (4.5) and whenever a chief factor is X-supplemented in G, then all its monolithic supplements are in X. Next we give a varied selection of examples of w-solid and solid sets. Examples 4.3.2. 1. The set Max∗ (G), of all monolithic maximal subgroups of a group G, is solid. 2. Consider a subgroup L of a group G; the set XL of all monolithic maximal subgroups of G containing L is w-solid. 3. Given a w-solid (respectively solid) set X of maximal subgroups of a group G and a class H of groups, then the set XaH of all H-abnormal subgroups in X and the set XnH of all H-normal subgroups in X are w-solid (respectively solid) as well. If X is a system of maximal subgroups, then XaH and XnH are subsystems of maximal subgroups. Let M be a monolithic maximal subgroup of G. Recall that the normal index of M in G, defined by W. E. Deskins in [Des59] and denoted by η(G, M ), is indeed η(G, M ) = Soc G/ CoreG (M ) . 3. The following families of monolithic maximal subgroups of a group G are w-solid:

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a) Fixed a prime p, the set Xp of all monolithic maximal subgroups M of G such that |G : M | is a p-power. In fact, if G is p-soluble, then Xp is indeed solid. However this is not true in the non-soluble case; in G = Alt(5) the set X5 is composed of all maximal subgroups isomorphic to Alt(4) and clearly it is not solid. b) Fixed a set of primes π, the set Xπ of all monolithic maximal subgroups M of G such that |G : M | is a π -number. c) the set of all monolithic maximal subgroups of G of composite index in G. d) the set of all monolithic maximal subgroups M of the group G such that η(G, M ) = |G : M |. If G is a group, the set S(G) composed of all systems of maximal subgroups of G is non-empty by Theorem 1.4.7. If X is a w-solid set of maximal subgroups of G and Y ∈ S(G), then X ∩ Y is a subsystem of maximal subgroups of G. Applying Theorem 1.4.7, we have that X = {X ∩ Y : Y ∈ S(G)}. Definitions 4.3.3. 1. Let G be a group. Let X be a non-empty subsystem of maximal subgroups of G. Define W(G, X) = {M : M ∈ X}. For convenience, we define W(G, ∅) = G. We will say that W is a subgroup of prefrattini type of G if W = W(G, X) for some subsystem X of maximal subgroups of G. 2. If X be a w-solid set of maximal subgroups of G, we say that Pref X (G) = {W(G, X ∩ Y) : Y ∈ S(G), X ∩ Y = ∅} is the set of all X-prefrattini subgroups of G. We show in the following that the known prefrattini subgroups are associated with w-solid sets of maximal subgroups. Examples 4.3.4. 1. The Max∗ (G)-prefrattini subgroups are simply called prefrattini subgroups of G. We write Pref(G) = {W(G, X) : X ∈ S(G)}. In other words, a prefrattini subgroup of a group G is a subgroup of the form W(G, X), where X is a system of maximal subgroups of G. If G is a soluble group, we can apply Corollary 1.4.18 and conclude that the prefrattini subgroups of G are those introduced by W. Gasch¨ utz in [Gas62] which originated this theory. 2. Let H be a Schunck class. The Max∗ (G)aH -prefrattini subgroups of a group G are the H-prefrattini subgroups defined in [BBE91]. If G is soluble, they are the H-prefrattini subgroups studied by P. F¨ orster in [F¨ or83] and, if H is a saturated formation, the Max∗ (G)aH -prefrattini subgroups of G are the ones introduced by T. O. Hawkes in [Haw67].


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3. If G is a soluble group, then Pref XL (G) is the set of all L-prefrattini subgroups introduced by H. Kurzweil in [Kur89]. 4. The Xp -prefrattini subgroups of a p-soluble group are the p-prefrattini subgroups studied by A. Brandis in [Bra88]. Notation 4.3.5. If H is a Schunck class, G is a group, and X is a system of maximal subgroups of G, we denote W(G, H, X) = W(G, XaH ), and say that W(G, H, X) is the H-prefrattini subgroup of G associated with X. We write Pref H (G) = {W(G, H, X) : X ∈ S(G)} for the set of all H-prefrattini subgroups of G. Theorem 4.3.6. Consider a group G, X a subsystem of maximal subgroups of G and W = W(G, X). Then W = {T(G, X, F ) : F is an X-supplemented chief factor of G}. Moreover W has the following properties. 1. Let 1 = G0 < G1 < · · · < Gn = G be a chief series of G; write I = {i : 1 ≤ i ≤ n such that Gi /Gi−1 is X-supplemented}; then, if I is nonempty, {Si : Si is an X-supplement of Gi /Gi−1 }. W = i∈I

2. If N is a normal subgroup of G, then W N/N = W(G/N, X/N ). Proof. Applying Proposition 1.3.11, we can deduce that W = {T(G, X, F ) : F is an X-supplemented chief factor of G}. Now Assertion 1 follows from Theorem 1.2.36 and Theorem 1.3.8. In proving Assertion 2, suppose first that N is a minimal normal subgroup of G and let 1 = G0 < G1 = N < · · · < Gn = G be a chief series of G. Clearly we can assume that X is non-empty. Then I = {i : 1 ≤ i ≤ n such that Gi /Gi−1 is X-supplemented} is non-empty and W = i∈I {Si : Si is an X-supplement of G/Gi−1 } by Statement 1. If N is an X-Frattini, then N is contained in Si for all i ∈ I and then W/N = W(G/N, X/N ). Otherwise, N is contained in Si for all i ∈ I \ {1} and G = N S1 . The case I = {1} leads to W = S1 and X/N = ∅. Then G = W N and W N/N = W(G/N, X/N ). Suppose that I \ {1} = ∅. Then W N = i∈I\{1} Si and then W N/N = W(G/N, X/N ). Therefore Assertion 2 holds when N is a minimal normal subgroup of G. A familiar inductive argument proves the validity of Statement 2 for any normal subgroup N of G.

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Remark 4.3.7. Theorem 4.3.6 does not hold when X is simply a JH-solid set (see Example 1.3.10). This is the reason why we introduce the prefrattini subgroups associated with subsystems of maximal subgroups and not with JH-solid sets of maximal subgroups. All classical examples of prefrattini subgroups in the soluble universe, including Kurzweil’s, enjoy the conjugacy and the cover-avoidance property. Now we prove that, roughly speaking, it can be said that conjugacy and cover-avoidance property of soluble chief factors are equivalent properties for subgroups of prefrattini type. In fact, conjugacy of prefrattini subgroups characterises solubility. The consideration of primitive non-soluble groups, whose core-free maximal subgroups are neither conjugate nor CAP-subgroups, causes that in the general non-soluble universe these properties fail. Proposition 4.3.8. Let G be a group and X a subsystem of maximal subgroups of G. Put W = W(G, X). Let H/K be a chief factor of G. 1. If H/K is X-Frattini, then W(G, X) covers H/K. 2. If H/K possesses X-complement in G, then W(G, X) avoids H/K. Proof. Assume that H/K is an X-Frattini chief factor of G. Then H/K ≤ M K/K for all M ∈ X. Hence, H/K ≤ {M K/K : M ∈ X} = W(G/K, X/K) = W K/K, by Proposition 4.3.6, and W(G, X) covers H/K. If a maximal subgroup M of G belongs to X, then W ≤ M . Hence, if M complements H/K, W avoids H/K. Corollary 4.3.9. Let G be a group, X a solid set of maximal subgroups of G and H/K an abelian chief factor of G. Then 1. H/K is either covered or avoided by all W ∈ Pref X (G); 2. H/K is covered by some W ∈ Pref X (G) if and only if H/K is an XFrattini chief factor of G. The above result justifies the following definition. Definition 4.3.10. Let G be a group and X a w-solid set of maximal subgroups of G. We say that Pref X (G) satisfies ACAP if whenever F is an abelian chief factor of G, 1. then F is either covered or avoided by all W ∈ Pref X (G), and 2. F is covered by some W ∈ Pref X (G) if and only if F is an X-Frattini chief factor of G. Clearly if Pref X (G) satisfies ACAP, any W ∈ Pref X (G) covers all abelian X-Frattini chief factors of G and avoids all abelian X-complemented. By the above corollary, if X is a solid set of maximal subgroups of a group G, then Pref X (G) satisfies ACAP. We give some more examples.


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Examples 4.3.11. 1. By Lemma 1.5 of [Kur89], if L is a subgroup of a soluble group G, the set Pref XL (G) of all L-prefrattini subgroups of G satisfies ACAP (note that XL is w-solid, but not solid in general). 2. Let G be the group as in Example 1.3.10. We consider the set X = {a, z, b, z, ab, z, a2 b, z}. Then X is a subsystem of maximal subgroups of G and W(G, X) = z. We consider the system Y of maximal subgroups defined by the Hall system Σ = {N, abz} (see Theorem 1.4.17). Then W(G, X ∩ Y) = ab, z. It is clear that the X-prefrattini subgroups of G do not satisfy ACAP. Proposition 4.3.12. Let G be a group, and let X be a w-solid set of maximal subgroups of G. Assume that Pref X (G) satisfies ACAP. Let X1 , X2 be two systems of maximal subgroups of G and H/K an abelian chief factor of G. Then, there exists an X-complement of H/K in X1 if and only if there exists an X-complement of H/K in X2 . Proof. Put {i, j} = {1, 2}. Suppose that Mi is an X-complement of H/K in Xi but for all maximal subgroups S ∈ X ∩ Xj such that K ≤ S, we have H ≤ S. Denote by Wk the (X ∩ Xk )-prefrattini subgroup of G, k = 1, 2. Applying Theorem 4.3.6, Wi ≤ Mi . Then K = Wi K ∩ H. Since Pref X (G) satisfies ACAP, we have K = Wj K ∩ H. However Wj K/K is the X/Kprefrattini subgroup of G/K associated with Xj /K by Theorem 4.3.6 (2). Then Wj K/K = {S/K : S ∈ X ∩ Xj , K ≤ S}. Our assumption implies H/K ≤ Wj K/K. This contradiction proves that H/K has an X-complement in Xj . Theorem 4.3.13. Let X be a w-solid set of maximal subgroups of group G. For Y = XnS , the set of all S-normal maximal subgroups in X, the following statements are equivalent: 1. Pref Y (G) satisfies ACAP; 2. Pref Y (G) is a set of conjugate subgroups of subgroups of G. Proof. 1 implies 2. Assume that Assertion 2 does not hold and choose for G a counterexample of least order. If H is any non-trivial normal subgroup of G, then X/H is w-solid set of maximal subgroups of G/H and (X/H)nS = Y/H. It is clear that Pref Y/H (G/H) satisfies ACAP. Hence the minimal choice of G implies that Pref Y/H (G/H) is a set of conjugate subgroups of G/H. Let N be a minimal normal subgroup of G. If N is Y-Frattini, then N is covered by every Y-prefrattini subgroup of G by Theorem 4.3.6. In that case, the Since the theorem holds in G/N , Pref Y (G) is a conjugacy class of subgroups of G, contrary to supposition. Hence N is Y-supplemented in G. In particular N is S-central in G and therefore N is abelian. Let M ∈ Y such that G = M N and M ∩ N = 1, and let S be a system of maximal subgroups of G such that M ∈ S (Theorem 1.4.7). Denote by A the (Y ∩ S)-prefrattini subgroup of G. Then A ≤ M by Theorem 4.3.6. Since by hypothesis Pref Y (G)

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is not a set of conjugate subgroups of G, there exists a system S0 of maximal subgroups of G such that A0 = W(G, Y ∩ S0) and A are not conjugate in G. Let ϕ be the isomorphism between G/N and M . We have (X/N )nS = Y/N and (Y/N )ϕ = (X ∩ M )nS = Y ∩ M by Lemma 1.2.23. Denote C = CoreG (M ) = CM (N ). Suppose that C = 1. Since the theorem holds in G/C, there exists x ∈ G such that Ax0 C = AC ≤ M . Without loss of generality we can assume that x = 1. In particular A0 ≤ M . Then AN ∩ M = A and A0 N ∩ M = A0 are (X ∩ M)nS -prefrattini subgroups of M. The minimal choice of G implies that A and A0 are conjugate in M . This contradiction leads to C = 1. Since M is S-normal in G, we have G is a primitive soluble group. By Corollary 1.4.18, there exists g ∈ G such that Sg0 = S. If U ∈ Y ∩ S, then U complements the chief factor Soc G/ CoreG (U ) . By Proposition 4.3.12, there exists V ∈ Y ∩ S0 such that V complements Soc G/ CoreG (U ) . Since G/ CoreG (U ) is a soluble primitive group, CoreG (U ) = CoreG (V ) and U and g V are conjugate in G by Theorem 1.1.10. This implies that Y ∩ S = Y ∩ S0 = g Theorem 4.3.6, A = {U : U ∈ Y ∩ S} = {U : U ∈ (Y ∩ S0 ) . Applying (Y ∩ S0 )g } = {V g : V ∈ Y ∩ S0 } = Ag0 . This contradiction proves the implication. 2 implies 1. Note that all non-abelian chief factors of G are Y-Frattini. This means that Y-prefrattini subgroups are conjugate CAP-subgroups indeed. Corollary 4.3.14. Let X be a w-solid set of maximal subgroups of a soluble group G. The following statements are equivalent: 1. Pref X (G) is a set of conjugate subgroups of G, and 2. every W ∈ Pref X (G) is a CAP-subgroup of G which covers all X-Frattini chief factors of G and avoids the X-complemented ones. In general the prefrattini subgroups of a group are not conjugate: in any non-abelian simple group the prefrattini subgroups are the maximal subgroups. We prove next that the solubility of a group is characterised by the conjugacy of its prefrattini subgroups. Theorem 4.3.15. A group G is soluble if and only if the set Pref (G) of all prefrattini subgroups is a conjugacy class of subgroups of G. Proof. If G is a soluble group, then the conjugation of the prefrattini subgroups of G follows directly from Theorem 4.3.6 and Corollary 1.4.18. Conversely, assume that G is a group such that the set Pref (G) of all prefrattini subgroups of G is a conjugacy class of subgroups of G. We prove that G is soluble by induction on the order of G. By Theorem 4.3.6 (2), we have that, for every normal subgroup N of G, the set Pref (G/N ) of all prefrattini subgroups of G/N is a conjugacy class of subgroups of G/N . Therefore G/N is soluble for each minimal normal subgroup N of G and G is a monolithic primitive group. Suppose that G is not soluble. Then N = Soc(G) is not abelian. Let W/N be a prefrattini subgroup of G/N associated with


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an arbitrary system of maximal subgroups X∗ of G/N . Let P1 be a nontrivial Sylow p1 -subgroup of N , for some prime p1 ; there exists a maximal subgroup M of G such that NG (P1 ) ≤ M . Clearly CoreG (M ) = 1. The set X1 = {H ≤ G : N ≤ H, H/N ∈ X∗ } ∪ {M } is a system of maximal subgroups of G. Applying Theorem 4.3.6, W ∩ M is the prefrattini subgroup of G associated with X1 . Let P2 be a non-trivial Sylow p2 -subgroup of N , for a prime p2 such that p1 = p2 . This is always possible since N is non-abelian. Consider now a maximal subgroup S of G such that NG (P2 ) ≤ S and the system of maximal subgroups X2 = {H ≤ G : N ≤ H, H/N ∈ X∗ } ∪ {S} of G. As above, we have that W ∩ S = W(G, X2 ). Consequently W ∩ M and W ∩ S are conjugate in G. This implies that W ∩ M contains a Sylow p2 -subgroup of N . Since p2 is arbitrary, we have that W ∩ M contains a Sylow p-subgroup of N for any prime p dividing the order of N . This implies that N ≤ M , which is a contradiction. Hence G is soluble. Finally in this section, we touch on the question of the description of the core and the normal closure of subgroups of prefrattini type. For solid sets X of maximal subgroups, the core of the X-prefrattini subgroups is the X-Frattini subgroup defined in Definition 1.2.18 (1). Proposition 4.3.16. If X is a solid set of maximal subgroups of a group G and W is an X-prefrattini subgroup of a group G, then CoreG (W ) = ΦX (G). Proof. Let Y be a system of maximal subgroups of G. Consider W = W(G, X ∩ Y). Since X is solid, we have that ΦX (G) = {CoreG (M ) : M ∈ X ∩ Y} = CoreG (W ). The classical Frattini subgroup of a group G, Φ(G), is clearly the Max∗ (G)Frattini subgroup of G. The Max∗ (G)aN -Frattini subgroup is denoted by L(G) in [Bec64]. H. Bechtell also denotes the Max(G)nN -Frattini subgroup by R(G). Following his notation, if H is a Schunck class and G is a group, we denote LH (G) = {M : M is H-abnormal monolithic maximal subgroup of G} the Max∗ (G)aH -Frattini subgroup of G, and similarly RH (G) = {M : M is H-normal monolithic maximal subgroup of G} the Max∗ (G)nH -Frattini subgroup of G. Theorem 4.3.17. Let F be a saturated formation and let X be a system of maximal subgroups of a group G, then CoreG W(G, XaF ) = ZF G mod Φ(G) = LF (G). Proof. Denote W = W(G, XaF ). Applying Theorem 4.3.6, LF (G) ≤ W . Hence LF (G) is contained in CoreG (W ). Conversely, if S is an F-abnormal monolithic maximal subgroup of G in X, then we have W ≤ S and CoreG (W ) ≤

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CoreG (S). Then CoreG (W ) is contained in every F-abnormal monolithic maximal subgroup of G. Hence Core G (W ) ≤ LF (G). To prove that ZF G/Φ(G) = LF (G)/Φ(G) suppose first that Φ(G) = 1. Since every chief factor of G below ZF (G) is F-central in G, it follows that ZF (G) ≤ LF (G). To prove the converse observe that if Φ(G) = 1, then LF (G) ∩ GF = 1. Assume not and let N be a minimal normal subgroup of G such that N ≤ LF (G) ∩ GF . Since Φ(G) = 1, it follows that N is supplemented in G by a monolithic F-normal maximal subgroup M . Hence GF ≤ M . This contradiction leads to LF (G) ∩ GF = 1. Consider a chief factor H/K of G such that H ≤ LF (G). Since GF ∩ LF (G) = 1, then HGF /KGF is a chief factor of G which is G-isomorphic to H/K. This means that H/K is F-central in G. Therefore LF (G) ≤ ZF (G) and equality holds. If Φ(G) = 1, then consider the quotient group G∗ = G/Φ(G). Since Φ(G∗ ) = 1, we obtain the required equality. Proposition 4.3.18. Let G be a group. If F is a saturated formation and X is a system of maximal subgroups of G, then CoreG W(G, XnF ) = RF (G) = Φ(G mod GF ). Proof. First notice that GF is contained in every F-normal maximal subgroup of G and if G ∈ F, then every maximal subgroup of G is F-normal. Therefore, RF (G)/GF = RF (G/GF ) = Φ(G/GF ). Since GF is contained in W(G, XnF ), we have W(G, XnF )/GF = W(G/GF , XnF /GF ) by Theorem 4.3.6 (2) and so CoreG W(G, XnF ) /GF = Φ(G/GF ). Definition 4.3.19. Let G be a group and suppose that X is a solid set of maximal subgroups of G. A normal subgroup N of G is said to be 1. an X-profrattini normal subgroup of G if either N = 1 or every chief factor of G of the form N/K is an X-Frattini chief factor of G, and 2. an X-parafrattini normal subgroup of G if either N = 1 or every chief factor of G of the form N/K is a non-X-complemented chief factor of G, that is, no maximal subgroup in X is a complement of N/K in G. For X = Max∗ (G), the solid set of all monolithic maximal subgroups of G, we say simply profrattini and parafrattini. Examples and remarks 4.3.20. 1. If N is an X-profrattini normal subgroup of G, then N is an X-parafrattini normal subgroup of G. The converse does not hold in general. It is enough to consider a non-abelian simple group S. It is clear that S is X-parafrattini for all solid sets X of maximal subgroups of S. However S is not X-profrattini. If N is soluble, N is X-profrattini if and only if N is X-parafrattini. 2. If F is a totally nonsaturated formation (see [BBE91]), then GF is a profrattini normal subgroup of G for every group G. 3. If X is a solid set of maximal subgroups of a group G, a quasinilpotent normal subgroup N of G is X-profrattini if and only if N ≤ ΦX (G).


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Proof. Assume that N is a quasinilpotent X-profrattini normal subgroup of G but N ≤ ΦX (G). Then there exists a maximal subgroup U of G such that K ≤ U , U ∈X and G = U N . We have that G/ CoreG (U ) = N CoreG (U )/ CoreG (U ) U/ CoreG (U ) and N CoreG (U )/ CoreG (U ) is quasinilpotent. Therefore N CoreG (U )/ CoreG (U ) = F∗ G/ CoreG (U ) = Soc G/ CoreG (U ) . But this contradicts N being X-profrattini. Hence N ≤ ΦX (G). The converse holds trivially. Theorem 4.3.21. Let G be a group and suppose that X is a solid set of maximal subgroups of G. 1. If N , M are both X-profrattini normal subgroups of G, then N M is an X-profrattini normal subgroup of G. 2. If N , M are both X-parafrattini normal subgroups of G, then N M is an X-parafrattini normal subgroup of G. Proof. Let (N M )/K be a chief factor of G. The normal subgroups KM and KN lie between K and N M . If K = KN = KM , then N M ≤ K, which is imimpossible. Hence, either NM = NK or NM = M K. Suppose that N M = N K (the other case is analogous). By Lemma 1.2.16, if S supplements (respectively, complements) N M/K = N K/K, then S also supplements (respectively, complements) the chief factor N/(N ∩ K). If N is a X-profrattini (respectively, X-parafrattini) normal subgroup of G, then S ∈ / X. Hence M N is also X-profrattini (respectively, X-parafrattini) normal subgroup of G. Remark 4.3.22. Let G be a group and X be a solid set of maximal subgroups of G. Suppose that N is a normal subgroup of G satisfying the property that either N = 1 or every chief factor N/K of G is X-complemented in G. If M is a normal subgroup of G with the same property, then M N does not have this property in general. For instance, consider G = A × B where A = a : a4 = 1, B = b : b2 = 1, and X = Max∗ (G). Then B and D = a2 b are two complemented minimal normal subgroups of G. However BD/B is a Frattini chief factor of G. Definitions 4.3.23. Let G be a group and X be a solid set of maximal subgroups of G. 1. The X-profrattini subgroup of G is the normal subgroup ProX (G) = N : N is an X-profrattini normal subgroup of G. 2. The X-parafrattini subgroup of G is the normal subgroup ParaX (G) = N : N is an X-parafrattini normal subgroup of G.

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For X = Max∗ (G), the solid set of all monolithic maximal subgroups of G, we write simply Pro(G) and Para(G). It is clear that ProX (G) ≤ ParaX (G). If X is a solid set of maximal subgroups of G composed of maximal subgroups of type 1, then ProX (G) = ParaX (G). In particular, the equality holds when G is soluble. There are nonsoluble groups such that ProX (G) = ParaX (G). Consider a prime p and a cyclic group Z of order p2 . Let G = S Z be the regular wreath product of S with Z, where S is a non-abelian simple group. Then Pro(G) = Para(G) is the unique maximal normal subgroup of G. It is clear that for each normal subgroup ParaX (G) < N (respectively, ProX (G) < N ) there is at least one G-chief factor N/K which is X-supplemented (respectively, X-complemented) in G. We can say much more than this. Proposition 4.3.24. Let G be a primitive group of type 2 which splits over Soc(G) = N by a maximal subgroup S of G. Then Soc(S) is non-abelian. Proof. Let A be an abelian minimal normal subgroup of S. Then A is an elementary abelian p-group for some prime p. Since S ≤ NG (A), then NG (A) = S since proper containment leads to a contradiction that A is normal in G, by maximality of S in G. Hence N ∩ CG (A) = 1. If p divides |N |, a contradiction arises since A would be contained in a Sylow p-subgroup P = [T ]A of N A with T = P ∩ N. Hence, T ∩ Z(P ) = 1 and there exists an element x ∈ CN (A) such that x = 1. This is not possible. Consequently p does not divide |N |. Let q be a prime dividing |N |. By [Gor80, 6.2.2], there exists a unique A-invariant Sylow q-subgroup Q of N . For any element s ∈ S, Qs is also A-invariant. Consequently, Q = Qs and S ≤ NG (Q). Since N ∩ S = 1, Q is not contained in S and so G = QS = N S. This implies N = Q, a contradiction. Corollary 4.3.25. Denote by K the class of all groups G such that every chief factor of G is complemented in G by a maximal subgroup of G. Then K is composed of soluble groups. Proof. Suppose that K is not contained in S and consider a group of minimal order G ∈ K \ S. Then G ∈ b(S) and G is a primitive group of type 2. By hypothesis, N = Soc(G) is a non-abelian minimal normal subgroup which is complemented in G by a core-free maximal soluble subgroup S of G. But Soc(S) abelian contradicts Proposition 4.3.24. Proposition 4.3.26. Let G be a group and let X be a solid set of maximal subgroups of G. 1. Denote by N the set of all normal subgroups N of G satisfying the property that every chief factor of G between N and G is X-supplemented in G. If N , M ∈ N , then N ∩ M ∈ N .


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2. Denote by K the set of all normal subgroups N of G satisfying the property that every chief factor of G between N and G is X-complemented in G. If N , M ∈ K, then N ∩ M ∈ K. Proof. Consider a chief series of G from M to M ∩ N . N ∩ M ≤ · · · ≤ M.

(4.6)

1. Consider a chief factor H/K of G in (4.6). Then HN/KN is a chief factor of G between N and G. Since N ∈ N , it follows that HN/KN is Xsupplemented in G by S ∈ X, say. This means that G = S(HN ) and KN ≤ S ∩ N H. Hence G = SH and K ≤ S ∩ H. Hence H/K is X-supplemented in G by S. Therefore Assertion 1 follows from Theorem 1.2.36. 2. Note that by Corollary 4.3.25, the groups G/N and G/M are soluble. Then G/(N ∩ M ) is soluble. Therefore all chief factors in (4.6) are abelian. The Assertion 2 now follows by applying the same arguments as those used in the proof of Statement 1 replacing “supplemented” by “complemented.” Corollary 4.3.27. Let G be a group and X a solid set of maximal subgroups of G. Then 1. ProX (G) = {N : N ∈ N } ∈ N and every chief factor of G between G is X-supplemented in G; ProX (G) and 2. ParaX (G) = {N : N ∈ K} ∈ K and every chief factor of G between ParaX (G) and G is X-complemented in G. Proof. 1. Denote K = {N : N ∈ N }. By Proposition 4.3.26, K ∈ N . If K/L is an X-supplemented chief factor of G, then L ∈ N by Theorem 1.2.34 and this is not possible. Therefore every chief factor of G of the form K/L is XFrattini. Hence K ≤ ProX (G). Assume that K < ProX (G). Let ProX (G)/N be a chief factor of G such that K ≤ N . Then ProX (G)/N should be XFrattini. This contradicts Proposition 4.3.26. The proof for 2 is analogous. Corollary 4.3.28. If X is a solid set of maximal subgroups of a group G, then G/ ParaX (G) is a soluble group. Proof. Note that G/ ParaX (G) ∈ K. Apply now Corollary 4.3.25.

S

It is clear from the above result that G , the soluble residual of G, is contained in ParaX (G). Corollary 4.3.29. Let X be a solid set of maximal subgroups of a group G. Then ParaX (G) = ProX (G)GS . Proof. It is clear that ProX (G)GS ≤ ParaX (G). Suppose there exists a chief factor F = ParaX (G)/N of G with ProX (G)GS ≤ N . By definition of ParaX (G), the chief factor F is non-X-complemented in G. On the other hand, F is abelian and X-supplemented in G because ProX (G)GS ≤ N . Such F cannot exist. Hence ParaX (G) = ProX (G)GS .

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Theorem 4.3.30. Let G be a group and let X be a solid set of maximal subgroups of G. Then N is an X-parafrattini normal subgroup of G if and only if N = N ∩ W g : g ∈ G for each W ∈ Pref X (G). Proof. Suppose that N = N ∩ W g : g ∈ G for each W ∈ Pref X (G). Let N/K be a chief factor of G. Assume that N/K is X-complemented in G. Then there exists a maximal subgroup M ∈ X of G such that G = M N and N ∩ M = K. If W is an X-prefrattini subgroup of G such that W ≤ M , it follows that W ∩ N ≤ M∩ N = K. Hence N = N∩ W g : g ∈ G ≤ K,contrary to supposition. Therefore N/K is non-X-complemented in G. Hence N is Xparafrattini. Conversely, assume that N is an X-parafrattini normal subgroup of G. We may suppose that N = 1. Let W ∈ Pref X (G) and L = N ∩ W g : g ∈ G. Suppose L < N . Let N/H be a chief factor of G such that L ≤ H. Since N is X-parafrattini, we have that N/H is non-X-complemented in G. Note that W ∩ N ≤ L ≤ H. Hence W avoids N/H. This implies that N/H is X-supplemented. Let S be the system of maximal subgroups of G such that W = W(G, X ∩ S) and M be an X-supplement of N/H in G such that M ∈ S. Consider a chief series of G passing through H and N . Let S1 , . . . , Sr be the X-supplements of thechief factors of G aboveN such that Si ∈ S r r (1 ≤ i ≤ r). Then W N/N = i=1 Si /N r and W H/H = i=1 (Si /H) ∩ (M/H) by Theorem 4.3.6. Therefore W H = i=1 (Si ∩ M) = W N ∩ M = W (M ∩ N ). Since W ∩ N ∩ M = W ∩ N = W ∩ H, it follows that |H| = |M ∩ N | and so H = M ∩ N. Hence M is an X-complement of N/H in G. This contradicts our assumption. Consequently N = N ∩ W g : g ∈ G for each W ∈ Pref X (G). The following result describes the normal closure of an X-prefrattini subgroup. Corollary 4.3.31. Let X be a solid set of maximal subgroups of group G. If W ∈ Pref X (G), we have that W G = W g : g ∈ G = ParaX (G). Proof. Write P = ParaX (G). Each abelian chief factor of G which is Xcomplemented in G is avoided by every X-prefrattini subgroup of G by Corollary 4.3.9. Since every chief factor H/K such that P ≤ K < H ≤ G is abelian and X-complemented in G, it follows that W ≤ ParaX (G) for all W ∈ Pref X (G). From Theorem 4.3.30, W G = P . In [Haw67] an elegant theorem of factorisation of prefrattini subgroups of soluble groups is proved. There T. O. Hawkes makes a strong use of the coveravoidance property. Here we present a similar factorisation in the general nonsoluble universe but, obviously, with no use of the cover-avoidance property. Theorem 4.3.32. Let G be a group and let H be a Schunck class of the form H = EΦ F, for some formation F. Consider a system of maximal subgroups X of G. Then, if Y is a w-solid set of maximal subgroups of G, we have


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a W G, (X ∩ Y)H = D W(G, X ∩ Y), where D is an H-normaliser of G associated with X. Proof. We argue by induction on the order of G. Obviously we can suppose a that Φ(G) = 1. Denote W ∗ = W G, (X ∩ Y)H and W = W G, X ∩ Y . By Theorem 4.3.6, W ∗ is contained in W . By Lemma 4.1.11, we know that D is contained in every H-abnormal maximal subgroup of G in X. Hence D, W ≤ W ∗ . If G ∈ H, then G = D and / H. (X ∩ Y)aH = ∅. Thus, W ∗ = G = D. Therefore we may assume that G ∈ Consider an H-critical maximal subgroup M of G in X such that D is an Hnormaliser of M associated with a system of maximal subgroups X(M ) such that XM ⊆ X(M ). Then M supplements a minimal normal subgroup N of G. If G is a simple group, then every maximal subgroup of G is H-abnormal and then W ∗ = W and the theorem is true in this case. Hence we can assume that N is a proper subgroup of G and N ∩ M = M . If N is an (X ∩ Y)-Frattini minimal normal subgroup, then N ≤ W ≤ W ∗ and the assertion follows by induction. Hence we may suppose that M ∈ X ∩ Y. Moreover, arguing as in Lemma 1.2.23, we have that YM /(M ∩ N ) = {(S ∩ M )/(M ∩ N ) : N ≤ S ∈ Y} is a w-solid set of maximal subgroups of M/(M ∩ N ). Thus YM = {S ∩ M : N ≤ S ∈ Y} is a w-solid set of maximal subgroups of M . By induction, (4.7) W M, (X(M ) ∩ YM )aH = D W M, X(M ) ∩ YM . Consider a chief series Γ of G through N . Then, by Theorem 4.3.6 (1), we have that W = M ∩ Si1 ∩ · · · ∩ Sir , where the Sij are (X ∩ Y)-supplements of chief factors in Γ over N . Observe that Γ ∩ M gives a piece of chief series of M over N ∩ M . Moreover, again by Theorem 4.3.6 (1), r (M ∩ Sij )/(N ∩ M ) W M, X(M ) ∩ YM (N ∩ M )/(N ∩ M ) = j=1

and then

W = W M, X(M ) ∩ YM (N ∩ M ).

Similarly,

W ∗ = W M, (X(M ) ∩ YM )aH (N ∩ M ).

Hence, by taking the product with N ∩ M in both sides of the equality (4.7) we obtain the required factorisation. Motivated by [Tom75, Theorem 5.3], we present the following factorisation involving H-normal maximal subgroups. Theorem 4.3.33. Let G be a group and let F be a saturated formation. Consider a system of maximal subgroups X of G. Then if Y is a w-solid set of maximal subgroups of G we have that W G, (X ∩ Y)nF = W(G, X ∩ Y)GF .

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Proof. Since GF is contained in M , for every F-normal maximal subgroup M of G, it follows that GF ≤ W G, (X ∩ Y)nF . Since G/GF ∈ F, it is clear that (X ∩Y)nF /GF = (X∩Y)/GF . Therefore W G, (X ∩ Y)nF /GF = W G/GF , (X ∩ Y)nF /GF = W(G, X ∩ Y)GF /GF by Theorem 4.3.6 (2). Corollary 4.3.34. Let G be a group and let F be a saturated formation. Consider a system of maximal subgroups X of G. Then if Y is a w-solid set of maximal subgroups of G we have that G = W G, (X ∩ Y)nF W G, (X ∩ Y)aF . Proof. Just notice that if D is an F-normaliser of G, then G = DGF . Now apply the factorisations presented in Theorem 4.3.32 and Theorem 4.3.33. The theory of prefrattini subgroups was continued by X. Soler-Escriv` a in her Ph. Doctoral Thesis at the Universidad P´ ublica de Navarra, [SE02]. Her work is another example of the progress produced by using non-arithmetical properties, even in soluble groups. In its place all relations between maximal subgroups of a group and maximal subgroups of its critical subgroups are used thoroughly (see [ESE05]). This leads to the existence and properties of some distributive lattices, generated by three types of pairwise permutable subgroups, namely hypercentrally embedded subgroups (see [CM98]), F-normalisers, and subgroups of prefrattini type (see [ESE]).

5 Subgroups of soluble type

Consider a subgroup H of a soluble group G. Since every minimal normal subgroup of G is abelian, the following implication holds: If G = M H, M is a minimal normal subgroup of G, and H is a proper subgroup of G, then H ∩ M = 1.

(5.1)

It is precisely this property which makes the theory of H-projectors and Hcovering subgroups, H a Schunck class, so much easier in the soluble universe than in the general finite one. Salomon, in his Doctoral Dissertation [Sal87], has introduced and studied notions of H-projectors and H-covering subgroups (different from the usual ones) which lead to a theory of these subgroups in arbitrary groups resembling the theory of H-projectors and H-covering subgroups in finite soluble groups. His first basic idea is to give a definition of H-projectors along the following lines: Recall that if H is a class of groups, a subgroup H of a group G is called an H-projector of G if H ∈ S(G) ∩ H and if N G and HN/N ≤ K/N ∈ S(G/N ) ∩ H, then HN/N = K/N . Here S(X) is the set of all subgroups of a group X. Salomon tries to replace the set S(G) of all subgroups of a group G by suitable subsets d(G) of S(G) which are such that any element H of d(G) enjoys the above property (5.1); he also tries to develop a theory of the “projectors” so obtained by following the classical approach. It is clear that in order to carry out this, one cannot take just any d(G) ⊆ S(G) satisfying (5.1). First of all, since the definition of H-projector involves not only the group G itself but also its quotients, such sets d(G) have to be chosen not only for G but at least for all quotients of G; in fact, the classical theory of projectors suggests that it would be reasonable to demand that such a choice be made for all finite groups simultaneously. Put differently, to begin with, one chooses a “subgroup functor” d which associates with any group G a set of subgroups d(G) (we refer to elements of d(G) as subgroups 2 05

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of soluble type) subject, of course, to the condition that (5.1) holds for any G and each H in d(G). Moreover, it is also plausible that a subgroup functor d ought to satisfy certain formal properties such as the following: If H ∈ d(G) and N G, then HN/N ∈ d(G/N ). Not surprisingly, it turns out that the properties relevant here are closely related to properties of ordinary projectors and covering subgroups. The theory developed in this Chapter is largely the work of Salomon [Sal87] and F¨ orster [Förb], [F¨ ora].

5.1 Subgroup functors and subgroups of soluble type: elementary properties The purpose of this section is to establish the necessary formal properties of the various functors for subgroups of soluble type. The functors t and t introduced by Salomon in [Sal87] are studied here. Two similar functors r and r defined by F¨ orster [Förb] are also studied. A subgroup functor is a function f which assigns group to each G a possibly empty set f(G) of subgroups of G satisfying θ f(G) = f θ(G) for any isomorphism θ : G −→ G∗ . Examples 5.1.1. 1. Functor S: it assigns to each group G the set S(G) of all subgroups of G. 2. Functor Sn : it associates with each group G the set Sn (G) of all subnormal subgroups of G. 3. Let p be a prime. Let Sp be the function assigning to each group G the set Sp (G) of all subgroups U of G containing a Sylow p-subgroup of G; Sp is a subgroup functor. 4. Let H be a Schunck class, then ProjH () and CovH () are subgroup functors. Given two subgroup functors e and f, we write e ≤ f if e(G) ⊆ f(G) for each group G. Definition 5.1.2. Let f be a subgroup functor. We say that f is inherited if f enjoys the following properties: 1. If U ∈ f(T ) and T ∈ f(G), U ≤ T ≤ G, then U ∈ f(G). 2. If U ∈ f(G) and N G, then U N/N ∈ f(G/N ). 3. If U/N ∈ f(G/N ), N G, then U ∈ f(G). If, moreover, f satisfies 4. U ≤ G and T ∈ f(G) implies U ∩ T ∈ f(U ), we say that f is w-inherited. Obviously the functors w-inherited.

S

and

Sn

are w-inherited.

Sp

is inherited but not

5.1 Subgroup functors and subgroups of soluble type

207

Lemma 5.1.3. Let f be an inherited functor. Then 1. If U ∈ f(G) and N G, then U N ∈ f(G). 2. If N is a subnormal subgroup of G and N = N1 N2 · · · Nk = G is a chain from N to G such that 1 ∈ f(Ni ) for all i ∈ {2, . . . , k}, then N ∈ f(G). In particular,

Sn (G)

⊆ f(G) for all groups G if 1 ∈ f(X) for all groups X.

Proof. 1. follows from Definition 5.1.2 (2 and 3). 2. By 1, Ni ∈ f(Ni+1 ) for all i ∈ {1, . . . , k − 1}. Applying Definition 5.1.2 (1), it follows that N ∈ f(G). Lemma 5.1.4. Let f be a w-inherited subgroup functor. If G is a group, U ∈ f(G) and N is a subnormal subgroup of G such that U ≤ NG (N ), then U N ∈ f(G). In particular, if 1 ∈ f(G), then Sn (G) ⊆ f(G). Proof. We argue by induction on |G|. Let N1 be the normal closure of N in G and, for i > 1, denote Ni = N Ni−1 , the normal closure of N in Ni−1 . Since N is subnormal in G, there exists n ≥ 1 such that N = Nn . Suppose that G = U N1 , then N2 = N1 because U normalises N . Repeating the argument with every Nk , it follows that N is normal in G. By Lemma 5.1.3 (1), we have U N ∈ f(G). Therefore we may assume that U N1 is a proper subgroup of G. By induction, U N ∈ f(U N1 ). Now U N1 ∈ f(G) yields U N ∈ f(G) by Definition 5.1.2 (1). Lemma 5.1.5. If f is a w-inherited subgroup functor and X1 , X2 ∈ f(G), then X1 ∩ X2 ∈ f(G). Proof. Since f is w-inherited, X1 ∩ X2 ∈ f(X1 ). Hence X1 ∩ X2 ∈ f(G) because X1 ∈ f(G) and f is inherited. Definition 5.1.6. Let f be a w-inherited subgroup functor and {Xi : i ∈ I} ⊆ f(G). Then and con the intersection of all subgroups of G belonging to f(G) taining i∈I Xi is the smallest subgroup of G in f(G) containing i∈I Xi . This subgroup is denoted by Xi : i ∈ If and called the f-join of {Xi : i ∈ I}. Theorem 5.1.7. Let f be a w-inherited subgroup functor. For each group G, f(G) is a lattice under the operations “∩” and “·, ·f .”

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Definitions 5.1.8. For any group G, we define the following subgroup functors: t(G) = {H ≤ G : T S ≤ G, S/T strictly semisimple, then (H ∩ S)T S} r(G) = {H ≤ G : T S ≤ G, S/T strictly semisimple, H ≤ NG (S), then (H ∩ S)T S} t (G) = {H ≤ G : T S sn G, S/T then (H ∩ S)T S} r (G) = {H ≤ G : T S sn G, S/T then (H ∩ S)T S} t (G) = {H ≤ G : T S sn G, S/T r (G) = {H ≤ G : T S sn G, S/T then (H ∩ S)T S}

strictly semisimple, strictly semisimple, H ≤ NG (S), simple, then (H ∩ S)T S} simple, H ≤ NG (S),

here strictly semisimple groups are those which can be written as direct products of isomorphic simple groups, while semisimple groups are direct products of not necessarily isomorphic simple groups. If H is a subgroup of G such that H ∈ e(G) for some e ∈ {t, r, t , r , t , r }, we shall say that H is a subgroup of soluble type. The functors t and t have been introduced by Salomon in his Dissertation [Sal87]. There are certain problems Salomon encounters with these two choices of e: the first of these functors does not really give sets t(G) “large enough” whenever G is “highly non-soluble,” while for t one of the crucial properties, namely, if H ∈ e(G) and N G, then H ∈ e(HN ), is missing. Later, Förster [F¨ orb] overcame these problems by introducing the remaining functors. As will be seen in the next section “the r-functors” enjoy the advantage of producing relevant subgroups of primitive groups. Remarks 5.1.9. 1. If G is soluble, then t(G) = r(G) = t (G) = r (G) = t (G) = r (G) = S(G). 2. Condition “(H ∩ S)T S” in the definitions of t and r implies that H either covers or avoids the simple section S/T , that is, (H ∩ S)T ∈ {S, T }. 3. In defining t, r, t , r (t , r , respectively) we might have taken the strictly semisimple (simple) groups S/T as direct products of non-abelian simple groups (as non-abelian simple, respectively). Moreover, since every subnormal subgroup of a direct product of non-abelian simple groups is in fact normal by [DH92, A, 4.13] the condition “(H ∩ S)T S” can be replaced by “(H ∩ S)T sn S.” 4. If H is a subgroup of G such that H ∈ e(G) for some e ∈ {t, r, t , r , t , r }, and N is a normal subgroup of G, then HN/N ∈ e(G/N ). 5. Sn ≤ t ≤ t ≤ t , r ≤ r ≤ r , t ≤ r, t ≤ r , t ≤ r . 6. r(G) (r (G), respectively) is the set of all subgroups H of G with the following property: If T S ≤ G (T S sn G, respectively), S/T is strictly semisimple, and H ≤ NG (S) ∩ NG (T ), then (H ∩ S)T S.


209

Proof. Only a proof for Statement 6 is needed. Let S/T denote a strictly semisimple section of G, with simple component E, say, and let H ≤ NG (S). Then T ∗ , the D0 (1, E)-residual of S, is a characteristic subgroup of S contained in T with strictly semisimple quotient S/T ∗ contained in D0 (1, E); in particular, H ≤ NG (T ∗ ). Now Statement 6 follows directly from this observation. Proposition 5.1.10. Let H and N be subgroups of a group G such that N is quasinilpotent and H normalises N . Suppose that the following condition holds: B A sn N , A/B strictly semisimple, and H ≤ NG (A) implies that (H ∩ A)B A. (5.2) Then H ∩ N is subnormal in N . Proof. Proceeding by induction on |G| + |N |, we may clearly assume that G = HN , and hence that N is normal in G. In the case N = 1 or N is a minimal normal subgroup of G, the claim is immediate from condition (5.2); so without loss of generality, there is a normal subgroup M of G such that 1 = M < N . Taking into account that the class of all quasinilpotent groups is an Sn -closed homomorph and that condition (5.2) is inherited by M and N/M , we may apply the inductive hypothesis twice to get that H ∩ M is a subnormal subgroup of M and HM ∩ N is a subnormal subgroup of N ; in particular, HM ∩ N is quasinilpotent. Therefore, if HM ∩ N is a proper subgroup of N , then another application of the inductive hypothesis yields that H ∩ N = H ∩ (HM ∩ N ) is subnormal in HM ∩ N , and hence that H ∩ N is subnormal in N . Thus, without loss of generality, N = HM ∩ N ≤ HM and G = HN = HM . Therefore we have: H < G = HM whenever M G and 1 = M < N .

(5.3)

Assume that the Fitting subgroup of N , F(N ), is non-trivial. Then N contains an abelian minimal normal subgroup M of G; since N = (H ∩ N )M , we can deduce that H ∩ N is normal in N because N centralises M . This is due to the fact that M is a direct product of minimal normal subgroups of M which are central in N because N is quasinilpotent. It remains to deal with the case when F(N ) = 1. Then N is a direct product of non-abelian simple groups by Proposition 2.2.22 (3). By Condition (5.2), H ∩ M is normal in M for every minimal normal subgroup of G contained in N . Therefore we obtain: N is a direct product of non-abelian simple groups, and H ∩ M = 1, whenever M is a minimal normal subgroup of G contained in N . (5.4) Now, N cannot have two non-isomorphic simple direct factors E and F : otherwise for each X in {E, F } we could find minimal normal subgroups MX of G contained in N with X as composition factor, so by Condition (5.3) and Condition (5.4) both ME and MF should be normal complements of H in G,

210


∼ MF . Consequently, leading to the contradiction that ME ∼ = (ME ×MF ) ∩ H = N is strictly semisimple, and our claim holds by assumption. Proposition 5.1.11. The functors t and t are inherited. The functor t is w-inherited. Proof. We give a proof for t. 1. Let G be a group and let X and U be subgroups of G such that X ∈ t(U ) and U ∈ t(G). Consider a strictly semisimple section S/T of G. We prove that (S ∩ X)T is normal in G. Clearly we may assume that S/T is a direct product of non-abelian simple groups. Since U ∈ t(G), it follows that (S ∩ U )T is normal in S. Hence (S ∩ U )/(T ∩ U) is either 1 or a strictly semisimple section of U . If S ∩ U = T ∩ U, then S ∩ X = T ∩ X and (S ∩ X)T = T . Thus we may assume that S ∩ U = T ∩ U . Since X ∈ t(U ), we have that (S ∩ X)(T ∩ U ) is normal in S ∩ U . This implies that (S ∩ X)T is normal in (S ∩ U )T S. Since S/T is a direct product of non-abelian simple groups, we have (S ∩ X)T is normal in S by [DH92, A, 4.13]. Consequently X ∈ t(G). 2. Let U ∈ t(G) and N normal in G. Suppose we have T /N S/N ≤ G/N and (S/N ) (T /N ) strictly semisimple. Then N ≤ T S ≤ G and S/T is strictly semisimple, so U ∈ t(G) gives that (U N ∩ S)T = (U ∩ S)N T = (U ∩ S)T S. Therefore U N/N ∈ t(G/N ). 3. Consider a normal subgroup N of G. Let U/N ∈ t(G/N ). Suppose that S/T is a non-abelian and strictly semisimple section of G. If T N = SN , then S = (S ∩ N )T = (S ∩ U )T and (S ∩ U )T is normal in S. Hence ∼ S/ (S ∩ N )T ) we may assume that T N = SN . In particular, SN/T N = and (SN/N ) (T N/N ) is a strictly semisimple section of G/N . Since U/N ∈ t(G/N ), it follows that (SN/N ) ∩ (U/N ) (T N/N ) is normal in SN/N and so (SN ∩ U )T is normal in SN . This implies that (SN ∩ U )T ∩ S = (S ∩ U )T is a normal subgroup of S. Therefore U ∈ t(G). The same statements can be obtained for the functors t and t just by adding the assumption S subnormal in G for t , and S subnormal in G and S/T simple for t in the above proof. Finally, it is clear that t is w-inherited because every strictly semisimple section of every subgroup of a group is actually a strictly semisimple section of the group itself. Example 5.1.12. Let E be a non-abelian simple group and let K = E E be the regular wreath product. Then K can be written as semidirect product K = F E, where F is the base group and E is its canonical complement in K. Consider now G = E E K, the wreath product of E with K with respect to the transitive permutation representation of K on the right cosets of E. Let B be the base group of G. Then B = E1 × · · · × En , n = |K : E|, where Ei ∼ = E (i = 1, . . . , n), E1 is E-invariant and E1 E = E1 × E. It is rather easy to see that F (B, respectively) is the unique minimal normal subgroup of K (G, respectively). Consequently, G has a unique maximal normal subgroup, namely BF .


211

Let H be the diagonal subgroup of E1 E. We claim that H ∈ t (G). Let T S sn G with S/T non-abelian and strictly semisimple. If S = G, S is contained in the unique maximal normal subgroup BF of G. Then (by construction of H and BF ) H ∩ BF = 1, whence H ∩ S = 1 and (H ∩ S)T = T S. If S = G, then T ∈ {G, BF } and (S ∩ H)T = HT = HBF = G = S. Assume, by way of contradiction, that H ∈ r (HB). Then HN/N ∈ r (HB/N ) whenever N HB. But B = E1 × C, where C = E2 × · · · × En and HB/C = (E×E1 )C/C ∼ = E×E1 via the canonical isomorphism. This maps the diagonal subgroup H of E × E1 onto the non-normal subgroup HC/C of EB/C, which contradicts HC/C ∈ r (EB/C) (see Proposition 5.1.10). / t (HB). This shows that t Therefore H ∈ / r (HB) and, in particular, H ∈ is not w-inherited. We do not know, however, whether r is w-inherited. Remark 5.1.13. Since t is w-inherited, then, by Theorem 5.1.7, t(G) is closed under the operations “∩” and “·, ·t .” In general, U, V t = U, V : let n ≥ 5 and let G = Σn be the symmetric group of degree n. Let X1 = (1, 2) and X2 = (2, 3). Then Xi ∈ t(G), i = 1, 2, X1 , X2 ∼ = Σ3 , and X1 , X2 t = G, as we can see by looking at the sections Alt(n)/1 of Sym(n). Proposition 5.1.14. If H is a proper subgroup of a group G = HM , with H ∈ r (G), and M is a minimal normal subgroup of G, then H ∩ M = 1. Proof. By Proposition 5.1.10, H ∩ M is normal in M . Since H is a proper subgroup of G and M is a minimal normal subgroup of G, we have that H ∩ M = 1. Remark 5.1.15. The statement of Proposition 5.1.14 does not hold for elements of t (G). Consider the regular wreath product G = E T of a non-abelian simple group E and a group T of order 2. Denote by D the T -invariant diagonal subgroup of the base group B, and put H = T D. Then G = HB, B is a minimal normal subgroup of G, H ∈ t (G), but H ∩ B = 1. Since the functors t and r do not have the crucial property described in Proposition 5.1.14, they are not so interesting for us as the functors t, r, t , and r . Nevertheless, we shall see at the end of the section (Theorem 5.1.25) that the composition factors of the subgroups in r (G) are composition factors of the whole group. Definition 5.1.16. A subgroup functor f is called inductive (respectively, weakly inductive) if it satisfies Conditions 1 and 2 (respectively, 1 and 2 ): 1. G ∈ f(G) 2. If H ≤ K ≤ G, N G, H ∈ f(K), N ≤ K, and K/N ∈ f(G/N ), then H ∈ f(G).

212


2 . H ≤ G, N G, H ∈ f(HN ), and HN/N ∈ f(G/N ) implies that H ∈ f(G). It is clear that w-inherited functors f such that f(X) is non-empty for all groups X are inductive and inductive functors are also weakly inductive. Proposition 5.1.17. t and t are inductive functors. Proof. We see that t is inductive. The same proof applies to t . First of all, the defining Condition 1 for inductivity of t holds trivially. To verify Condition 2, let H ≤ K ≤ G and N G such that N ≤ K, H ∈ t (K) and K/N ∈ t (G/N ). We show that (S ∩ H)T is normal in G whenever S/T is a non-abelian simple section of G such that S is subnormal in G. Suppose that SN = T N . Then S = (S ∩ N )T = (S ∩ K)T and (S ∩ K)/(T ∩ K) is a simple section of K such that S ∩ K is subnormal in K. Since H ∈ t (K), it follows that (S ∩ H)(T ∩ K) is normal in S ∩ K. Hence (S ∩ H)T is normal in (S ∩ K)T = S. Suppose that SN = T N . Then (SN/N ) (T N/N ) is a simple section of G/N and SN/N is subnormal in G/N . Since K/N ∈ t (G/N), we have that (K ∩ SN )T is normal in SN . Hence (K ∩ S)T = S ∩ (K ∩ SN ) T = S ∩ (K ∩ SN )T is normal in S ∩ SN = S. Further, if K ∩ T = K ∩ S, then (S ∩ K)/(T ∩ K) is a simple section of K and S ∩ K is subnormal in K. As H ∈ t (K), this gives that H ∩ (K ∩ S) (K ∩ T ) is normal in K ∩ S. This is also true if K ∩ T = K ∩ S. Consequently we obtain: (H ∩ S)T = (K ∩ H ∩ S)(K ∩ T )T (K ∩ S)T S. This proves the inductivity of t .

Note that, in the special case when K = HN , the same argument still works for the functors r, r and r , for H ≤ NG (S) and N G imply that HN ≤ NG (SN ); hence we get: Proposition 5.1.18. r, r , and r are weakly inductive. Moreover, if H ∈ r(G) and L is a subgroup of G containing H, then H ∈ r(L). Lemma 5.1.19. Let f ∈ {t , r } and let G be a group. 1. If H is a subgroup of G and N is a soluble subgroup of G normalised by H, then H ∈ f(HN ). 2. Let H ≤ K ≤ G = HN , where N is a direct product of non-abelian simple groups. If H ∈ f(G) and K ∩ N is normal in N , then H ∈ f(K). Proof. 1. For a proof of H ∈ f(H N ), we may use induction on |G| and the properties of f verified earlier in this section (Propositions 5.1.17 and 5.1.18) to see that without loss of generality 1 = G = HN , N is a minimal normal subgroup of G, H ∩ N = 1, and H is a core-free maximal subgroup of G. Hence G is a primitive group of type 1 and N is the unique minimal normal subgroup of G. Let T S be a non-abelian strictly semisimple section of G


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such that S is subnormal in G. Assume that S is a proper subgroup of G. Then S is contained in some maximal normal subgroup M of G. Since M = 1, it follows that N is contained in M and M = N (H ∩ M ). Since M is a proper subgroup of G, H ∩ M ∈ f(M ) by induction. Hence (S ∩ H)T = (S ∩ H ∩ M )T is normal in S. Therefore we may assume that S = G. Then T = 1 because otherwise G would be non-abelian and simple. Consequently N is contained in T and so S = (S ∩ H)T . This means that H ∈ f(G). 2. By hypothesis K ∩ N is normal in N , so K ∩ N is a normal subgroup of KN = G. Since N is a direct product of non-abelian simple groups, it follows that N = (K ∩ N ) × M for a normal subgroup M of G. Then G/M = KM/M ∼ = K. It is clear therefore that H ∈ f(K) because HM/M ∈ f(G/M ). Theorem 5.1.20. Let f ∈ {t , r } and let G = H F∗ (G) be a group which is a product of a subgroup H ∈ f(G) and F∗ (G), the generalised Fitting subgroup of G. If M is a normal subgroup of G and M is quasinilpotent, then H ∈ f(HM ). Proof. We argue by induction on |G|. If M is soluble, the result follows from Lemma 5.1.19 (1). We may suppose that M is not contained in F = GS , the soluble radical of G. If F = 1, then F∗ (G) and M are both direct products of non-abelian simple groups by Proposition 2.2.22 (3). Moreover HM ∩ F∗ (G) is normal in F∗ (G) because H ∩ F∗ (G) is normal in F∗ (G) by Proposition 5.1.10 and [DH92, A, 4.13]. Applying Lemma 5.1.19 (2), we conclude that H ∈ f(HM ). Therefore we can suppose that F = 1. By induction HF/F ∈ f (HF/F )(M F/F ) . This yields H(HM ∩ F )/(HM ∩ F ) ∈ f HM/(HM ∩ F ) . Since H ∈ f H(HM ∩ F ) by Lemma 5.1.19 (1), we conclude that H ∈ f(HM ) by weakly inheritness. Lemma 5.1.21. Let G = E1 × · · · × En be a direct product of n copies of a non-abelian simple group E. Let H be a subgroup of G such that H covers or avoids every simple section of G. Then H is a normal subgroup of G. Proof. We argue by induction on |G|. It is clear we can assume CoreG (H) = 1. Since either Ej ∩ H = 1 or Ej ≤ H for all j ∈ {1, . . . , n}, it follows that Ej ∩ H = 1 because H does not contain any normal subgroup of G. Denote Nj = Xi=j Ei , 1 ≤ j ≤ n. Suppose that G = HNj for each j ∈ {1, . . . , n}. In particular, H is a subdirect subgroup of G. Hence H = R1 × · · · × Rt , where Rk ∼ = Rl ∼ = E, 1 ≤ k, l ≤ t. Let g = 1 be an element of R1 . Without loss of generality, we may assume that π1 (g) = 1, where π1 : G −→ E1 is the projection of G over its first component. Then 1 = π1 (R1 ) is a normal subgroup of π1 (H) = E1 , and so π1 (R1 ) = E1 . Let 1 = h ∈ E1 such that g h = g. Since g ∈ R1 ∩ R1h , it follows that R1h is contained in H. Assume [h, R1 ] = 1. Then 1 = π1 ([h, R1 ]) = [h, E1 ] and h ∈ Z(E1 ). This contradiction shows that [h, R1 ] = 1. Let t be an element of R1 such that th = t. Then 1 = th t−1 ∈ H ∩ E1 = 1 (note that if t = (e1 , e2 , . . . , en ), th = (eh1 , e2 , . . . , en )).

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This contradiction proves that HNj is a proper subgroup of G for some j ∈ {1, . . . , n}. Then G/Nj is a simple section of G which is not covered by H. Hence H is contained in Nj . By induction H is normal in Nj and so H is subnormal in G. This implies that H is a normal subgroup of G [DH92, A, 4.13] and the result follows. Proposition 5.1.22. Let H be a subgroup of G. Then H ∈ t(G) if, and only if, every simple section of G is covered or avoided by H. Proof. The cover-avoidance property dealt with here is obviously a special case of the defining property of H ∈ t(G). Conversely, let H be a subgroup of G enjoying this cover-avoidance property, and let T S ≤ G and S/T strictly semisimple and non-abelian. Let X = (H ∩ S)T /T . Then X is a subgroup which covers or avoids every simple section of S/T . By Lemma 5.1.21, X is normal in S/T . Consequently H ∈ t(G). Remark 5.1.23. r is not characterised by the corresponding property: consider the direct product of two copies of a non-abelian simple group E and a diag/ r(G) by Proposition onal subgroup H of G: as H is not normal in G, H ∈ 5.1.10, yet H covers or avoids any (non-abelian) simple section S/T of G with H ≤ NG (S). In soluble groups, each composition factor of a subgroup is a composition factor of the whole group. It is not true in general. However, this property holds for subgroups of soluble type as the next result shows. We need first a lemma. Lemma 5.1.24. Let G be a group and H ∈ r (G). Assume that M is a n subnormal subgroup of G such that M = Xi=1 Ei , where the Ei (i = 1, . . ., n) are pairwise isomorphic non-abelian simple groups; further assume that H denote the projection of M normalises M . Let πi : M −→ Ei (i = 1, . . ., n)

over Ei . Put I = i ∈ {1, . . . , n} : (H ∩ M )πi = 1 . Then H ∩ M is subdirect in Xi∈I Ei . Proof. Note that M/ Ker(πi ) is a simple section of G. Since H ∈ r (G), it follows that H covers or avoids M/ Ker(π i ). If M ∩ H = M ∩ Ker(πi ), we (M ∩ H) = 1. Therefore I = have π i

i ∈ {1, . . . , n} : πi (M ∩ H) = 1 = i ∈ {1, . . . , n} : M = (M ∩ H) Ker(πi ) . It follows that H ∩ M is a subdirect subgroup of Xi∈I Ei . Theorem 5.1.25. Let G be a group. Every composition factor of a subgroup H ∈ r (G) is isomorphic to a composition factor of G. Proof. Proceeding by induction on |G|, we consider a minimal normal subgroup of G. Since HM/M ∈ r (G/M ), we may assume that our claim holds for H/(H ∩ M ) ( ∼ = HM/M ). If M is abelian, the result follows. Hence we may assume that M is non-abelian. By Lemma 5.1.24, H ∩ M is a subdirect subgroup of a suitable normal subgroup of M . Then H ∩ M ∈ D0 (1, E), where E is the composition factor of M .

5.2 Existence criteria

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5.2 Existence criteria The main goal in this section is to prove some existence results for various subgroups of soluble type. We begin with a result on t-subgroups which may be viewed as a non-existence result: as a consequence of this theorem, in a monolithic primitive group with non-abelian socle the minimal normal subgroup cannot be complemented by a t-subgroup unless the corresponding quotient of the group is soluble. In fact, the results on this type of subgroups show that the non-soluble t-subgroups share some properties with non-soluble subnormal subgroups. This is already apparent from the first theorem of this section, which may be thought as a partial generalisation of the following theorem of Wielandt ([Wie39], see Theorem 2.2.19): If H and K are subnormal subgroups of a group G and H is a perfect comonolithic group, then either H is contained in K or K normalises H. We supplement this result by a theorem which leads to a criterion for the existence of t-complements of the minimal normal subgroup in primitive groups of type 1; it turns out that in order that such a complement exists, the corresponding quotient of the group has to be p-soluble, p the prime dividing the order of the minimal normal subgroup. The remaining part of this section deals with results on the functors r, t , and r . Theorem 5.2.1. Let H = H ∈ t(G), and consider a non-abelian, strictly semisimple normal subgroup M of G. Then [H, M ] ≤ H ∩ M HM. Proof. Let (H, M, G) represent a counterexample such that |G| + |H| is minimal. Then, since t is w-inherited, it follows that: 1. G = HM and H is a proper subgroup of G. Now, since H ∈ t(G) and M is strictly semisimple, H ∩ M HM = G. Since G is a counterexample, [H, M ] is not contained in H ∩ M . 2. H ∩ M = 1, M is a minimal normal subgroup of G and CoreG (H) = 1 = CG (M ). Suppose that H ∩ M = 1. Then the triple H/(H ∩ M ), M/(H ∩ M), G/(H ∩ M ) satisfies the hypotheses of the theorem. By minimality of G, [H, M ] is contained in H ∩ M , contrary to our supposition. Hence H ∩ M = 1. Note that M = M1 × · · · × Mt , where Mi are minimal normal subgroups of G for all i ∈ {1, . . . , s}. Suppose that M is not a minimal normal subgroup of G. Then HMj is a proper subgroup of G and so [H, Mj ] ≤ H ∩ Mj = 1 for all j ∈ {1, . . . , s}. Therefore [H, M ] = 1. This contradiction proves that M is a minimal normal subgroup of G. If CoreG (H) = 1, then the minimality of G yields [H, M ] ≤ H ∩M = 1, and we have a contradiction. Thus CoreG (H) = 1 and CH (M ) = 1 because CH (M ) is a normal subgroup of HM = G. Now put

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C = CG (M ) and D = H ∩ CM. Then CM = CM ∩ HM = DM . Suppose that D = 1. Then C is contained in M and so C = 1 because M is non-abelian. Assume now that D = 1. By Lemma 5.1.3, HC ∈ t(G). Hence HC ∩ M is normal in (HC)M = G. Since M is a minimal normal subgroup of G, either HC ∩ M = 1 or HC ∩ M = M . If HC ∩ M = 1, then HC = H. This implies C = 1. Suppose that HC ∩ M = M . Then G = HC = HM , DC = (H ∩ CM )C = HC ∩ CM = CM , and C∼ =D∼ = DC/C = CM/C ∼ = M, = CM/M = DM/M ∼ whence CM ∼ = M ×M is a strictly semisimple subgroup of G. Since H ∈ t(G), D = H ∩ CM is a normal subgroup of CM . Hence D is normal in (CM )H = G. This yields a contradiction against CoreG (H) = 1. Therefore CG (M ) = 1. 3. Let HS denote the soluble radical of H. Then H/HS is a non-abelian simple group. Moreover there is no proper t-subgroup K of G with H = KHS . Let A/HS be a minimal normal subgroup of H/HS . It is clear that A is not soluble. Hence there exists an integer n such that (A(n) ) = A(n) = 1. Moreover A(n) ∈ t(A) and so A ∈ t(G). By the inheritness of t, it follows A(n) ∈ t(G). If A were a proper subgroup of H, then A(n) < H. The minimal choice of (G, H) would yield A(n) ≤ CG (M ) = 1, contrary to our supposition. Therefore A = H and H/HS is a non-abelian simple group. Suppose there exists a subgroup K ∈ t(G) such that H = KHS . Then K is not soluble and so there exists an integer b such that 1 = K (b) is a perfect subgroup of K. Moreover K (b) ∈ t(G). Since |G|+|K (b) | < |G|+|H|, it follows K (b) is contained in CG (M ) = 1. This is a contradiction. Applying [Hup67, VI, 4.7], there exists a Sylow 2-subgroup H2 of H and a Sylow 2-subgroup M2 of M such that G2 = M2 H2 is a Sylow 2-subgroup of G. By the Odd Order Theorem ([FT63]), H2 = 1 and M2 = 1. Moreover M2 is a normal subgroup of G2 and HS is a proper subgroup of H2 HS . 4. [M2 , H] = 1. By way of contradiction, assume that T = CM2 (H) is a proper subgroup of M2 . Then T is a proper subgroup of N = NM2 (T ). It is clear that N/T is a normal subgroup of H2 N/T because H2 normalises N and T . Hence Z(H2 N/T ) ∩ (N/T ) = 1. Let g ∈ N such that 1 = gT ∈ Z(H 2 N/T ) ∩(N/T ). Clearly HS T /T < (H2 T /T )(HS T /T ) ≤ (H g T ∩ HT )/T HS T /T . Thus HS is a proper subgroup of HS (H ∩ H g T ). We prove that H ∩ H g T ∈ t(G). Let C = N M (T ). Then T is normal in C and C is H-invariant. Since t is w-inherited, it follows that HT /T ∈ t(HN/T ) and Hg T/N ∈ t(HN/T). Hence (HT /T )∩(H g T /T ) ∈ t(HN/T ) because t has the intersection property. Then (HT ∩ H g T )/T ∈ t(HT /T ) and hence HT ∩ H g T ∈ t(HT ). This implies that H ∩ H g T ∈ t(H). By the inheritness of t, H ∩ H g T ∈ t(G). Now H ∩ H g T does not avoid the simple section H/HS . Hence H = HS (H ∩ H g T ) by Proposition 5.1.22. Applying Step 3, H ∩ H g T = H. Therefore g ∈ NG (HT ). Moreover [g, H, H] = [H, g, H] ≤ [HT, g, H] ≤ [HT ∩ C, H] = [T, H] = 1 by Step 1. By the Three Subgroups Lemma ([Hup67, III, 1.10]), we infer that


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[H, g] = [H , g] = [H, H, g] = 1. Thus g ∈ CN (H) ≤ CM2 (H) = T . This contradiction against the choice of g establishes Step 4. 5. Final contradiction. Since CG (M ) = 1, it follows that G is, up to isomorphism, a subgroup of Aut(M ). Now O2 (M ) = 1. Hence a theorem of G. Glauberman [Gla66] applies. It says that CAut(M ) (M2 ) is 2-nilpotent. In particular, CAut(M ) (M2 ) is soluble by the Odd Order Theorem [FT63]. By Step 4, H is a subgroup of CG (M2 ) ≤ CAut(M ) (M2 ). This is a contradiction because H is perfect and non-trivial. Before proving a result similar to Theorem 5.2.1 for abelian normal subgroups M , we have to recall some terminology and results. Let G be a group. If H ∗ is a normal subgroup of a proper subgroup H of G and H g ∩ H is a subgroup of H ∗ for all g ∈ G \ H, then G is called a Frobenius-Wielandt group with respect to H and H ∗ . H. Wielandt [Wie58] (cf. [Hup67, V, 7.5]) has shown that the Frobenius-Wielandt kernel, defined by G∗ = G\ g∈G (H \H ∗ )g , is a normal subgroup of G = HG∗ and H ∩ G∗ = H ∗. If H ∗ = 1, then G is called simply a Frobenius group. Let Ω = {xH : x ∈ G} and let H act on Ω by left multiplication. We have that |Ω| = |G : H| and if x ∈ G \ H, then the orbit of xH has size divisible by |H : H ∗ |. Consequently gcd(|H/H ∗ |, |G : H|) = 1. Lemma 5.2.2. Suppose that H = H ∈ t(G), and let M be an abelian normal subgroup of G. Further, assume that H is comonolithic, with maximal normal subgroup L = Cosoc(H), and that there is no a proper subgroup K of H such that K ∈ t(H) and H = KL. Then one of the following holds: 1. [H, M ] is contained in H ∩ M . 2. H ∩ M is a subgroup of CH (M/N ), with N defined by N/(H ∩ M ) = CM/(H∩M ) (H). Moreover CH (M/N ) is a proper subgroup of H and HM/N is a Frobenius-Wielandt group with respect to HN/N and LN/N , with Frobenius-Wielandt kernel LM/N . Proof. Assuming that [H, M ] is not contained in H ∩ M , we will verify Statement 2. As in the proof of Theorem 5.2.1, an induction argument yields that without loss of generality G = HM and N = H ∩ M = 1. Consider g ∈ G such that H g ∩ H is not contained in L; put g = hm, where h ∈ H and m ∈ M . Then H = L(H ∩ H g ). Moreover H ∩ H g ∈ t(H). The hypothesis of the Lemma yields H ∩ H g = H. Hence H = H m ∩ H = CH (m) by [DH92, A, 16.3]; that is, m ∈ CM (H) = N = 1. Thus g ∈ H. We have shown that H g ∩ H is a subgroup of L for all g ∈ G \ H, and G is a Frobenius-Wielandt group with respect to H and L. Note that, if G∗ is the Frobenius-Wielandt kernel, then M L is contained in G∗ . Therefore G∗ = M (G∗ ∩ H) = M L and Statement 2 follows. Lemma 5.2.3. Let G = HM be a group, where M is an abelian normal p-subgroup of G for some prime p. Assume that H is a subgroup of G such that

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H ∩ M = 1. If H is p-soluble and Hp ∈ t(Hp M ) for every Hall p -subgroup Hp of H, then H ∈ t(G). Proof. We use induction on |G|. Let A be a minimal normal subgroup of G contained in M . Clearly HA/A satisfies the hypothesis of the Lemma in G/A. By induction, HA/A ∈ t(G/A). If HA is a proper subgroup of G, it follows that H ∈ t(HA) by induction. Inductivity of t implies that H ∈ t(G). Hence we may assume that M is a minimal normal subgroup of G and CoreG (H) = 1. That is to say that G is a primitive group of type 1. Let T S ≤ G where S/T is a non-abelian simple group. In order to show that (S ∩ H)T ∈ {T, S}, it will suffice to deal with the case when G = SM . This can be seen by means of a reduction argument since SM = (H ∩ SM )M , H ∩ SM is p-soluble and every Hall p -subgroup B of H ∩ SM belongs to t(BM ). Suppose that S = G. Then either T = 1 or M is contained in T . In both cases (H ∩ S)T ∈ {S, T }. Hence we may assume that S is a proper subgroup of G. Applying Remark 1.1.11 (1), it follows that S = H m for some m ∈ M . On the other hand, S/T is a p -group because G is p-soluble. This implies that T contains every Sylow p-subgroup of S. Let K be a Hall p -subgroup of H such that K m contains a Hall p -subgroup of H ∩ S, (H ∩ S)p say. Since S = K m T , it follows that K m /(K m ∩ T ) is a simple section of K m M = KM . Since K ∈ t(KM ) either (K ∩ K m )(K m ∩ T ) = K m or K ∩ K m ≤ K m ∩ T . Assume the latter holds. Then (S ∩ H)T = (S ∩ H)p T ≤ (K ∩ K m )T = T . Suppose that (K ∩ K m)(K m ∩ T ) = K m . Then S = S m T = (K ∩ K m)T ≤ (H ∩ S)T ≤ S and so S = (H ∩ S)T . Applying Proposition 5.1.22, H ∈ t(G) and the proof of the lemma is complete. Theorem 5.2.4. Let G = HM be a factorised group, where M is an abelian normal p-subgroup of G for some prime p. Assume that H ∩ M = CH (M ) = 1. Then the following three conditions are pairwise equivalent: 1. H ∈ t(G). 2. a) H is p-soluble and b) if L K ≤ H, K/L non-abelian simple, and m ∈ M, then (K ∩ K m )L ∈ {L, K}. 3. a) H is p-soluble and b) Hp ∈ t(Hp M ) for every Hall p -subgroup Hp of H. Proof. 1 implies 2. First of all, if K is a subgroup of H, then K = (H ∩ M)K = H ∩ MK ∈ t(M K) as t is w-inherited. Hence, if L K ≤ H, K/L non-abelian simple, and m ∈ M , then (K ∩ K m )L ∈ {L, K} because K m ∈ t(M K). Next we see that H is p-soluble. Let S/T be a non-abelian simple section of H. Then S ∈ t(SM ) by the above argument. Choose R as a supplement of T in S contained in t(SM ) of minimal order. As Sn ≤ t, R ∩ T is the unique maximal normal subgroup of R; the corresponding quotient is a nonabelian simple group: R/(R ∩ T ) ∼ = RT /T = S/T . In particular, R is perfect. Hence the above lemma applies to R, R ∩ T , and SM . Since the hypothesis


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that CH (M ) = 1 and R = 1 rule out the case that [R, M ] ≤ R ∩ M = 1, it follows that RM/ CM (R) is a Frobenius-Wielandt group with respect to on R CM (R)/ CM (R) and (R ∩ T ) CM (R)/ CM(R). Now from our comments Frobenius-Wielandt groups, we get that gcd |R/(R ∩ T )|, |M/ CM (R)| = 1 = gcd(|S/T |, p). Consequently S/T is a p -group and H is p-soluble. 2 implies 3. Applying the well-known theorem of Hall-Chunikhin ([Hup67, VI, 1.7]), H has Hall p -subgroups. Let Hp be one of them. We prove that Hp ∈ t(Hp M ). Suppose that T S ≤ Hp M where S/T is non-abelian and simple. Clearly S = Sp Op (S) for all Sp ∈ Hallp (S), and Op (S) = S ∩ M is contained in T . Put K = Hp ∩ SM ∈ Hallp (SM ) and note that SM = KM as well as Hp ∩ S = K ∩ S. Suppose that (Hp ∩ S)T = T (i.e. (K ∩ S)T = T ); we prove that in this case (Hp ∩ S)T = S (i.e. (K ∩ S)T = S). The p -subgroup K ∩ S of S is contained in some Hall p -subgroup of S. Since every Hall p -subgroup of S is a Hall p subgroup of SM , the latter can be written as K m for some m ∈ M because the Hall p -subgroups of SM are conjugate. In particular, K ∩ S = K ∩ S ∩ K m = K ∩ K m . From Op (S) = S ∩ M ≤ T we infer that K m /(K m ∩ T M ) ∼ = K m (T M )/T M = SM/T M ∼ = S/T , so K m /(K m ∩ T M ) is a non-abelian simple group. Since T is proper in (K ∩ S)T , it follows that M T is a proper subgroup of (K ∩ S)T M . Assume that Km ∩ T M = (K ∩ K m )(K m ∩ T M ) = m m (K ∩ S)(K ∩ T M ) = K ∩ (K ∩ S)T M . Then K ∩ S would be contained in K m ∩ T M . This would imply that (K ∩ S)T ≤ (K m ∩ T M )T = K m T ∩ T M = S ∩ T M = T (S ∩ M ) = T (T ∩ M ) = T , contrary to our supposition. Hence K m ∩ T M is properly contained in (K ∩ K m )(K m ∩ T M ). By Statement 2b, K m = (K ∩ K m )(K m ∩ T M ). Consequently S = K m Op (S) = K m T = (K ∩ K m )(K m ∩ T M )T = (K ∩ K m )(K m T ∩ T M ) = (K ∩ K m )(S ∩ T M ) = (K ∩ S)T (S ∩ M ) = (K ∩ S)T (M ∩ T ) = (K ∩ S)T . Consequently Hp either covers or avoids each simple section of Hp M . Applying Proposition 5.1.22, Hp ∈ t(Hp M ) and Statement 3 follows. 3 implies 1. It follows from Lemma 5.2.3. The circle of implications is now complete. Remark 5.2.5. Under the hypothesis of Theorem 5.2.4, H cannot have a nonabelian simple p -subgroup. For a non-abelian simple p -subgroup K of H, Lemma 5.2.2 and Theorem 5.2.4 together with the fact that t is w-inherited would give the conclusion that KM/ CM (K) is a Frobenius group with respect to K CM (K)/ CM (K) ( ∼ = K). Then 2 divides |K| by the Odd Order Theorem [FT63]. In addition, applying [Hup67, V, 8.7], the Sylow 2-subgroups of K are cyclic or generalised quaternion. This contradicts [Hup67, V, 22.9]. The next result shows that non-soluble t-subgroups are close to subnormal subgroups. Corollary 5.2.6. If H ∈ t(G) and all composition factors of H are nonabelian, then H is subnormal in G.

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Proof. Consider a counterexample G of least order. Clearly, 1 < H < G. A reduction argument based on the w-inheritness of t yields G = HM and H ∩ M = 1 for every minimal normal subgroup M of G. In particular, CoreG (U ) = 1 for each maximal subgroup U of G containing H and G is a primitive group. Let K be a minimal subnormal subgroup of H. By hypothesis, K is a nonabelian group. Applying Theorem 5.2.1, it follows that K centralises every minimal normal subgroup of G. If G is a primitive group of type 2 or 3, then K is contained in CG Soc(G) = 1 by Theorem 1. This is a contradiction. If G is a primitive group of type 1, then H is p-soluble for the prime p dividing |Soc(G)| by Theorem 5.2.4 and so K is a p -subgroup. This contradicts Remark 5.2.5. Therefore no counterexample exists and H is a subnormal subgroup of G. A standard argument proves the following: Corollary 5.2.7. If G = HM where H ∈ t(G) and M is an abelian normal p-subgroup of G such that H ∩ M = CH (M ) = CM Op (H) = 1, then: Any subgroup K of G with G = KM and K ∩ M = 1 is a conjugate of H in G. Examples 5.2.8. 1. Any involution in Sym(6) \ Alt(6) generates a t-subgroup of Sym(6) complementing the non-abelian minimal normal subgroup Alt(6); however, not all of these involutions are conjugate. 2. Any involution in Sym(5)\Alt(5) also generates a t-subgroup of Sym(5) complementing the non-abelian minimal normal subgroup Alt(5). In this case, these involutions are conjugate in Sym(5). 3. Let n ≥ 7, f = (n, n − 1) and the section S/T of Sym(n), where S = f × Alt(n − 2) and T = f . Let g be an involution of Alt(n − 2). Then gf ∈ Sym(n) \ Alt(n). Put H = gf . Thus (H ∩ S)T = H × f , which is not normal in S. Therefore H is not a t-subgroup (compare with Beispiel 1.14 of [Sal87]). Corollary 5.2.7 shows that in a primitive group of type 1 all t-subgroups complementing the minimal normal subgroup are conjugate, while in a primitive group of type 2 this need not be true (Example 5.2.8 (1)). However, there is an important restriction on the t-subgroups supplementing the unique minimal normal subgroup in these primitive groups. Remark 5.2.9. If H < G = H Soc(G) is a primitive group of type 2 and H ∈ t(G), then H ∩ Soc(G) = 1 and H is soluble. Proof. Since H ∩ Soc(G) is normal in H Soc(G) = G and Soc(G) is a minimal normal subgroup of G, it follows that H ∩ Soc(G) = 1. Now the soluble residual S H S is a perfect t-subgroup of G. Applying Theorem 5.2.1, [H , Soc(G)] ≤ S S H ∩ Soc(G) = 1. Hence H ≤ CG Soc(G) = 1 and H is a soluble group. There are, however, primitive groups of type 1 with non-soluble t-subgroups H such that H < G = H Soc(G).


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Lemma 5.2.10. Assume that G is a Frobenius-Wielandt group with respect to the subgroups H and H ∗ such that H is non-soluble and complemented by the elementary abelian minimal normal p-subgroup Soc(G) of G. If H ∗ is contained in Φ(H), then G is a primitive group of type 1 and H ∈ t(G). Proof. Applying Theorem 1, G is a primitive group of type 1. It is known that gcd(p, |H/H ∗ |) = 1. Hence H is a p -group because H ∗ is contained in Φ(H). This implies that H is a Hall p -subgroup of G and Condition 2a of Theorem 5.2.4 holds. Let L K ≤ H, where K/L is non-abelian and simple. Assume that (K ∩ K x )L = L for some x ∈ Soc(G). Then K ∩ K x is not contained in H ∗ (otherwise K ∩ K x ≤ K ∩ H ∗ = L ∩ H ∗). Hence the definition of Frobenius-Wielandt group yields that x ∈ H; thus x ∈ H ∩ Soc(G) = 1, K = K x , and (K ∩ K x )L = K. Consequently Condition 2b of Theorem 5.2.4 holds and H ∈ t(G). Example 5.2.11. Let G = M H be the semidirect product of H = SL(2, 5) with an elementary abelian 11-subgroup of order 112 on which H acts fixed-pointfreely (see [Hup67, V, 8.8]). Then G is a Frobenius group. By Lemma 5.2.10, H ∈ t(G). Moreover H is perfect and comonolithic. If 1 = x ∈ M , then neither H ≤ H x nor H x ≤ NG (H). Therefore the analogue of the result of Wielandt, mentioned at the beginning of the section, does not hold t-subgroups. To conclude the investigations on t-subgroups, we mention the following: Proposition 5.2.12. Let G = HN be a group such that N is a normal subgroup of G, H ∩ N is a t-subgroup of N and G/N is soluble. Suppose that gcd(|N |, |G/N |) = 1. Then H ∈ t(G). Proof. Put K = H ∩ N and π = π(N ). Let S/T a non-abelian simple section of G such that T is a proper subgroup of (H ∩ S)T . We will show that (H ∩ S)T = S. Since G/N is soluble, N covers S/T ; thus S/T is isomorphic to (S ∩ N )/(T ∩ N ). Hence S/T is a π-group. On the other hand, the π-subgroup (H ∩ S)T /T of S/T is covered by the normal Hall π-subgroup K of H; thus (H ∩S)T = (K ∩S)T . Therefore T ∩N is a proper subgroup of (K ∩ S)(T ∩ N). Hence K ∈ t(N ) implies that S ∩ N = (K ∩ S)(T ∩ N ), and (H ∩ S)T = (K ∩ S)T = (S ∩ N )T = S follows. Therefore H either covers or avoids each simple section of G and so H ∈ t(G) by Proposition 5.1.22. To get more general results than those above appears to be rather difficult: Open question 5.2.13. Let G = HM be a group with a soluble subgroup H and a non-abelian minimal normal subgroup M of G satisfying H ∩ M = CH (M ) = 1. What are the precise conditions for H ∈ t(G)? In the following, we turn to prove some results for the remaining subgroup functors.

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Theorem 5.2.14. Let G be a monolithic primitive group and let H be a complement of Soc(G) in G. 1. If Soc(G) is abelian, H is an r-subgroup of G. 2. If Soc(G) is non-abelian, H is an r -subgroup of G. Consequently, any complement of the minimal normal subgroup of a monolithic primitive group is an r -subgroup. Proof. 1. To verify that H ∈ r(G), let T S ≤ G where S/T is nonabelian and strictly semisimple and H ≤ NG (S); we have to check that (S ∩ H)T is normal in S. By Remark 5.1.9 (6), we may also assume that H ≤ NG (T ). Note that H is a maximal subgroup of G because Soc(G) is abelian. If S is contained in H, then the result follows. Hence we may assume that NG (S) = G. This implies that Soc(G) is contained in S and normalises T . Hence T ∩ Soc(G) is a normal subgroup of H Soc(G) = G and so T ∩ Soc(G) = 1 or Soc(G) is contained in T . If T ∩ Soc(G) = 1, then T is contained in CG Soc(G) = Soc(G) by Theorem 1. This would imply T = 1 and so Soc(G) would be non-abelian, contradicting our supposition. Therefore Soc(G) is contained in T . Then G = HT and (H ∩ S)T = S. 2. In order to show H ∈ r (G), let T S sn G with S/T non-abelian and strictly semisimple and H ≤ NG (S); we have to prove that (H ∩S)T is normal in S. As in Case 1, we may assume that H normalises T . Since T and S are subnormal subgroups of G, they are normalised by Soc(G) by [DH92, A, 14.3]. Hence T and S are both normal in H Soc(G) = G. Since Soc(G) is a minimal normal subgroup of G, it follows that either T = 1 or Soc(G) ≤ T . If T = 1, then Soc(G) is contained in S. On the other hand, by [DH92, A, 4.14], S is contained in Soc(G). Thus S = Soc(G) and (H ∩ S)T = 1. Assume that Soc(G) ≤ T . Then G = HT and so (H ∩ S)T = S. The next aim is to obtain a similar result for t . Lemma 5.2.15. Let G be a monolithic primitive group. Assume that G has a subgroup H such that G = H Soc(G) and H ∩ Soc(G) = 1. If G = S Soc(G) with T S sn G and S/T semisimple, then (H ∩ S)T ∈ {S, T }. Proof. First consider the case when S = G. Then T is a normal subgroup of G. If T = 1, then Soc(G) is contained in T and then G = HT . Hence (H ∩ S)T = S. If T = 1, then G = Soc(G) and H = 1. Suppose that 1 = S is a proper subnormal subgroup of G. Then a maximal normal subgroup M of G would contain S as well as the unique minimal normal subgroup Soc(G) of G; so the contradiction that G = S Soc(G) = M < G would emerge. Theorem 5.2.16. Let G be a group. Assume that G = HM and H ∩ M = 1 where M is a strictly semisimple normal subgroup of G. Then the following statements are equivalent:


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1. If E is a simple direct factor of M , and K is a subnormal subgroup of H normalising E, then IK (E) = {k ∈ K : K induces an inner automorphism in E} = CK (E). 2. H ∈ t (G). Proof. 1 implies 2. We argue by induction on |G|. Let A be a minimal normal subgroup of G contained in H. Then it is straightforward to verify that the subgroup H/A satisfies the hypotheses of the theorem in G/A. We can conclude by induction that H/A ∈ t (G/A). In this case we have H ∈ t (G) because t is inherited. Therefore we can assume that CoreG (H) = 1 and, in particular, CH (M ) = 1. Next we show that we can suppose that M is a minimal normal subgroup of G. Let B be a minimal normal subgroup of G contained in M . If B < M , then HB is a proper subgroup of G for which the hypotheses of the theorem clearly hold. Therefore by induction H ∈ t (HB). We also have that HB/B ∈ t (G/B) by induction. Then H ∈ t (G). Thus we can suppose that M is a minimal normal subgroup of G. Let S/T be a non-abelian and strictly semisimple section of G such that S is subnormal in G. If SM is a proper subgroup of G, then SM = (H ∩ SM )M and the hypotheses of the theorem hold in SM because H ∩ SM is subnormal in H. Hence by induction H ∩ SM ∈ t (SM ) and so (H ∩ S)T = (H ∩ SM ∩ S)T S. Therefore we can assume that G = SM . If M is abelian, then CG (M ) = M . This implies that M is the unique minimal normal subgroup of G and G is a monolithic primitive group by Theorem 1. Applying Lemma 5.2.15, (S ∩ H)T ∈ {S, T }. Assume that M is non-abelian. We prove that CG (M ) = 1. To this end, put C = CG (M ), D = H ∩ CM , and suppose that C = 1. Observe that D = 1 because M < CM = C × M = HM ∩ CM = DM . Let E denote a simple component of M . Since both C and M act upon E by inner automorphisms, from Statement 1 we obtain that D ≤ CH (E). On the other hand, H (acting by conjugation) permutes the simple components h of the minimal normal subgroup M of G transitively, so D = h∈H D is h h = h∈H CH (E ). This yields the contained in CH (M ) = CH h∈H E contradiction that 1 = D ≤ CH (M ) = CoreG (H) = 1. Therefore CG (M ) = 1 and M = Soc(G). Hence G is a monolithic primitive group by Theorem 1. Applying Lemma 5.2.15, (S ∩ T )T ∈ {S, T }. Consequently H ∈ t (G). 2 implies 1. Assume, arguing by contradiction, that there exists a subnormal subgroup K of H ∈ t (G) such that K normalises a simple direct factor E of M and I = IK (E) is not contained in CK (E). Then E is non-abelian. Note that I is a normal subgroup of K. Hence I is subnormal in H. Since Sn ≤ t and t is inherited, it follows that I ∈ t (G). Moreover IM is subnormal in G. These facts clearly imply that I ∈ t (IM ). On the other hand, M = E × A for some I-invariant normal complement A of E in M . Since IA/A ∈ t (IM/A) by Remark 5.1.9 (4), we conclude that I ∈ t (IE). Denote Z = IE. Then

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Z = E CZ (E) and E ∩ CZ (E) = 1. Hence Z/ CZ (E) is isomorphic to E. As I is not contained in CZ(E), I CZ (E)/ CZ (E) = 1 and so I CZ (E) = Z because I CZ (E)/ CZ (E) ∈ t Z/ CZ (E) . Hence I ∩ CZ (E) = CK (E) is a normal to E × E. This contradicts the subgroup of Z and Z/ C K (E) is isomorphic fact that I/ CK (E) ∈ t Z/ CK (E) . Corollary 5.2.17. If G is a primitive group of type 1, then any complement of Soc(G) in G is a t -subgroup. The corresponding statement for subgroups of primitive groups of type 2, though, does not always hold. Example 5.2.18. Let H = E F , the regular wreath product of a non-abelian simple group E and a cyclic group F of order 3. Its base group can be written as B = E1 ×E2 ×E3 with Ei ∼ = E. A subnormal subgroup K of H is defined by K = E1 × E2 . Let K act on E such that E1 induces all inner automorphisms, while E2 acts trivially, and form the twisted wreath product corresponding to this action: G = E K H. One checks that G is a primitive group of type 2, which, by its very construction, violates Theorem 5.2.16 (1) (taking M as the base group of G).

5.3 Projectors of soluble type The purpose of this section is to develop a general framework for a theory of Schunck class projectors in finite (not necessarily soluble) groups. A specialisation of that theory will yield a theory resembling the theory of projectors in finite soluble groups. Hypothesis 5.3.1. In the sequel, d will denote a subgroup functor subject to the following conditions: 1. If H ∈ d(G) and N G, then HN/N ∈ d(G/N ). 2. d is weakly inductive, that is, G ∈ d(G) and if H ∈ d(HN ) and HN/N ∈ d(G/N ), then H ∈ d(G). 3. If H ∈ d(G) and M is a minimal normal subgroup of G, then H < G = HM implies H ∩ M = 1. 4. If H is a core-free maximal subgroup of a primitive group G of type 3, then H ∈ / d(G). 5. If G = H F∗ (G), H ∈ d(G), and M is a quasinilpotent normal subgroup of G, then H ∈ d(HM ). 6. If H ∈ d(G) and N is a normal quasinilpotent subgroup of G, then H ∩ N is subnormal in N . Remark 5.3.2. The results of the previous sections show that any of the functors Sn , t, r, t , and r satisfies the above conditions.

5.3 Projectors of soluble type

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Definitions 5.3.3. 1. A subgroup U of a group G is said to be d-maximal in G if U is a proper subgroup of G, U ∈ d(G), and U < H, and H ∈ d(G) implies that H = G. 2. A group G is said to be a d-primitive group if G has a d-maximal subgroup U such that CoreG (U ) = 1. Lemma 5.3.4. Assume that G is a d-primitive group. Then G is a monolithic primitive group. If U is a core-free d-maximal subgroup of G, then G = U Soc(G) and U ∩ Soc(G) = 1. Proof. Let H be a maximal subgroup of G containing U . Put T = CoreG (H). Since d is weakly inductive, it follows that U T ∈ d(G). Hence U = U T because U T is a proper subgroup of G. This implies that T = 1 and G is a primitive group. Let M be a minimal normal subgroup of G. Then U is a proper subgroup of U M and U M ∈ d(G). This yields G = U M by d-maximality of U . Applying Hypothesis 5.3.1 (3), U ∩ M = 1. If G were a primitive group of type 3, then CG (M ) would be non-trivial and so U CG (M ) = U by the d-maximality of U . Then U would be a core-free maximal subgroup of G. This would be a contradiction against Hypothesis 5.3.1 (4). Therefore G is a monolithic primitive group. We say that G is a d-primitive group of type i (i = 1, 2) if G is a primitive group of type i. Remarks 5.3.5. 1. Applying Remark 5.2.9, a t-primitive group G of type 2 satisfies that G/ Soc(G) is soluble and if G is a t-primitive group of type 1, then G/ Soc(G) is p-soluble, where p is the prime dividing |Soc(G)|, by Theorem 5.2.4. 2. By Theorem 5.2.14, any complement of the minimal normal subgroup of a monolithic primitive group G is an r -subgroup. In particular, monolithic primitive groups with complemented socle are r -primitive. 3. Let E be any non-abelian simple group. Applying Theorem 1.1.36, there is a group G with a minimal normal subgroup M such that M is the direct product of copies of Alt(6), G/M is isomorphic to E, and G does not split over M . It is clear that G is a primitive group of type 2. Applying Lemma 5.3.4, G is not d-primitive. Definitions 5.3.6. Let H be a class of groups. 1. A subgroup X of a group G is said to be H-d-maximal subgroup of G if X ∈ H ∩ d(G) and if X ≤ K ∈ d(G) ∩ H, then X = K. Denote by MaxHd (G) the (possibly empty) set of all H-d-maximal subgroups of G. 2. A subgroup U of a group G is called an H-d-projector of G if U N/N is H-d-maximal in G/N for all N G. We shall use ProjHd (G) to denote the (possibly empty) set of H-dprojectors of a group G.

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3. An H-d-covering subgroup of a group G is a subgroup E of G satisfying the following two conditions: a) E ∈ MaxHd (G), and b) if T ∈ d(G), E ≤ T , N T , and T /N ∈ H, then T = N E. The set of all H-d-covering subgroups of a group G will be denoted by CovHd (G). It is clear that MaxHd , ProjHd , and CovHd are subgroup functors and CovHd ≤ ProjHd ≤ MaxHd . Definitions 5.3.7. 1. A class H is called d-projective if ProjHd (G) = ∅ for each group G. 2. A class H is said to be a d-Schunck class if H is a homomorph that comprises precisely those groups whose d-primitive epimorphic images are in H. Clearly a class H is a d-Schunck class if and only if H is a homomorph whose boundary is composed of d-primitive groups. Remark 5.3.8. Every d-Schunck class is a Schunck class whose boundary is composed of monolithic primitive groups (Lemma 5.3.4). The converse does not hold: the Schunck class H of all groups without epimorphic images isomorphic to the group G of Remark 5.3.5 (3) is not a d-Schunck class. Our next goal is to prove that the d-projective classes are exactly the d-Schunck classes. We need some previous results. Lemma 5.3.9. Let H be a homomorph. Let G be a group and N a normal quasinilpotent subgroup of G. If H ∈ MaxHd (HN ) and HN/N ∈ MaxHd (G/N ), then H ∈ MaxHd (G). Proof. Let T be a subgroup of G such that H is contained in T and T ∈ d(G) ∩ H. Since T N/N ∈ d(G/N ) ∩ H by Hypothesis 5.3.1 (1) and HN/N ∈ MaxHd (G/N ), it follows that HN = T N . By Hypothesis 5.3.1 (6), T ∩ N is subnormal in N . Hence T ∩ N is a normal quasinilpotent subgroup of HN . By Hypothesis 5.3.1 (5), we know that H(T ∩ N ) = T ∈ d(HN ). Therefore T = H by the H-d-maximality of H in HN . Remark 5.3.10. Lemma 5.3.9 holds for any normal subgroup N of G if the functor d has the following property: 5*. If H ∈ d(G) and N G, then H ∈ d(HN ). Note that t and r satisfy Property 5* but t and r do not enjoy this property (see Example 5.1.12). Lemma 5.3.11. Let H be a homomorph and let G be a group. Assume that N is a quasinilpotent normal subgroup of G and K is a subgroup of G such that N is contained in K and K/N ∈ ProjHd (G/N ). If H ∈ ProjHd (K), then H ∈ ProjHd (G).


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Proof. We must show that HM/M ∈ MaxHd (G/M ) for all normal subgroups M of G. First of all, K = HN because HN/N ∈ MaxHd (K/N ). Moreover, since H ∈ ProjHd (K), it follows that H(K ∩ M )/(K ∩ M ) is H-d-maximal in K/(K ∩ M ). This implies that HM/M is H-d-maximal in KM/M . On theother hand, (KM/N ) (M N/N ) is an H-d-maximal subgroup of (G/N ) (M N/N ) because K/N ∈ Proj Hd (G/N ). In particular, (KM/M ) (M N/M ) is H-d-maximal in (G/M ) (M N/M ). Since N M/M is quasinilpotent and normal in G/M , we apply Lemma 5.3.9 to conclude that HM/M is H-d-maximal in G/M . Remark 5.3.12. Lemma 5.3.11 holds for any normal subgroup N of G if d satisfies Property 5*. Lemma 5.3.13. Let G ∈ b(H) for some homomorph H. 1. ProjHd (G) = CovHd (G), and this set coincides with {U ∈ d(G) : U < G = U M for all minimal normal subgroups M of G}, here U ∩ M = 1 for all H ∈ ProjHd (G) and any minimal normal subgroup M of G. 2. ProjHd (G) = ∅ if and only if G is d-primitive. Proof. 1. If ProjHd (G) = ∅, then CovHd (G) = ∅. Assume ProjHd (G) = ∅ and let U be an H-d-projector of G. Since G ∈ / H, it follows that U is a proper subgroup of G. Let M be a minimal normal subgroup of G. Then G/M ∈ H by definition of the boundary, and therefore by definition of an H-d-projector we have G = U M . Applying Hypothesis 5.3.1 (3), U ∩ M = 1. It is clear that in this case U is a d-maximal subgroup of G. Therefore U ∈ CovHd (G). On the other hand, if G = HM and H ∩ M = 1 for a proper subgroup H ∈ d(G) of G and for all minimal normal subgroup of G, then H ∼ = G/M ∈ H and G and H are the only d-subgroups of G above H. Consequently H ∈ CovHd (G) and Statement 1 holds. 2. Suppose that ProjHd (G) = ∅ and, as in 1, we take U ∈ ProjHd (G). Since G∈ / H, it follows that U is a proper subgroup of G. Let H be a d-maximal subgroup of G containing U , and let K = CoreG (H). If K = 1, then G/K ∈ H because G belongs to the boundary of H. By definition of an H-d-projector we have G = U K ≤ H < G. This contradiction shows that K = 1 and hence that G is d-primitive. Conversely, assume that G is d-primitive. Let U be a core-free d-maximal subgroup of G. Then G = U Soc(G) and U ∩ Soc(G) = 1. It is clear then that U ∈ H. Moreover U and G are the only d-subgroups of G above U . Consequently U ∈ MaxHd (G) and so U ∈ ProjHd (G). This completes the proof of the lemma.

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Theorem 5.3.14. A class H = ∅ is d-projective if and only if it is a d-Schunck class. Proof. Let H be a d-projective class, and let G ∈ H. Since ProjHd (G) = ∅, it follows from the definition of a projector that ProjHd (G) = {G} and hence that for all N G the quotient G/N = GN/N is Hd -maximal in G/N , in other words, that G/N ∈ H. Therefore H is a homomorph. By Lemma 5.3.13, b(H) is composed of d-primitive groups, and consequently H is a d-Schunck class. Conversely, let H be a d-Schunck class. We prove that ProjHd (G) = ∅ by induction on |G|. Clearly we may suppose that G ∈ / H (in particular G = 1) and that ProjHd (A) = ∅ for all groups A such that |A| < |G|. Let N be a minimal normal subgroup of G. Then ProjHd (G/N ) contains a subgroup, K/N say, and if |K| < |G|, there exists a subgroup H ∈ ProjHd (K). Then from Lemma 5.3.11 we conclude that H ∈ ProjHd (G). There remains the possibility that G/N ∈ ProjHd (G/N ), which implies that G/N ∈ H. But if G/N ∈ H for all minimal normal subgroups N of G, it follows that G ∈ b(H), and then ProjHd (G) = ∅ by Lemma 5.3.13. The induction argument is therefore complete. Corollary 5.3.15. d-Schunck classes are precisely those Schunck classes H such that ProjHd (G) = ∅ for all G ∈ b(H). Now we turn our attention to the conjugacy problem for d-projectors and d-covering subgroups. As in the classical case, primitive groups of type 3 have to be excluded from the boundaries. In this context, the information given in the next lemma is useful. Lemma 5.3.16. Let H be a Schunck class such that b(H) ∩ P3 = ∅. Let G = HN , where N is a quasinilpotent normal subgroup G. If H is an H-dmaximal subgroup of G, then H ∈ ProjHd (G). Proof. We proceed by induction on |G|. Without loss of generality we may assume that N = F∗ (G). Suppose there exists a minimal normal subgroup M of G such that HM < G, and G/M ∈ / H. Then, by [DH92, III, 2.2 (c)], there exists a normal subgroup T of G such that M ≤ T and G/T ∈ b(H). By hypothesis, G/T is a monolithic primitive group; as G/N = HN/N ∈ H, we have that N is not contained in T and so N T /T is a non-trivial quasinilpotent normal subgroup of G/T . This implies that N T /T = Soc(G/T ) by Theorem 1 and Proposition 2.2.22 (3). Since G/T ∈ / H but G/N T ∈ H, the d-subgroup H of G cannot possibly cover N T /T , and so must avoid this factor. Consequently HT ∩ N = HT ∩ N T ∩ N = T (H ∩ N T ) ∩ N = T ∩ N , and HT = HT ∩ HN = H(HT ∩ N) = H(T ∩N) follows. As N is not contained in T , it follows that HT is a proper subgroup of G. Moreover T ∩ N is a normal quasinilpotent subgroup of HT = H(T ∩ N) and hence H ∈ d(HT ) by Hypothesis 5.3.1 (5). Let H1 be an H-d-maximal subgroup of HT containing H. Then HT = H1 T . Note that HT /T is H-d-maximal in G/T because HT /T ∈ d(G/T ) ∩ H and (HT /T ) ∩ (N T /T ) = 1. By induction, H1 ∈ ProjHd (H1 T ) and H1 T /T ∈ ProjHd (G/T ). Applying Lemma 5.3.11, H1 ∈ ProjHd (G). Then H = H1 because of H-dmaximality of H, and the result follows.


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Consequently we can suppose that G/M ∈ H for all minimal normal subgroups M of G. Then either G ∈ H — in which case nothing remains to be shown —, or else G ∈ b(H). In the latter case our claim is immediate from the assumption that G = H F∗ (G): then F∗ (G) is the minimal normal subgroup of the monolithic primitive group by Theorem 1 and Proposition 2.2.22 (3). The induction argument is therefore complete. The further results in this section will all require that d has the properties stated in Hypothesis 5.3.1 together with Property 5* introduced in Remark 5.3.10. In this case, ProjHd is an inductive functor by Lemma 5.3.11 and Remark 5.3.12. Lemma 5.3.17. Let H be a d-Schunck class such that b(H) ∩ P3 = ∅. If H ∈ ProjHd (G) and H is contained in U , where U ∈ d(G), then H ∈ ProjHd (U ). Proof. We proceed by induction on |G|. Clearly we can assume that U < G. Let N be a minimal normal subgroup of G. By induction, HN/N is an Hd-projector of U N/N because U N/N ∈ d(G/N ) Hypothesis 5.3.1 (1) and HN/N ∈ ProjHd (G/N ). The standard isomorphism from U N/N to U/(U ∩N ) transforms HN/N into H(U ∩ N )/(U ∩ N ). This yields H(U ∩ N )/(U ∩ N ) ∈ ProjHd U/(U ∩ N ) . Assume that HN is a proper subgroup of G. By Property 5*, H ∈ d(HN ). Let H1 be an H-d-maximal subgroup of HN containing H. By Lemma 5.3.16, H1 ∈ ProjHd (HN ). Since HN/N = H1 N/N is an H-d-projector of G/N , it follows that H1 ∈ ProjHd (G) by Lemma 5.3.11. In particular, H = H1 because H ∈ MaxHd (G). On the other hand, N ∩ U is normal in N as N ∩ U is subnormal in N by Hypothesis 5.3.1 (6). Hence N ∩ U is a normal subgroup of HN . This implies that H(N ∩ U ) ∈ d(HN ) by Hypothesis 5.3.1 (5). Applying induction, H ∈ ProjHd H(N ∩ U ) . Consequently H ∈ ProjHd (U ) by Hypothesis 5.3.1 (2). Therefore we can suppose that G = HN . Since U ∈ d(G) is a proper subgroup of G, it follows that U ∩ N = H ∩ N = 1 by Hypothesis 5.3.1 (3). This implies U = H. Therefore H ∈ ProjHd (U ) and the proof of the lemma is complete. We can state an important consequence of this lemma. Theorem 5.3.18. Let H be a Schunck class such that b(H) ∩ P3 = ∅. Then ProjHd (G) = CovHd (G) for every group G. Proof. Let H be an H-d-projector of a group G. Let T be a subgroup of G such that T ∈ d(G) and H ≤ T . By Lemma 5.3.17, H is an H-d-projector of T . Therefore if N is a normal subgroup of T and T /N ∈ H, it follows that T = N H. Consequently H satisfies Conditions 3a and 3b of Definition 5.3.6 (3) and so H ∈ CovHd (G). The other inclusion is clear. We prove in the following the main conjugacy theorem for projectors.

230


Theorem 5.3.19. For any Schunck class H such that b(H) ∩ P3 = ∅, the following statements are equivalent: 1. For every group G ∈ b(H), ProjHd (G) is a conjugacy class of G. 2. For every group G, ProjHd (G) is a conjugacy class of G. Proof. Only the implication “1 implies 2” is in doubt. We argue by induction on |G| that ProjHd (G) is a conjugacy class of G. Let N be a minimal normal subgroup of G and H1 and H2 H-d-projectors of G. Since Hi N/N ∈ ProjHd (G/N ), by induction we have H1 N = H2g N for some g ∈ G. By Theorem 5.3.18, H1 and H2g are H-d-projectors of H1 N . If |H1 N | < |G|, by induction H1 is conjugate to H2g and hence to H2 . We can therefore suppose that G = H1 N and hence G/N ∈ H for every minimal normal subgroup of G. Thus, either G ∈ H and then H1 = G = H2 , or G ∈ b(H), and H1 and H2 are conjugate by the hypothesis. Two H-t-projectors of a group need not be conjugate.

Example 5.3.20. Let H be the Schunck class defined by b(H) = Sym(6) . Since Sym(6) is t-primitive, it follows that H is a t-Schunck class; Sym(6) has two non-conjugate t-subgroups complementing Alt(6), say X and Y , where the involution in X is a 2-cycle, while the involution in Y is a product of three disjoint 2-cycles. Clearly X, Y ∈ ProjHt Sym(6) . As we have mentioned earlier in this section, the subgroup functors t, r enjoy the properties stated in Hypothesis 5.3.1 (1)–(6). They also enjoy Property 5*. Moreover if d is one of these functors, d(G) = S(G) for all G ∈ S. Therefore ProjHd (G) = ProjH (G) = ProjH∩S (G) for all G ∈ S; thus, we have obtained an alternative approach to projectors and covering subgroups in finite soluble groups. On the other hand, a conjugacy criterion for ProjHr (or ProjHt ) analogous to Theorem 5.3.19 does not hold; moreover CovHt ProjHt . Example 5.3.21. Let E be any non-abelian simple group and take an involution t in E. Define T = τ with τ being the inner automorphism of E corresponding to t. We consider the regular wreath product H = E T and form the twisted wreath product G = E T H where T acts naturally on E as a group of inner automorphisms. The base group of G will be denoted by M = E1 × · · · × En , n = |H : T |. Applying Theorem 5.2.16, T ∈ t (H) and H ∈ t (G). Therefore T ∈ t (G) ⊆ r (G) because t is inherited. Let D = mτ , where m is the product of the involutions ti ∈ Ei such that Eiti = Eiτ and iτ = i. Then D centralises Ei for all i such that iτ = i. This implies that D ∈ t (M D) by Theorem 5.2.16, and hence D ∈ t (G) by inductivity. Let H be the Schunck class of all groups without quotients isomorphic to E or H. Then b(H) = {E, H}. It is clear that H is a t -Schunck class. / ProjHt (T M ) Moreover {T, D} ⊆ ProjHt (G). Applying Theorem 5.2.16, T ∈ and D ∈ ProjHt (DM ). Thus, from T M = DM , we get that T and D cannot possibly be conjugate in G. Similarly {T, D} ⊆ ProjHr (G).


231

We have shown that H is a d-Schunck class and G is a group with two nonconjugate H-d-projectors for d ∈ {t , r }. Note that ProjHd (A) is a conjugacy class for all A ∈ b(H). Moreover T ∈ ProjHt (G) \ CovHt (G). We have mentioned in Chapter 2 that Hall π-subgroups, π a set of primes, have been among the motivating examples for the theory of projectors in finite soluble groups: they are, in the soluble universe, the projectors with respect to the class Sπ of all soluble π-groups. When applying the theory of projectors of finite (not necessarily soluble) groups to the class Eπ of all finite π-groups, one obtains that every finite group has Eπ -projectors. However, in a non-soluble group, the set of Eπ -projectors does not normally form a set of conjugate subgroups, and it does not appear to be true that every π-subgroup is contained in some Eπ -projector; certainly, a π-subgroup need not be contained in a conjugate of a given Eπ -projector. In the following example, we see that for each set π of primes there is some class E(π), which may be thought of as being generated by Eπ in a certain sense, with the following properties: the E(π)-projectors of an arbitrary finite group G form a non-empty set of conjugate subgroups of G and each π-subgroup of G is contained in some E(π)-projector of G. A worked example Let π be a set of primes. Define E(π) := G : if N G and G/N is t-primitive, then π Soc(G/N ) ∩ π = ∅ . It is clear that E(π) = Q E(π) is contained in E(π), and the boundary of E(π) is given by b E(π) = G : G is t-primitive, G/ Soc(G) is an E(π)-group

and Soc(G) is a π -group .

Therefore E(π) is a t-Schunck class. Since t(G) = S(G) for all soluble groups G, E(π) ∩ S = Sπ , the class of all soluble π-groups. Assume that G is a t-primitive group of type 2 in b E(π) . Then G/ Soc(G) is a soluble π-group by Remark 5.2.9. Therefore Soc(G) is a Hall π -subgroup of G. By the Schur-Zassenhaus theorem [Hup67, I, 18.1 and 18.2], any two complements of Soc(G) in G are conjugate. They are the Hall π-subgroups of G and the E(π)-t-projectors of G. If G is a t-primitive group of type 1 in b E(π) , then Soc(G) is a pgroup for some prime p and it is complemented in G by a p-soluble subgroup H of G which is an E(π)-projector of G. Applying Corollary 5.2.7, any two complements of Soc(G) in G are conjugate. By Theorem 5.3.18 and Theorem 5.3.19, we have:

232


Theorem 5.3.22. ProjE(π)t (G) = CovE(π)t (G) is a class of conjugate subgroups in every group G. For each group G, we put hπ (G) = {H ∈ t(G) : H ∈ E(π) and π(|G : H|) ⊆ π }. Lemma 5.3.23. If G ∈ E(π) and H ∈ t(G), π(|G : H|) ⊆ π , then G = H. Proof. Assume the result is false and let G be a counterexample of minimal order. This choice requires that G = HM for any minimal normal subgroup M of G. Hence H is a t-maximal subgroup of G and CoreG (H) = 1. In particular G is t-primitive and π Soc(G) = π(|G : H|) ⊆ π . We conclude then that G ∈ b E(π) ∩ E(π) = ∅. This contradiction proves the lemma. Corollary 5.3.24. hπ (G) = {G} if and only if G ∈ E(π). Theorem 5.3.25. ProjE(π)t (G) = hπ (G) for every group G. Proof. We argue by induction on |G|. If G ∈ E(π), we have that hπ (G) = / E(π). By [DH92, III, 2.2 (c)] ProjE(π)t (G) = {G}. Therefore suppose that G ∈ there exists a normal subgroup K of G such that G/K ∈ b E(π) . Assume that K = 1. Let H be an E(π)-t-projector of G. Then HK/K ∈ ProjE(π)t (G/K). By induction, HK/K ∈ hπ (G/K) and hence π(|G : HK|) ⊆ π . On the other hand, HK ∈ t(G). By Theorem 5.3.22, H ∈ ProjE(π)t (HK). Since HK is a proper subgroup of G, we conclude that H ∈ hπ (HK) by induction. Thus π(|G : H|) ⊆ π and H ∈ hπ (G). The same argument, using Remark 5.3.12, shows that hπ (G) ⊆ ProjE(π) (G). Therefore the result holds in this case. K = 1. Then G ∈ b E(π) and so G is t-primitive, Suppose that π Soc(G) ⊆ π , G = Soc(G)H and H ∩Soc(G) = 1 for every E(π)-t-projector H of G. It is clear that ProjE(π)t (G) ⊆ hπ (G). Let A ∈ hπ (G) and let H be an E(π)-t-projector of G. If G = A Soc(G), then A is clearly an E(π)-tprojector of G. Assume that X = A Soc(G) is a proper subgroup of G. Then A ∈ ProjE(π)t (X) by induction (note that A ∈ hπ (X) because t is w-inherited and Soc(G) is a π -group). On the other hand, X = Soc(G)(X ∩ H). Since X ∩ H ∈ t(X) and π(|X : X ∩ H|) ⊆ π , it follows that X ∩ H ∈ hπ (X). By induction, X ∩ H ∈ ProjE(π)t (X) and therefore X ∩ H and A are conjugate in X by Theorem 5.3.22. Hence, without loss of generality, we may suppose that A is contained in H. Then A ∈ hπ (H) because π(|G : A|) ⊆ π and t is inherited. Applying Corollary 5.3.24, it follows that A = H and A ∈ ProjE(π)t (G). The proof of the theorem is complete. Theorem 5.3.26. ProjE(π)t (G) = ProjE(π) (G) = CovE(π) (G) for every group G. Proof. We only prove that ProjE(π)t (G) = ProjE(π) (G). Arguing by induction on |G|, we may assume that G ∈ b E(π) . Then Soc(G) is a π -group which is complemented in G by every E(π)-t-projector of G.


233

Let H ∈ ProjE(π)t (G) and let T be an E(π)-maximal subgroup of G containing H. Applying Proposition 2.3.16, T ∈ ProjE(π) (G). Since H ∈ hπ (T ) and T ∈ E(π), we conclude that H = T by Corollary 5.3.24. Consequently ∅ = ProjE(π)t (G) ⊆ ProjE(π) (G). Assume that G is a t-primitive group of type 1. Then the complements of Soc(G) in G are precisely the E(π)-t-projectors of G by Corollary 5.2.7. Since every E(π)-projector of G complements Soc(G) by [DH92, III, 3.9 (i)], it follows that ProjE(π) (G) = ProjE(π)t (G). Suppose that G is a t-primitive group of type 2. Then Soc(G) is a Hall π -subgroup of G and ProjE(π)t (G) is the set of all Hall π-subgroups of G. Let T be an E(π)-projector of G. Then T Soc(G) = G and T = T ∩ Soc(G) T0 for some Hall π-subgroup T0 of T . It is clear that T0 is a Hall π-subgroup of G. Therefore T0 is an E(π)-t-projector of G. Then T0 ∈ hπ (T ) by Theorem 5.3.25. Since T ∈ E(π), we conclude that T0 = T by Corollary 5.3.24. Consequently ProjE(π)t (G) = ProjE(π) (G). Theorem 5.3.27. Every π-subgroup of a group G is contained in some E(π)projector of G. Proof. Let A be a π-subgroup of G. We prove by induction on |G| that A is contained in an E(π)-projector of G. It is clear that we may assume that G ∈ b E(π) . If G is a t-primitive group of type 2, then the complements of N = Soc(G) in G are the Hall π-subgroups of G and the E(π)-projectors of G. Let H be one of them. If AN = G, then A is an E(π)-projector of G. Assume that X = AN is a proper subgroup of G. Then X = Soc(G)(X ∩ H). By the Schur-Zassenhaus theorem [Hup67, I, 18.1 and 18.2], A and X ∩ H are conjugate. Hence A is contained in an E(π)-projector of G. Assume that G is a t-primitive group of type 1 and let p denote the prime dividing |Soc(G)|. By Theorem 5.2.4, G is p-soluble and so, applying the theorem of Hall-Chunikhin ([Hup67, VI, 1.7]), G has Hall p -subgroups, all of them are conjugate, and every p -subgroup — in particular, every π-subgroup — is contained in some Hall p -subgroup. Since the elements of ProjE(π) (G) are precisely the complements in G of Soc(G), they contain Hall p -subgroups of G and every Hall p -subgroup of G is contained in some E(π)-projector of G. Therefore A is contained in an E(π)-projector of G. Final remark 5.3.28. Earlier in this example we have made the observation that E(π) ∩ S = Sπ for which one derives that in soluble groups ProjE(π)t coincides with Hallπ , for in such groups t coincides with S. We ought to alert the reader that — in contrast to this observation — in groups from Tπ , the class of all π-separable groups, ProjE(π)t = Hallπ , that is, Eπ is not equal to E(π) ∩ Tπ in general: take π = p , for some prime p not dividing the order of some non-abelian simple group E. Let G = [V ]E be the semidirect product a faithful and irreducible E-module over GF(p). Then of E with G ∈ E(π) ∩ Tπ \ Eπ .

6 F-subnormality

How a subgroup can be embedded in a group is always a question of particular interest for clearing up the structure of finite groups. One of the most important subgroup embedding properties is the subnormality, transitive closure of the relation of normality. This property was extensively studied by H. Wielandt (see [Wie94a]). For an excellent survey of the theory of subnormal subgroups, we refer the reader to J. C. Lennox and S. E. Stonehewer [LS87]. In finite groups the significance of the subnormal subgroups is apparent since they are precisely those subgroups which occur as terms of composition series, the factors of which are of great importance in describing the group structure. Let F be a saturated formation of full characteristic. If G is a soluble group, the F-normaliser D of G associated with a Hall system Σ of G is contained in the F-projector E of G in which Σ reduces (see [DH92, V, 4.11] and Theorem 4.2.9). In 1969, T. O. Hawkes [Haw69] analysed how D is embedded in E. It turns out that D can be joined to E by means of a maximal chain of F-normal subgroups, that is, D is F-subnormal in E ([DH92, V, 4.12]). The F-subnormality could be regarded, in the soluble universe, as the natural extension of the subnormality to formation theory. In fact, most of the results concerning subnormal subgroups can be read off by specialising to the case where F is the formation of all nilpotent groups. Our objective in this chapter is to present the main results of the F- subnormal subgroups. They are primarily connected with the study of subnormal subgroups properties by the methods of formation theory.

6.1 Basic properties In the sequel, F will denote a non-empty formation. A subgroup U of a group G is called F-normal in G if G/ CoreG (U ) ∈ F;

235

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6 F-subnormality

otherwise U is said to be F-abnormal in G. This definition was introduced in Definition 2.3.22 (1) for maximal subgroups. Illustrations 6.1.1. 1. A subgroup U is F-normal in G if and only if GF is contained in U . 2. A maximal subgroup is normal in G if and only if it is N-normal in G. In general, a subgroup U is subnormal in G provided that U is N-subnormal in G. 3. If F = LF(F ) is a saturated formation, a maximal subgroup M of G is F-normal in G if and onlyif G/ CoreG (M ) ∈ F (p) for every prime p divi ding Soc G/ CoreG (M ) . Definition 6.1.2. A subgroup H of a group G is said to be F-subnormal in G if either H = G or there exists a chain of subgroups H = H 0 < · · · < Hn = G such that Hi−1 is an F-normal maximal subgroup of Hi for i = 1, . . . , n. We shall write H F-sn G; SnF (G) will denote the set of all F-subnormal subgroups of a group G. It is clear that Sn F is a subgroup functor. Remark 6.1.3. Assume that F = N, the formation of all nilpotent groups. Then SnN (G) ⊆ Sn (G) for all groups G by Illustration 6.1.1 (2). However the equality does not hold in general because if G = Alt(5), then 1 ∈ Sn (G) \ SnN (G). Nevertheless, if G is soluble, then Sn (G) = SnN (G). To avoid the above situation, O. H. Kegel [Keg78] introduced a little bit different notion of F-subnormality. It unites the notions of subnormal and F-subnormal subgroup. Definition 6.1.4. A subgroup U of a group G is called K-F-subnormal subgroup of G if either U = G or there is a chain of subgroups U = U0 ≤ U1 ≤ · · · ≤ Un = G such that Ui−1 is either normal in Ui or Ui−1 is F-normal in Ui , for i = 1, . . . , n. We shall write U K-F-sn G and denote Sn K-F (G) the set of all K-Fsubnormal subgroups of a group G. Clearly SnK-F is a subgroup functor. Remark 6.1.5. SnK-N (G) = Sn (G) for every group G. Let e be one of the functors

SnF

or

SnK-F .

Lemma 6.1.6. e is inherited, that is, if G is a group, we have 1. If H ∈ e(K) and K ∈ e(G), then H ∈ e(G). 2. If N G and U/N ∈ e(G/N ), then U ∈ e(G). 3. If H ∈ e(G) and N G, then HN/N ∈ e(G/N ).

6.1 Basic properties

237

Proof. It is obvious from the definitions that Statements 1 and 2 are fulfilled in both cases. We show that Statement 3 is satisfied when e = Sn F . Let H be an F-subnormal subgroup of G and let N be a normal subgroup of G. Proceeding by induction on |G|, we may clearly suppose that H = G. Let X be an F-normal maximal subgroup of G such that H is contained in X and H is F-subnormal in X. If N ≤ X, then HN/N is F-subnormal in X/N by induction. Since X/N is F-normal in G/N , it follows that HN/N is F-subnormal in G/N by Assertion 1.Therefore we may assume that N is not contained in X and so G = N X. By induction, H(X ∩ N )/(X ∩ N ) is F-subnormal in X/(X ∩ N ) ∼ = G/N . Hence HN/N is F-subnormal in G/N . Lemma 6.1.7. Assume that F is subgroup-closed. 1. If H is a subgroup of a group G and GF ≤ H, then H ∈ e(G). 2. If H ∈ e(G) and K ≤ G, then H ∩ K ∈ e(K), that is, e is w-inherited. n 3. If {Hi : 1 ≤ i ≤ n} ⊆ e(G), then i=1 Hi ∈ e(G). Proof. 1. It follows at once from the fact that X F ≤ GF for all subgroups X of G. 2. Let e = SnF . Proceeding by induction on |G|, we may clearly assume that H = G. Then there exists an F-normal maximal subgroup M of G such that H ≤ M and H is F-subnormal in M . Since K F ≤ GF ≤ M , it follows that M ∩K is F-subnormal in K by Assertion 1. On the other hand, H ∩K is F-subnormal in M ∩ K by induction. Therefore H ∩ K is F-subnormal in K. 3. It follows at once applying Lemma 5.1.5, as e is w-inherited, and using induction on n. Example 6.1.8. Lemma 6.1.7 (2) does not remain true if F is not subgroupclosed. Let F = LF(f ), where f (2) = S2 Q R0 Sym(3) , f (3) = S3 S2 and f(p) = ∅ for all p > 3. If G = Sym(4) and H is a Sylow of G, then 3-subgroup / SnF Alt(4) . H ∈ SnF (G) (H ≤ Sym(3) ≤ G). However H ∈ The theory of F-subnormal subgroups is relevant only in the case of persistence in intermediate subgroups. Therefore Unless otherwise stated, we stipulate that for the rest of the chapter the formation F is closed under the operation of taking subgroups. Lemma 6.1.9. Let G be a group. 1. If A is a K-F-subnormal subgroup of G, then AF is subnormal in G. 2. Let H = E K(F). Then AH = GH for every F-subnormal subgroup of G. 3. If 1 ∈ SnF (G), then Sn (G) ⊆ SnF (G). 4. If G is a p-group for some prime p and 1 ∈ SnF (Cp ), then SnF (G) = Sn (G) = S(G).

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6 F-subnormality

Proof. 1. We argue by induction on |G|. If A = G, then GF is normal in G, and the statement is true. Suppose A < G and let X be an F-normal maximal subgroup of G containing A such that A is K-F-subnormal in X. Then AF is subnormal in X by induction. Since AF ≤ X F , it follows that AF is subnormal in X F . Moreover GF is contained in X. Hence X F is subnormal in GF . This implies that X F is subnormal in G, hence so is AF . A similar argument could be applied if X is a normal subgroup of G. 2. Proceeding by induction on |G|, we may assume that A < G. We argue as in Assertion 1 and use the same notation. It follows that AH = X H for an F- normal maximal subgroup X of G such that A is F-subnormal in X. Moreover GH is contained in X as G/ CoreG (X) ∈ F ⊆ H. Now X F GH /GH belongs to H because it is subnormal in G/GH , by Statement 1 and Lemma 6.1.6 (3), and H is closed under taking subnormal subgroups. Hence X/X F ∩ GH belongs to H. It implies that X H is contained in GH . Note that every composition factor of GH /X H belongs to K(F). Therefore GH = (GH )H is contained in X H and so AH = X H = GH . 3. Since SnF is a w-inherited functor, the result follows from Lemma 5.1.4. 4. It is enough to show that 1 ∈ SnF (G). Assume that it is not true and let G be a counterexample of minimal order. Let M be a maximal subgroup of G. The minimal choice of G implies that 1 ∈ SnF (M ). Since |G/M | = p, it follows that M/M ∈ Sn F (G/M ). Hence M ∈ Sn F (G) by Lemma 6.1.6 (2). Therefore 1 ∈ SnF (G). This contradiction shows that no counterexample exists. Proposition 6.1.10. If G ∈ E K(F), then

Sn F (G)

= Sn K-F (G).

Proof. The inclusion SnF (G) ⊆ SnK-F (G) follows from the definitions. Let H ∈ SnK-F (G). We prove that H ∈ SnF (G) by induction on |G|. We may assume that H = G. Let N be a minimal normal subgroup of G. Then G/N ∈ E K(F) and HN/N ∈ SnK-F (G/N ) by Lemma 6.1.6 (3). Consequently HN/N ∈ SnF (G/N ) by induction. This implies that HN is F-subnormal in G by Lemma 6.1.6 (2). Moreover HN ∈ E K(F) by Lemma 6.1.9 (1). Assume that HN is a proper subgroup of G. Since H is K-F-subnormal in HN by Lemma 6.1.7 (2), it follows that H is F-subnormal in HN by induction. Hence H ∈ Sn F (G), as required. Hence we may suppose that G = HN for every minimal normal subgroup N of G. In particular, CoreG (H) = 1. On the other hand, H F is subnormal in G by Lemma 6.1.9 (1) and so N normalises H F by [DH92, A, 14.3]. Thus H F is normal in G. This implies that H F ⊆ CoreG (H) = 1. Consequently G/N ∈ F for each minimal normal subgroup N of G. If G ∈ F, then H is clearly F-subnormal in G. Hence we may assume that G ∈ / F and therefore G ∈ b(F). This means that G is a monolithic group, and GF = Soc(G) is the unique minimal normal subgroup of G. Let M be a proper subgroup of G such that H ∈ SnK-F (M ) and either M G or GF is contained in M . If the second condition holds, then N H = G is contained in M , contrary to supposition. Therefore M G. Since GF is not contained in M , it follows that M = 1 = H and G = Soc(G) is a simple group. Therefore

6.2 F-subnormal closure

239

G ∈ F ∩ b(F). This contradiction leads to G ∈ F and so H is F-subnormal in G. Proposition 6.1.11. Let F be a saturated formation and let G be a group with an F-subnormal subgroup H such that G = H F∗ (G). If H ∈ F, then G ∈ F. Proof. We argue by induction on |G|. Suppose that H is a proper subgroup of G and let M be an F-normal maximal subgroup of G such that H ≤ M and H is F-subnormal in M . Then M = H F∗ (M ). By induction, M ∈ F. Assume G∈ / F. By Proposition 2.3.16, M is an F-projector of G. This is impossible because G = GF M and GF is contained in M . Consequently G ∈ F.

6.2 F-subnormal closure Let F be a formation. By Lemma 6.1.7 (3), intersections of F-subnormal subgroups are F-subnormal. Therefore for any subset X of a group G, there exists a unique smallest F-subnormal subgroup of G containing X, the F-subnormal closure of X in G. We write SG (X; F) to denote this subgroup. It is clear that the same argument can be applied to K-F-subnormal subgroups. Consequently there exists a unique K-F-subnormal subgroup of G containing X, the K-F-closure of X in G. It is denoted by SG (X; K-F). When F = N, the formation of all nilpotent groups, the subgroup SG (X) = SG (X; K-F) is the subnormal closure of X in G, that is, the smallest subnormal subgroup of G containing X. The normal closure of X in G is generated by all of the conjugates of X in G and we might wonder whether or not the subnormal closure is generated by some natural subset of the set of these conjugates. Let us say that two subsets X, Y ⊆ G are strongly conjugate if they are conjugate in X, Y . It is rather clear that SG (X) must contain all strong conjugates of X. In fact, the following powerful result, due to D. Bartels, is true. Theorem 6.2.1 ([Bar77]). Let X be a subset of a group G. Then SG (X) = Y ⊆ G : Y is strongly conjugate to X in G. The first part of this section is devoted to prove this theorem. First of all, we introduce some notation. Notation 6.2.2. Let X and Y be subsets of a group G. We write: • X σ Y if X and Y are strongly conjugate in G. • X σ ∞ Y if there are subsets X = X0 , X1 , . . . , Xn = Y such that Xi σ Xi+1 for all i, 0 ≤ i < n (n natural number). • X =U Y if X and Y are conjugate in the subgroup U of G. • X =¨G Y if SG (X) = SG (Y ) = S and X =S Y . • KG (X) = Y ⊆ G : X σ Y .

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It is clear that σ ∞ and =¨G are equivalence relations on the set of all subsets of G. Lemma 6.2.3. Let X and Y be subsets of a group G such that X σ Y . Then X =¨G Y . Proof. Denote J := X, Y . Since X σ Y , there exists an element g ∈ J such that Y = X g . In particular, X J , the normal closure of X in J, is equal to J. Applying [DH92, A, 14.1], SG (X) ∩ J is subnormal in J and contains X. Since J = X J , it follows that SG (X) ∩ J = J and so SG (J) = SG (X). Analogously SG (J) = SG (Y ). Therefore X =¨G Y . Lemma 6.2.4. Let X be a subset of a group G. Then SG (X) = Y ⊆ G : X =¨G Y .

Proof. Denote A = Y ⊆ G : X =¨G Y . Then A = X g : g ∈ SG (X) by Lemma 6.2.3. It is clear that A is normal in SG (X). Hence A is subnormal in G. Since A contains X, it follows that A = SG (X). By Lemma 6.2.3, X σ Y implies X =¨G Y . Hence KG (X) ⊆ SG (X) for every subgroup X of G. Lemma 6.2.5. Let X be a subset of a group G. Then 1. KG (X) = Y ⊆ G : X σ ∞ Y . 2. X σ ∞ X g for all g ∈ KG (X). Proof. 1. It is clear that KG (X) ≤ Y ⊆ G : X σ ∞ Y . Let Y ⊆ G such that X σ ∞ Y . We have to show that Y ⊆ KG (X). There is a natural number n and there are subsets X = X0 , X1 , . . . , Xn = Y such that Xi σ Xi+1 for all i, 0 ≤ i < n. Suppose inductively that we have already shown that X0 , X1 , . . . , Xn−1 are contained in KG (X). Since KG (X) = Z : X σ Z, we may assume that n > 1. There exists an element g ∈ X0 , X1 , . . . , Xn−1 ⊆ KG (X) such that X g = X0g = Xn−1 . Then Y ≤ KG (X g ), and since σ is G-invariant, it follows that KG (X g ) = KG (X)g = KG (X), and the induction step is complete. 2. Let Y be a subset of G. Let y be an element of Y ∪ Y −1 and assume that X σ Y . Then X y σ Y y and Y y σ Y , whence X y σ ∞ X. If g ∈ KG (X), then g = g1 · · · gt , where gi ∈ Yi ∪ Yi−1 , X σ Yi , for all i, 1 ≤ i ≤ t. If t = 1, then X g1 σ ∞ X by the above argument. Suppose inductively −1 −1 that X (g1 ···gt−1 ) σ ∞ X. Then X gt σ ∞ X (g1 ···gt−1 ) because X gt σ ∞ X. Hence ∞ g Xσ X . Proposition 6.2.6. For any subset X of a group G, the following statements are equivalent:


241

1. KG (X) = SG (X). 2. The equivalence relations σ ∞ and =¨G coincide when restricted to the conjugacy class of X in G. Proof. Assume that KG (X) = SG (X). Then X =¨G Y implies that Y = X g for some g ∈ SG (X). By Lemma 6.2.5 (2), X σ ∞ Y . Since =¨G is a transitive relation, X σ ∞ Y implies X =¨G Y by Lemma 6.2.3. Thus Statement 2 holds. Conversely, assume Statement 2. Since KG (X) = Y ⊆ G : X σ ∞ Y by Lemma 6.2.5 (1), it follows that KG (X) = Y ⊆ G : X =¨G Y , which is equal to SG (X) by Lemma 6.2.4. Lemma 6.2.7. Let X0 and X1 be subsets of a group G such that X0 ⊆ X1 . Then KG (X0 ) ≤ KG (X1 ). Proof. Let t be an element of G such that t ∈ X0 , X0t . Then obviously t ∈ X1 , X1t . Hence X0 σ Y for some Y ⊆ G implies that there is a subset W of G such that Y ⊆ W and X1 σ W . The lemma follows by definition of KG (X1 ). Lemma 6.2.8. Let G be a group and let N be a normal subgroup of G. Let X ⊆ G and let Y1 /N be a subset of G/N such that XN/N σ Y1 /N . Then there exists a subset Y of G such that X σ Y and Y1 = Y N . Proof. Let A := {V ⊆ G : V N/N = Y1 /N and X =¨G V }. Since XN/N σ Y1 /N , it follows that XN/N and Y1 /N are conjugate in SG/N (XN/N ) = SG (X)N/N . Hence Y1 = X z N for some z ∈ SG (X). It is clear that X =¨G X z and so X z = V ∈ A. This shows that A is non-empty. Let W be an element of A such that X, W has minimal order. Since XN/N σ W N/N , there exists an element t ∈ X, W such that W N/N = X t N/N = Y1 /N . It is clear that X =¨G X t . Hence X t belongs to A. The minimal choice of X, W implies that X, X t = X, W and so X σ X t ( = Y ). Corollary 6.2.9. For any subset X of a group G and for any N G, KG (X)N/N = KG/N (XN/N ). Proposition 6.2.10. For any subset X of a group G, the relations σ ∞ and = ¨G coincide on the conjugacy class of X in G. Proof. Assume that the result is false, and let (G, X) be a counterexample with |G| + |X| as small as possible. Clearly X = ∅ and the conjugacy class of X in G splits into σ ∞ -equivalence classes; we denote the set of these equivalence classes by Ω. Since X σ ∞ Y implies X =¨G Y for all Y ⊆ G by Lemma 6.2.3, it follows from our choice of (G, X) that Ω contains at least two elements. It is clear that G acts transitively by conjugation on Ω in the obvious way.

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Let K = KG (X). By Proposition 6.2.6, K is a proper subgroup of G. For any non-trivial normal subgroup N of G, the relations σ ∞ and =¨G coincide on the conjugacy class of XN/N in G/N by minimality of G. Hence KG/N (XN/N ) = SG/N (XN/N ) = SG (X)N/N by Proposition 6.2.6, and so KN/N = KG/N (XN/N ) is subnormal in G/N . In particular, KN is subnormal in G. Suppose that Z = KN is a proper subgroup of G. Then K = KZ (X) = SZ (X) by the choice of G. Hence K is subnormal in Z and so is in G. Proposition 6.2.6 implies that the relations σ ∞ and =¨G coincide on the conjugacy class of X in G. This is a contradiction against the choice of (G, X). Consequently, G = KN for any non-trivial normal subgroup N of G. From this we conclude that CoreG (K) = 1 and X G , the normal closure of X in G, is equal to G. Let p be a prime dividing |X| and let Q be a Sylow p-subgroup of X. By Lemma 6.2.7, KG (Q) is contained in K. Suppose that Q is a proper subgroup of X. The minimal choice of (G, X) implies that KG (Q) is subnormal in G. Let N be a minimal normal subgroup of G. By [DH92, A, 14.3], N normalises KG (Q). Since G = KN , it follows that KG (Q)G = KG (Q)K is a subgroup of K. Hence KG (Q)G is contained in CoreG (K) = 1. This contradiction shows that Q = X and X is a p-group. For any subgroup U of G, let [U ] denote the set [U ] = {ω ∈ Ω : there exists X g ∈ ω such that X g ⊆ U }. The following statements hold: 1. For any proper subgroup U of G and for every Sylow p-subgroup P of U , [U ] = [P ]. It is clear that [P ] ⊆ [U ]. Conversely, let ω ∈ [U ] and let Y ∈ ω be a subset of U . Let L = SU (Y ). Since L is subnormal in U , it follows that L ∩ P is a Sylow p-subgroup of L. Hence Y z is contained in P for some z ∈ L. It is clear that Y =¨G Y z . Since =¨G and σ ∞ coincide on the conjugacy class of Y in U by induction, we have that Y σ ∞ Y z . Hence ω ∈ [P ]. 2. [U ] is a proper subset of Ω for any proper subgroup U of G. Assume that [U ] = Ω. Then Ω = [P ] for some Sylow p-subgroup P of U . Since Ω = ∅, it follows P = 1 and so Z(P ) = 1. Note that if x ∈ Z(P ) and ω ∈ Ω, then ω x = ω because x centralises an element of ω. Hence Z(P ) acts trivially on Ω. Since Ω = [P ] = [P g ] for all g ∈ G, it follows that Z(P g ) acts trivially on Ω. This implies that N = Z(P )G acts trivially on Ω. Let ω0 be the element of Ω such that X ∈ ω0 . If z ∈ K, then X z ∈ ω0 by Lemma 6.2.5 (2). Hence ω0z = ω0 . Let g be an element of G. There exist z ∈ K and n ∈ N such that g = zn. It follows that ω0g = ω0 and so X g σ ∞ X for all g ∈ G. Therefore KG (X) = X G = G. This contradiction shows that [U ] = Ω. 3. Any maximal subgroup M of G such that [M ] = ∅ contains a Sylow p-subgroup of G. Let P be a Sylow p-subgroup of M . By Statement 1 and Statement 2, [M ] = [P ] = Ω. Note that if ω ∈ [P ] and g ∈ M ∪ NG (P ), then ω g ∈ [P ].


243

Hence M, NG (P ) is not transitive on Ω. Therefore M, NG (P ) is a proper subgroup of G. In particular, NG (P ) ≤ M and so P is a Sylow p-subgroup of G. 4. Any Sylow p-subgroup P of G is contained in a unique maximal subgroup of G. Obviously G is not a p-group. Let P be contained in L ∩ M , where L and M are maximal subgroups of G. Then [L] = [M ] = [P ] by Statement 1 and [P ] = Ω by Statement 2. This implies that L, M is not transitive on Ω. Hence G = L, M and L = M . 5. X is contained in a unique maximal subgroup of G. Suppose that X is contained in at least two maximal subgroups L and M of G. Choose L and M such that the Sylow p-subgroups of L ∩ M have maximal order. There exist Sylow p-subgroups R and S of L and M respectively such that R ∩ S is a Sylow p-subgroup of L ∩ M containing X. By Statement 4, R and S are Sylow p-subgroups of G. Moreover R = S by Statement 4. From this we conclude that R ∩ S is a proper subgroup of R1 = NR (R ∩ S). Since N = NG (R ∩ S) is a proper subgroup of G, this implies N is contained in M by our choice of M and L. The same argument with L and S replacing M and R yields N ≤ L. But then R ∩ S < R1 ≤ M ∩ L and R ∩ S is a Sylow p-subgroup of M ∩ L. This contradiction proves Statement 5. Now from Statement 5 we deduce the final contradiction, thus proving the lemma. We know that K is a proper subgroup of G. Let M be the unique maximal subgroup of G containing X. Since X G = G, it follows that M = NG (M ). Let g ∈ G\M . Then G = X, X g . This implies X σ X g and therefore we have G = K. Combining Proposition 6.2.6 and Proposition 6.2.10, we have: Theorem 6.2.11. SG (X) = KG (X) for any subset X of G. g Let X be a subset of G and g ∈ G such that g ∈ X, X . Then g ∈ g SG (X), SG (X) ≤ SG SG (X) = SG (X). Hence the following result is true. Corollary 6.2.12. SG (X) = g ∈ G : g ∈ X, X g .

Let H be a subgroup of a group G. If A is a subgroup of G, containing H, then HAN is a subnormal subgroup of A containing H. Now if g ∈ G and g ∈ H, H g = J, then the normal closure of H in J is equal to J. The subnormality of HJ N in J implies that J = HJ N and g ∈ HH, H g N . Moreover there exists z ∈ H, H g N such that J = H, H z . Thus we have shown the following: Theorem 6.2.13. Let H be a subgroup of a group G. Then SG (H) = H g : g ∈ H, H g N = g ∈ G : g ∈ HH, H g N . The descriptions of the subnormal closure provide a proof of the following subnormality criterion due to Wielandt.

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Theorem 6.2.14 ([Wie74]). Let H be a subgroup of a group G. The following statements are pairwise equivalent: 1. H is subnormal in G. 2. H is subnormal in H, g for all g ∈ G. 3. H is subnormal in H, H g for all g ∈ G. 4. If g ∈ G and g ∈ H, H g , then g ∈ H. Moreover, they are equivalent to: 5. If g ∈ G and g ∈ H, H g N , then g ∈ H. Remark 6.2.15. Theorem 6.2.11 does not provide a description of the N-subnormal closure. Let G = Alt(5) and H = {1}. Then SG (H) = H and SG (H; N) = G. If G is a soluble group, then SG (H) = SG (H; K-N) = SG (H; N) by Proposition 6.1.10. In this context, the following conjecture arises. Conjecture 6.2.16 (K. Doerk). Let F be a saturated formation and π = char F. closure of Given a subgroup H of a soluble group G ∈ Sπ , the F-subnormal H in G is the subgroup SG (H; F) = g ∈ G : g ∈ HH, H g F . A. Ballester-Bolinches and M. D. Pérez-Ramos [BBPR91] confirmed Conjecture 6.2.16. In fact, they showed that the conjecture is valid for groups with soluble F-residual, that is, groups in the class SF. Henceforth in the rest of the section F = LF(F ) will denote a subgroup-closed saturated formation of characteristic π. The proof of Doerk’s conjecture depends heavily on the following extension of Theorem 6.2.14 to subgroup-closed saturated formations. Theorem 6.2.17 ([BBPR91]). For a subgroup H of a π-group G ∈ SF, the following statements are pairwise equivalent: 1. H is F-subnormal in G 2. H is F-subnormal in H, x for every x ∈ G. 3. H is F-subnormal in H, H x for every x ∈ G. 4. If T is a subgroup of G such that T is contained in H, T F , then T is contained in H. 5. If x ∈ G and x ∈ H, xF , it follows that x ∈ H. 6. If x ∈ G and x ∈ H, H x F , it follows that x ∈ H. Proof. 3 implies 1. We argue by induction on |G|. We can assume that GF = 1 by Lemma 6.1.7 (1). Let N be a minimal normal subgroup of G such that N is contained in GF . By induction, HN/N is F-subnormal in G/N and so HN is F-subnormal in G by Lemma 6.1.6 (2). If HN were a proper subgroup of G, then H would be F-subnormal in HN ∈ SF by induction. Applying


245

Lemma 6.1.6 (1), H is F-subnormal in G and the implication is true. Hence we can suppose G = HN and G = H. Since N is soluble, H is a maximal subgroup of G. If H is a normal subgroup of G, then H is F-subnormal in G because G ∈ E K(F). If H is not normal in G, there exists an element x ∈ G such that H = H x . Then G = H, H x and H is F-subnormal in G by Statement 3. By Lemma 6.1.7 (2), 1 implies 2 and 2 implies 3. Consequently, 1, 2, and 3 are pairwise equivalent. It is clear that 4 implies 5 and 5 implies 6 because X F ≤ Y F ≤ GF if X ≤ Y ≤ G. 1 implies 4. Suppose that H is F-subnormal in G and T is a subgroup of G such that T is contained in H, T F . Then H, T = HH, T F . If H were a proper subgroup of H, T , there would exist an F-normal maximal subgroup M of H, T containing H. Since H, T F ≤ M , we would have M = H, T . This contradiction yields H = H, T and T is contained in H. To complete the proof we now show that 6 implies 1. We proceed by induction on |G|. Let x ∈ G and T = H, H x . If T is a proper subgroup of G, then by induction H is F-subnormal in T . Since 3 is equivalent to 1, we may assume that T = G for some x ∈ G. By Lemma 6.1.7 (1), HGF is F-subnormal in G. Hence, if HGF were a proper subgroup of G, then H would be F-subnormal in HGF by induction. Therefore H would be F-subnormal in G by Lemma 6.1.6 (1). Therefore we may suppose G = H, H x = HGF = HH, H x F . In particular, x = ht for some h ∈ H and t ∈ H, H x F = H, H t F . Applying Statement 6, it follows that t ∈ H. Hence x ∈ H and H = G is F-subnormal in G. The circle of implications is now complete. If H is a subgroup of a group G, denote TG (H; F) = x ∈ G : x ∈ HH, H x F . Lemma 6.2.18. If N is a normal subgroup of a group G and H is a subgroup of G, then TG/N (HN/N ; F) = TG (H; F)N/N. ¯ = G/N . It is clear that TG (H; F) = Proof. Denote with bars the images in G ¯ H ¯ g¯ F . TG (H; F)N/N is contained in TG/N (HN/N ; F). Consider now g¯ ∈ H, Then there exists an element z ∈ H, H g F such that z¯ = g¯. Hence the set L = {z ∈ H, H g F : z¯ = g¯} is non-empty. Let t ∈ L such that H, H t F has minimal order. Then H, H g F N = H, H t F N and t = xn for some ¯ = t¯. Hence x ∈ H, H t F and n ∈ N . It is clear that x ∈ H, H g F and x x F t F x ∈ L. The minimal choice of t implies that H, H = H, H . Therefore g¯ ∈ TG (H; F) and the equality holds. Theorem 6.2.19 ([BBPR91]). Let G be a π-group with soluble F-residual. Let H be a subgroup of G. Then SG (H; F) = TG (H; F) = T ≤ G : T ≤ HH, T F .

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6 F-subnormality

Proof. Write S = TG (H; F). If L is an F-subnormal subgroup of G containing H, then S is contained in L by Theorem 6.2.17. Thus the first equality holds if we prove that S is F-subnormal in G. We argue by induction on |G|. Since GF is soluble and G is a π-group, it follows that GF is a proper subgroup of G. Of course, it may be assumed that GF = 1. Let N be a minimal normal subgroup of G contained in GF . Then, by Lemma 6.2.18, SN/N = TG/N (HN/N ; F). Hence SN/N is F-subnormal in G/N . By Lemma 6.1.6 (2), SN is F-subnormal in G. Suppose that SN = X is a proper subgroup of G. Then S = TX (H; F) is F-subnormal in X by induction. By Lemma 6.1.6 (1), S is F-subnormal in G. Therefore we must have G = SN and thus S is a maximal subgroup of G because N is abelian and S is a proper subgroup of G. This argument also yields CoreG (S) = 1. Therefore G is a primitive group of type 1 and N = Soc(G) = CG (N ). Suppose, by way of contradiction, that S is not F-subnormal in G and let us choose H of minimal order among those subgroups of G such that TG (H; F) is not F-subnormal in G. If M is a maximal subgroup of H satisfying H = TH (M ; F), then H ≤ TG (M ; F) and TG (M ; F) is F-subnormal in G. Consequently S is contained in TG (M ; F) and S is F-subnormal in G, contrary to the choice of H. Therefore, each maximal subgroup of H is F-subnormal in H. This implies that every primitive epimorphic image of H belongs to F. Hence H ∈ F because F is saturated. Let N0 be a minimal H-invariant subgroup of N . Put A = HN0 . If A = G, then H = S. By Theorem 6.2.17, S is F-subnormal in G, contrary to supposition. Hence A is a proper subgroup of G. Suppose that A is not an F-group. Then N0 = AF and A = SA (H; F) = TA (H; F) ≤ S. This is a contradiction. Therefore A ∈ F. Let SocH (N ) be the product of all minimal H-invariant subgroups of N . Since F is a formation, / F and let L be an it follows that H SocH (N ) ∈ F. Suppose that HN ∈ F-maximal subgroup of HN containing H SocH (N ). Then HN = L(HN )F and L ∩ (HN )F = 1 by Theorem 4.2.17. But then, since (HN )F = 1, we have that 1 = (HN )F ∩ SocH (N ) ≤ (HN )F ∩ L, which is a contradiction. Therefore HN ∈ F. Let 1 = N0 N1 N2 · · · Nr = N be an Hcomposition series of N . Then H CH (Ni /Ni−1 ) ∈ F (p), for i = 1, . . . , r, and p the prime dividing |N |. Hence H F (p) ≤ {CH (Ni /Ni−1 ) : i = 1, . . . , r} and so that H F (p) / CH F (p) (N ) = H F (p) is a p-group by [DH92, A, 12.4]. Therefore H ∈ Sp F (p) = F (p). Consider now g ∈ H, H g F \ H. It is clear that H is a proper subgroup / F. Obviously T = HT F is contained in S. Denote T F = R. of T = H, H g ∈ Let 1 = K0 K1 · · · Ks = N be a T -composition series of N . If every T -chief factor Kj /Kj−1 , j ∈ {1, . . . , s}, is centralised by R, it follows, arguing as above, that R is a p-group and since H ∈ F (p), it follows that T ∈ F (p) ⊆ F, contrary to the choice of T . Consequently, there exists a T -chief factor Ki /Ki−1 , i ∈ {1, . . . , s}, such that R is not contained in CT (Ki /Ki−1 ). Write L = Ki T = Ki (RH), and denote with bars the images ¯i ≤ L ¯ F , because otherwise K ¯i ∩ L ¯ F = 1, and ¯ = L/Ki−1 . We have that K in L F ¯ by ¯ ¯ ¯ ¯ iR ¯ then R ≤ CL¯ (Ki ), contradicting our choice of Ki . Therefore L = K

6.3 Lattice formations

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¯ < |G|. Then SL¯ (H, ¯ F) = TL¯ (H, ¯ F) = L ¯ by Proposition 2.2.8. Assume that |L| ¯ F) = TL (H; F)Ki−1 /Ki−1 . Hence induction. Applying Lemma 6.2.18, TL¯ (H; TL (H; F)Ki−1 = Ki RH. If TL (H; F) ∩ Ki = 1, then TL (H; F) = RH and Ki = Ki−1 . This is a contradiction. Thus 1 = Ki ∩ TL (H; F) ≤ S ∩ N , which ¯ = |G|, that is, G = N T and S = T = H, g. is also impossible. Therefore |L| Let n ∈ N such that [H, n] = 1 and consider M = H, H ng . Since G = HGF , it follows that M < G, because otherwise ng ∈ S, and so n ∈ S, contradicting our supposition. Let L be a maximal subgroup of G containing M . If L = S, then H n ≤ S. Hence 1 = [h, n] = h−1 hn ∈ S ∩ N , for some h ∈ H. This is impossible. If N were contained in L, then L would contain H g , and so S = H, H g ≤ L. This would be a contradiction. Hence CoreG (L) = 1 and L = H(L ∩ GF ) = HLF . Our choice of G implies that L = TL (H; F) ≤ S, and we have reached the desired contradiction. Therefore TG (H; F) is F-subnormal in G and SG (H; F) = TG (H; F). On the other hand, it Fis clear that SG (H; F) is contained in LG (H; F) = T ≤ G : T ≤ HH, T . Now, if K is an F-subnormal subgroup of G containing H and T is a generator of LG (H; F), it follows that T ≤ KK, T F . Thus, if t ∈ T , then t = kt xt with kt ∈ K, xt ∈ K, T F . Denote by R = xt : t ∈ T . Then K, T = K, R and R ≤ K, RF . Since K is F-subnormal in G, it follows that R ≤ K by Theorem 6.2.17. Consequently T is contained in K and LG (H; F) ≤ K. Since SG (H; F) is F-subnormal in G, SG (H; F) contains LG (H; F) and the proof of the theorem is complete. Open question 6.2.20. Let F be a saturated formation of characteristic π. Is it possible to find a useful description for the F-subnormal closure of a subgroup H of a π-group G?.

6.3 Lattice formations One of the most striking results in the theory of subnormal subgroups is the celebrated “join” theorem, proved by H. Wielandt in 1939: the subgroup generated by two subnormal subgroups of a finite group is itself subnormal. As a result, the set of all subnormal subgroups of a group is a sublattice of the subgroup lattice. Let F be a formation. One might wonder whether the set of F-subnormal subgroups of a group forms a sublattice of the subgroup lattice. The answer is in general negative. Example 6.3.1 ([BBPR91]). Let F be the formation of all 2-nilpotent groups and G = Sym(4). By [DH92, A, 10.9], G has an irreducible and faithful module V over GF(3). Let R = [V ]G be the corresponding semidirect product. If P is a Sylow 2-subgroup of G, then V P is an F-normal maximal subgroup of R. Since V P ∈ F, it follows that P is F-subnormal in R. However, if x ∈ G \ NR (P ), then G = P, P x is not F-subnormal in R.

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Therefore the following question naturally arises: Which are the formations F for which the set SnF (G) is a sublattice of the subgroup lattice of G for every group G? This question was first proposed by L. A. Shemetkov in his monograph [She78] in 1978 and it appeared in the Kourovka Notebook in 1984 as Problem 9.75 [MK84]. In 1992, A. Ballester-Bolinches, K. Doerk, and M. D. Pérez-Ramos gave in [BBDPR92] the answer to that question in the soluble universe for saturated formations. On the other hand, O. H. Kegel [Keg78] showed that if F is a subgroupclosed formation such that FF = F, then the set of all K-F-subnormal subgroups of a group G is a sublattice of the subgroup lattice of G for every group G. He also asks for other formations enjoying the lattice property for K-F-subnormal subgroups. In 1993, A. F. Vasil’ev, S. F. Kamornikov, and V. N. Semenchuk [VKS93] published the extension of the lattice results of [BBDPR92] to the general finite universe. They also proved that the problems of O. H. Kegel and L. A. Shemetkov are equivalent for saturated formations. Our objective in this section is to give a full account of the above results. In the sequel, F will be a (subgroup-closed) formation. Definition 6.3.2. We say that F is a lattice (respectively, K-lattice) formation if the set of all F-subnormal (respectively, K-F-subnormal) subgroups is a sublattice of the lattice of all subgroups in every group. The next result provides a criterion for a saturated formation to be a lattice formation. Theorem 6.3.3. Any two of the following assertions about a saturated formation F are equivalent: 1. F is a lattice formation. 2. If A and B are F-subnormal F-subgroups of a group G, then A, B is an F-subgroup of G. 3. F is a Fitting class and the F-radical GF of a group G contains every F-subnormal F-subgroup of G. Proof. Assume, arguing by contradiction, that F is a lattice formation such that F does not satisfy Statement 2. Let G be a group of minimal order among the groups X having two F-subnormal F-subgroups H and K such that H, K is not an F-group. Among the pairs (H, K) of F-subnormal F-subgroups of G such that H, K ∈ / F, we choose a pair (A, B) with |A|+|B| maximal. Because of Lemma 6.1.7 (2) and the choice of G, it must be G = A, B. Moreover if N is a minimal normal subgroup of G, it follows that G/N ∈ F because G/N is generated by the F-subnormal F-subgroups AN/N and BN/N . Therefore G is in the boundary of F. In particular, G is a monolithic primitive group.


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Put N = Soc(G) = GF . By Lemma 6.1.7 (2) and Proposition 6.1.11, AN = AF ∗ (AN ) and BN = BF ∗ (BN ) are F-groups. Applying Lemma 6.1.7 (1), we have that AN and BN are F-subnormal subgroups of G. The choice of the pair (A, B) yields N ≤ A ∩ B. Let H be a minimal supplement to N in G. By [DH92, A, 9.2(c)], we have H ∩ N ≤ Φ(H); since H/(H ∩ N ) ∼ = HN/N = G/N ∈ F, it follows that H ∈ EΦ F = F. On the other hand, A = N (A ∩ H) and B = N (B ∩ H). By Lemma 6.1.6 (1) and Lemma 6.1.7 (1), A ∩ H is F-subnormal in G. Hence the normal closure (A ∩ H)H of A ∩ H in H is F-subnormal in G. Note that N (A ∩ H)H is normal in G and A is contained in N (A ∩ H)H . Therefore G = N (A ∩ H)H (B ∩ H) . Since (A ∩ H)H and B∩H are F-subnormal in G, it follows that (A ∩ H)H (B ∩ H) is an F-subnormal F-subgroup of G. Applying Proposition 6.1.11, G ∈ F and we have reached the desired contradiction. Therefore G ∈ F. We have proved that 1 implies 2. 2 implies 3. Suppose that G is a group such that G = N1 N2 with Ni G and Ni ∈ F for i = 1, 2. Then Ni ∈ E K(F), i = 1, 2, and so G ∈ E K(F). Applying Proposition 6.1.10, Ni are F-subnormal in G for i = 1, 2. By Statement 2, G ∈ F and we have shown that F is N0 -closed. Therefore F is a Fitting class because F is subgroup-closed. Let G be a group and A = X ∈ F : X is F-subnormal in G. Then A is normal in G and A ∈ F by Statement 2. Hence A is contained in the F-radical GF of G. 3 implies 1. Suppose that F is not a lattice formation and derive a contradiction. Let G be a counterexample with least possible order. Then G has two F-subnormal subgroups U and V such that U, V is not F-subnormal. If N is a minimal normal subgroup of G, then U, V N/N is F-subnormal in G/N by Lemma 6.1.6 (3). Hence U, V N is F-subnormal in G by Lemma 6.1.6 (2). Assume that U, V N is a proper subgroup of G. Then U and V are F-subnormal in U, V N by Lemma 6.1.7 (2). Hence U, V is F-subnormal in U, V N by the minimal choice of G. Therefore U, V is F-subnormal in G, contrary to supposition. Hence G = U, V N for every minimal normal subgroup N of G. On taking N contained in CoreG (U, V ), if this is non-trivial, we can conclude G = U, V . This is not possible. Thus CoreG (U, V ) = 1. On the other hand, U F and V F are subnormal in G by Lemma 6.1.9 (1) and so N normalises U F , V F by [DH92, A, 14.3 and 14.4]. Hence U F , V F G = F F U,V ≤ CoreG (U, V ) = 1. This yields U ∈ F and V ∈ F. By U , V Statement 3, U and V are contained in GF and so G = GF N . On taking N ≤ GF , we conclude that G = GF ∈ F. In particular, U, V is F-subnormal in G. This is the final contradiction. Corollary 6.3.4. Let F be a saturated lattice formation. If G ∈ GF = X ∈ F : X is F-subnormal in G.

E K(F),

then

Proof. Applying Proposition 6.1.10, every subnormal subgroup of G is F- subnormal. Hence GF ≤ X ∈ F : X is F-subnormal in G and the equality holds by Theorem 6.3.3 (3).

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Remark 6.3.5. If F = Sp for some prime p, there exist groups G such that 1 = X ∈ F : X is F-subnormal in G < GF = Op (G) < GF = Op (G). A well-known result of Baer asserts that if p is a prime, then a p-element x of a group G lies in Op (G) if, and only if, any two conjugates of x generate a p-subgroup of G. As a consequence a subgroup H of a group G is contained in the Hall π-subgroup of F(G), π a set of primes, if, and only if, H, H g is a nilpotent π-group for every g ∈ G ([DH92, A, 14.11]). This result does not hold for saturated Fitting formations. For instance, if F is the class of all groups with nilpotent length at most 2 and G = Sym(4), then H, H g ∈ F for every subgroup H generated by a transposition and every g ∈ G. However H is not contained in Alt(4) = GN2 . Our next theorem shows that lattice formations F do enjoy the above property in groups with soluble residual. This result was proved in the soluble universe in [BBDPR92]. Theorem 6.3.6. Let F be a lattice formation of characteristic π. For a subgroup H of a π-group G ∈ SF, the following statements are equivalent: 1. H is contained in the F-radical GF of G; 2. H, H g is an F-group for every g ∈ G. Proof. 1 implies 2. If H is contained in GF , then H, H g ≤ GF for all g ∈ G. Hence H, H g is an F-group for all g ∈ G. 2 implies 1. By Lemma 6.1.7 (1), the subgroup H is F-subnormal in H, H g for all g ∈ G. By Theorem 6.2.17, H is F-subnormal in G. Since H ∈ F, it follows that H ≤ GF by Theorem 6.3.3 (3). Lemma 6.3.7. Let F be a K-lattice formation. Then F is a lattice formation. Proof. Assume the result is false and let G be a group of minimal order among the groups X for which SnF (X) is not a sublattice of the subgroup lattice of X. Then G has two F-subnormal subgroups U and V such that U, V is not F-subnormal in G. Let N be a minimal normal subgroup of G. Then U, V N is F-subnormal in G by Lemma 6.1.6 (3) and Lemma 6.1.6 (2). Put / H, then H = E K(F). Applying Lemma 6.1.9 (2), U H = V H = GH . If G ∈ N ≤ GH and so U, V = U, V N . This contradiction yields G ∈ H and so Sn F (G) = Sn K-F (G) by Proposition 6.1.10. We have reached a contradiction. Therefore F is a lattice formation. Lemma 6.3.8. Let F be a saturated K-lattice formation. Then every KF-subnormal F-subgroup of a group G is contained in the F-radical of G. Proof. We proceed by induction on |G|; we may clearly suppose that G ∈ / F. Let 1 = H be a K-F-subnormal subgroup of G such that H ∈ F. Let N be a minimal normal subgroup of G. By Lemma 6.1.6 (3), H N/N is KF-subnormal in G/N. Applying induction HN/N ≤ (G/N )F = A/N . If A is a


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proper subgroup of G, then H is contained in AF because H is K-F-subnormal in A by Lemma 6.1.7 (2). Since AF is a normal F-subgroup of G, it follows that AF ≤ GF . Therefore H is contained in GF . There remains the possibility that G/N ∈ F for all minimal normal subgroups N of G. Then G is in the boundary of F and so G is a primitive group and N = GF is the unique minimal normal subgroup of G. Assume that N ∈ / F. Then (HN )F ∩ N = 1. This implies that (HN )F ≤ CG (N ) ≤ N . On the other hand, H is a proper subgroup of G and therefore H is contained in a proper subgroup M of G such that either M G or GF ≤ M . In both cases N ≤ M and so HN is a proper subgroup of G. By induction, H ≤ (HN )F ≤ (HN )F ∩ N = 1. This contradiction implies that N ∈ F. Therefore G ∈ E K(F) and so H is F-subnormal in G by Proposition 6.1.10. In this case, H is contained in GF by Lemma 6.3.7 and Theorem 6.3.3 (3). This is the final contradiction. Theorem 6.3.9. Let F be a saturated formation. Then F is a lattice formation if and only if F is a K-lattice formation. Proof. Only the necessity of the condition is in doubt. Assume, arguing by contradiction, that F is a lattice formation and there exists a group G for which SnK-F (G) is not a sublattice of the subgroup lattice of G. Furthermore let G be a group of smallest order with this property. Then G has two KF-subnormal subgroups U and V such that U, V is not K-F-subnormal in G. Let N be a minimal normal subgroup of G. Since U N/N and V N/N are K-F-subnormal in G by Lemma 6.1.6 (3), it follows that U, V N/N is K-F-subnormal in G/N by the minimal choice of G. Hence U, V N is KF- subnormal in G by Lemma 6.1.6 (2). If U, V N were a proper subgroup of G, then U, V would be K-F-subnormal in U, V N by minimality of G (note that U and V are K-F-subnormal in U, V by Lemma 6.1.7 (2)). Applying Lemma 6.1.6 (1), U, V is K-F-subnormal in G. This contradiction yields G = U, V N for every minimal normal subgroup N of G. In particular, CoreG (U, V ) = 1. By Lemma 6.1.9 (1), U F and V F are subnormal subgroups of G. Therefore U F , V F = D is subnormal in G and Soc(G) ≤ NG (D) by [DH92, A, 14.3 and 14.4]. Hence DG = D U,V N = D U,V ≤ U, V . This means that DG ≤ CoreG (U, V ) = 1. Hence U and V belong to F. Applying Lemma 6.3.8, U, V is contained in GF . Hence G = GF N for every minimal normal subgroup N of G. In particular, G = GF and U, V is K-F-subnormal in G by Lemma 6.1.7 (1). This is the desired contradiction. Lemma 6.3.10. Let {Fi : i ∈ I} be a family of saturated lattice formations. Then F = i∈I Fi is a saturated lattice formation. Proof. It is sufficient to see that F satisfies Statement 3 of Theorem 6.3.3. It is clear that F is a saturated Fitting formation. Moreover X Fi is contained in X F for every group X, i ∈ I. Hence every F-subnormal subgroup is Fi -subnormal for all i ∈ I by Lemma 6.1.7 (1). Let G be a group and let H be an F-subnormal F-subgroup of G. Then H is an Fi -subnormal Fi -subgroup of G for every i ∈ I. By Theorem 6.3.3 (3),

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H is contained in i∈I GFi , which is a normal F-subgroup of G because Fi is subgroup-closed for every i ∈ I. Therefore H is contained in GF and F is a lattice formation. Lemma 6.3.11. Let I be a non-empty set. For each i ∈ I, let Fi be a subgroup-closed saturated lattice formation. Assume that π(Fi ) ∩ π(Fj ) = ∅ for all i, j ∈ I, i = j. Then F = Xi∈I Fi is a subgroup-closed saturated lattice formation. Proof. By Remark 2.2.13, F is a subgroup-closed saturated formation. Assume that F does not satisfy Statement 2 of Theorem 6.3.3 and derive a contradiction. Let G be a counterexample of minimal order. Then G has two F-subnormal F-subgroups A and B such that A, B is not an F-group. Then obviously A = 1 and B = 1. Observe that A, B and any epimorphic image of G inherits the conditions of G. Therefore G = A, B and G/N ∈ F for every minimal normal subgroup N of G. Since G ∈ / F, it follows that N = GF is the unique minimal normal subgroup of G and CG (N ) ≤ N . Since A is F-subnormal in AN by Lemma 6.1.7 (2) and N is a quasinilpotent normal subgroup of G, it follows that AN belongs to F by Proposition 6.1.11. Hence there exists i ∈ I such that N ∈ Fi . Moreover, CG (N ) ≤ N forces AN ∈ Fi . The same arguments can be applied to B. We then conclude that AN , BN ∈ Fi . Since G/N ∈ F and F = Xi∈I Fi , it follows that G/N has a normal π(Fi )-Hall subgroup. Since AN/N and BN/N are π(Fi )-groups, we have that G/N is a π(Fi )-group. In particular, G is a π(Fi )-group and so A and B are Fi -subnormal Fi -subgroups of G. Therefore G = A, B ∈ Fi ⊆ F by Theorem 6.3.3 (2). This contradiction confirms that F is a lattice formation. Let Z be a class of groups. A group G is called S-critical for Z, or simply Z- critical, if G is not in Z but all proper subgroups of G are in Z. Critical groups associated with some classes of groups will play a central role in Section 6.4. Lemma 6.3.12. Let F be a saturated Fitting formation. Assume that each of the following conditions holds: 1. F = Sp F for all p ∈ char F. 2. F is an F2 -normal Fitting class. 3. Every F-critical group G with Φ(G) = 1 is either cyclic or G is monolithic such that Soc(G) is non-abelian and G/ Soc(G) is a cyclic group of prime power order. Then F is a lattice formation. Proof. It will be established that every F-subnormal F-subgroup H of a group G is contained in the F-radical of G. This will be accomplished by induction on |G|, which we suppose greater than 1. Obviously we may suppose G ∈ / F and 1 = H < G. Let N be a minimal normal subgroup of G. Then HN/N is an


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F-subnormal F-subgroup of G/N by Lemma 6.1.6 (3). By induction, HN/N is contained in (G/N )F = T /N . If T is a proper subgroup of G, then H is F-subnormal in T by Lemma 6.1.7 (2) and H is contained in TF by induction. Since T is normal in G, it follows that TF ≤ GF and H is contained in GF . Hence we may assume that G/N ∈ F for every minimal normal subgroup N of G. This implies that G is a monolithic primitive group and N = GF is the unique minimal normal subgroup of G and CG (N ) ≤ N . By Lemma 6.1.7 (2) and Proposition 6.1.11, HN = HF ∗ (HN ) is an F-group. Hence N ∈ F and G ∈ F2 . By Statement 2, GF is the F-injector of G. If N were abelian, then N would be a p-group for some prime p ∈ char F. Then G ∈ Sp F = F, contrary to supposition. Hence N is non-abelian. If HGF were a proper subgroup of G, then H would be contained in (HGF )F . Thus HGF ∈ F and HGF = GF by the F-maximality of GF in G. Consequently we may assume that G = HGF . Let M be a maximal subgroup of G containing GF . Then M = (H ∩ M )GF and H ∩ M is F-subnormal in M by Lemma 6.1.7 (2). Since H ∩ M ∈ F, it follows that H is contained in MF by induction. This forces M ∈ F and so M = GF by the F-maximality of GF in G. Hence G/GF is a cyclic group of order p, for a prime number p ∈ char F. Let Hp and J be Sylow p-subgroups of H and GF , respectively, such that P = Hp J is a Sylow p-subgroup of G ([Hup67, VI, 4.7]). Then G = P GF . Consider the subgroup P N of G. Since N = GF , it follows that P N is F-subnormal in G by Lemma 6.1.9 (1). Moreover, P N is the product of its F-subnormal Fsubgroups Hp N and JN . If G = P N , then Hp N is subnormal in G. Since Hp N ∈ F, it follows that Hp N ≤ GF . Consequently G = GF , contrary to the choice of G. Hence we may assume that P N is a proper subgroup of G. By induction P N ∈ F. This implies that P is F-subnormal in G. Let A be a maximal subgroup of G such that A = GF . Then G = AGF and A = Ap (A ∩ GF ) for some Sylow p-subgroup A p of G. Without loss of generality we may assume that Ap is contained in P . Then, by Lemma 6.1.6 (1), Ap is F-subnormal in G because Ap is F-subnormal in P by Lemma 6.1.9 (4). Since Ap and A ∩ GF are two F-subnormal F-subgroups of A, it follows that A ∈ F by induction. Therefore G is an F-critical group. By Statement 3, G/N is a cyclic group of order pα for some α ≥ 1. But then G = P N . This contradicts our supposition. Therefore G satisfies Statement 3 of Theorem 6.3.3 and F is a lattice formation. Example 6.3.13. Let F = Sπ be the class of all soluble π-groups for a set of primes π. Then F is a lattice formation as F satisfies Statements 1–3 of Lemma 6.3.12. There exist non-soluble saturated lattice formations as the next example due to A. F. Vasil’ev, S. F. Kamornikov, and V. N. Semenchuk [VKS93] shows: Example 6.3.14. Let S be a non-abelian simple group with the property that if T < S, then T is soluble (e. g., G = Alt(5)). Let F = Sπ D0 (1, S), for π = π(S). By Proposition 2.2.11, F is a formation. Moreover, by [DH92, II,

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1.9], F is Sn -closed. It is not difficult to prove that F is also N0 -closed and saturated. Hence F is a saturated Fitting formation contained in Eπ . Suppose, by way of contradiction, that F is not subgroup-closed, and choose a group G of minimal order such that G ∈ F and G has a subgroup H ∈ / F. Let N be a normal soluble π-subgroup of G such that G/N ∈ D0 (1, S) ⊆ F. If N = 1, then HN/N ∈ F by the minimal choice of G. Hence H ∈ Sπ F = F, contrary to supposition. Therefore N = 1 and G = S1 × · · · × Sn is a direct product of copies of S. For i = 1, . . . , n, let πi denote the projection of G onto the ith component of the direct product. Let A denote the subset of {1, . . . , n} defined by i ∈ A if and only if πi (H) = Si . Set K = i∈A Ker (πi )H and K ∗ = i∈A / Ker (πi )H . Then H/K ∈ D0 (1, S) and H/K ∗ is soluble. Since H/KK ∗ ∈ D0 (1, S) and H/KK ∗ is soluble, it follows that H = KK ∗ = K × K ∗ as K ∩ K ∗ = 1. Hence H ∈ F. This contradiction shows that F is subgroup-closed. Assume that F is not a lattice formation and choose a group G of minimal order having an F-subnormal F-subgroup H which is not contained in GF . Clearly H = 1. By familiar arguments, G ∈ b(F) and so G is a monolithic primitive group. Let N be the unique minimal normal subgroup of G. If N is abelian, then G ∈ Sπ F = F, which contradicts our assumption. Hence N is non-abelian and CG (N ) = 1. By Lemma 6.1.7 (2) and Proposition 6.1.11, HN = HF ∗ (HN ) is an F-group. Since CG (N ) = 1, it follows that HN has no normal soluble π-subgroups. Thus HN ∈ D0 (1, S) and HN = N ≤ GF . This is the final contradiction. Applying Theorem 6.3.3 (3), F is a lattice formation. We have now arrived at our first main objective, namely the classification of the subgroup-closed saturated lattice formations. Theorem 6.3.15. Let F = LF(F ) be a saturated formation. Then F is a lattice formation if and only if F = M × G for some subgroup-closed saturated formations M and G satisfying the following conditions: 1. π(M) ∩ π(G) = ∅. 2. There exists a set of prime numbers π ∗ and a partition {πi : i ∈ I} of π ∗ such that G = Xi∈I Sπi . 3. M = Sp M for all p ∈ π(M) and M is an M2 -normal Fitting class. 4. Every non-cyclic M-critical group G with Φ(G) = 1 is a primitive group of type 2 such that G/ Soc(G) is a cyclic group of prime power order. Proof. First of all, applying Proposition 3.1.40, F (p) is a subgroup-closed formation for every prime p ∈ π = char F. Assume that F is a lattice formation. For the ease of reading we break the argument into separately-stated steps.


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1. For each p ∈ π, every primitive group G of type 1 in F ∩ b F (p) is cyclic. It is clear that N = Soc(G), the unique minimal normal subgroup of G, is a q-group for some prime q = p. By [DH92, B, 10.9], G has an irreducible and faithful G-module V over GF(p). We claim that G has a unique core-free maximal subgroup, which provides the result. Suppose that M1 and M2 are maximal subgroups of G such that M1 = M2 and CoreG (Mi ) = 1 for i = 1, 2 and derive a contradiction. Then Mi ∈ F (p), i = 1, 2. Consider the semidirect product H = [V ]G, with respect to the action of G on V . Clearly H ∈ / F because G ∈ / F (p). Hence H F = V and G is not F-subnormal in H. But for i = 1, 2, V Mi is an F-normal maximal subgroup of H, and Mi is F-subnormal in V Mi because V Mi ∈ Sp F (p) = F (p) ⊆ F, that is, Mi is F-subnormal in H (Lemma 6.1.6 (1) and Lemma 6.1.7 (1)). Since F is a lattice formation, it follows that G = M1 , M2 is F-subnormal in H, contrary to supposition. 2. If p and q belong to π, and q ∈ char F (p), then p ∈ char F (q). / F (q) and consider an irreducible and faithful Cq -module Assume that Cp ∈ V over GF(p) ([DH92, B, 10.9]). Then the semidirect product [V ]Cq , with respect to the action of Cq on V , is a non-cyclic primitive group of type 1 in F ∩ b F (p) . This contradicts Step 1. Therefore Cp ∈ F (q) and p ∈ char F (q). 3. If p, q ∈ π and p ∈ char F (q), then char F (p) = char F (q). If r ∈ char F (q) \ char F (p), then r = q and Cq ∈ F (r), because of Step 2. Consider an irreducible and faithful Cq -module V over GF(r). Then [V ]Cq ∈ F ∩ b F (p) and [V ]Cq is non-cyclic primitive group of type 1. This contradicts Step 1. Therefore char F (q) ⊆ char F (p) and analogously char F (p) ⊆ char F (q). 4. If p, q ∈ π and p ∈ char F (q), then Sp ⊆ F (q). n Since F (q) is subgroup-closed, and a p-group of order p is isomorphic with a subgroup of the n-fold iterated wreath product Hn = . . . (Cp Cp ) . . . Cp , it is enough to prove that Hn ∈ F (q) for all n ∈ N. Denote inductively H1 = Cp and Hn = Hn−1 Cp for n ≥ 2. We can assume that p = q. Since Z(Hn ) is cyclic, Hn has a unique minimal normal subgroup, and consequently there exists an irreducible and faithful Hn -module V over GF(q) by [DH92, B, 10.9]. Consider the semidirect product G = [V ]Hn , with respect to the action of Hn on V . If (Hn−1 ) denotes the base group of Hn , then Hn = (Hn−1 ) Cp . Since (Hn−1 ) and Cp are F (q)-groups, it follows that V (Hn−1 ) and V Cp belong to F (q). Moreover they are F-subnormal in G. Hence G ∈ F by Theorem 6.3.3 (2) and so Hn ∈ F (q). 5. If p, q ∈ π and q ∈ char F (q), then Sp F (q) = F (q). Assume that F (q) = Sp F (q) and derive a contradiction. Let G be a group of minimal order in the supposed non-empty class Sp F (q) \ F (q). Then, since F (q) is a subgroup-closed formation, G has a unique minimal normal subgroup, M say, and G/M ∈ F (q). Moreover M is a p-group and every maximal subgroup of G belongs to F (q). If M ≤ Φ(G), then G ∈ F and we may argue as in Step 1 to obtain that G is cyclic. Consequently G ∈ F (q) by Step 4.

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This contradiction implies that M is not contained in Φ(G). Let R be a maximal subgroup of G such that G = M R. Then M ∩ R = 1, R ∈ F (q) and M = GF . Clearly we may assume that p = q. Hence considering a faithful and irreducible G-module over GF(q), it is rather clear that R must be a cyclic r-group for some prime r, r ∈ π. From Step 3 and Step 4, it follows that G ∈ Sp Sr ⊆ Sp F (p) = F (p) ⊆ F. This is the desired contradiction. Therefore F (q) = Sp F (q). Calling two elements p, q ∈ π equivalent if and only if char F (p) = char F (q), we obtain an equivalence relation on π whose equivalence classes {πi : i ∈ I} form a partition of π. Let p ∈ πi , i ∈ I. Since F (p) is a subgroup-closed formation, it follows / πi , then every group in Eπi is soluble by the Odd that F (p) ⊆ Eπi . If 2 ∈ Order Theorem [FT63]. Therefore F (p) ⊆ Sπi . In fact, we have: / πi , then F (p) = Sπi . 6. If p ∈ πi , i ∈ I, and 2 ∈ Assume that F (p) = Sπi and choose a group G ∈ Sπi \ F (p) of minimal order. Then G has a unique minimal normal subgroup N , N is a q-group for some prime q ∈ πi , and G/N ∈ F (p). By Step 5, G ∈ Sq F (p) = F (p). This contradiction forces F (p) = Sπi . Put M = (1) if 2 ∈ / π and M = F ∩ Eπi0 if 2 ∈ πi0 for some i0 ∈ I. Assume that 2 ∈ π. Then {πi : i = i0 } is a partition of π ∗ = π \ πi0 . Let G = Xi∈I\{i0 } Sπi . Then G is a subgroup-closed lattice-formation by Lemma 6.3.11 and Example 6.3.13. 7. F = M × G, π(M) ∩ π(G) = ∅. It is clear that M × G is contained in F. Suppose, for a contradiction, that this inclusion is proper, and choose a group G of minimal order in F \ (M × G). Then G is a monolithic primitive group because M × G is a saturated formation. Let N be the unique minimal normal subgroup of G. Suppose that N is non-abelian. Then 2 divides |N | and G ∈ F (2) ⊆ F ∩ Eπi0 = M, contrary to our choice of G. Therefore N is abelian. Let p be the prime dividing |N |. Then G ∈ F (p). If p ∈ πi0 , it follows that G ∈ M. If p ∈ πi for some i ∈ I\{i0 }, we have G ∈ G. In both cases, G ∈ M × G, another contradiction. Evidently, πi0 = π(M) and π ∗ = π \ πi0 = π(G). Hence π(M) ∩ π(G) = ∅. 8. M = Sp M for all p ∈ π(M). Let p ∈ π(M). Assume that Sp M is not contained in M and derive a contradiction. Let G ∈ Sp M \ M be a group of minimal order. By familiar reasoning, G is a primitive group of type 1 and N = Soc(G) is an abelian p-group. Our eventual goal is to show that every core-free maximal subgroup M of G is cyclic. Suppose that M1 and M2 are maximal subgroups of M such that M1 = M2 . Since Sp M is subgroup-closed, it follows that N Mi ∈ M ⊆ F for i = 1, 2. Moreover, N Mi are F-subnormal in G because N = GF (Lemma 6.1.7 (1)). By Theorem 6.3.3 (2), G = N M1 , M2 ∈ F, which contradicts the assumption that G ∈ / M. Therefore M has a unique maximal subgroup and so M is a cyclic group of prime power order. By Step 4, M ∈ F (q), where q ∈ π(M ). Therefore G ∈ Sp F (q) = F (q) by Step 5. We conclude then that G ∈ F ∩ Eπi0 . This final contradiction completes the proof.


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9. M is an M2 -normal Fitting class. It is clear that M is a Fitting class. Let G ∈ M2 be a group and let J be an M-maximal subgroup of G containing GM . Since G/GM ∈ M ⊆ F, it follows that GF ≤ GM . Consequently J is F-subnormal in G by Lemma 6.1.7 (1). Applying Theorem 6.3.3 (3), J ≤ GF = GM because G ∈ Eπi0 . Therefore GF is F-maximal in G. Let H be a subnormal subgroup of G. Then H is actually F-subnormal because G ∈ E K(F) (Proposition 6.1.10). In addition, H ∩ GF is contained in HF as H ∩ GF is an F-subnormal F-subgroup of H (Lemma 6.1.7 (2) and Theorem 6.3.3 (3)). Consequently H ∩ GF = HF = HM and HM = H ∩GM is M-maximal in H. This means that GM is an M-injector of G. 10. If G is a non-cyclic M-critical group and Φ(G) = 1, then G is a primitive group of type 2 such that Soc(G) is non-abelian and G/ Soc(G) is a cyclic group of prime power order. Let G be a non-cyclic M-critical group such that Φ(G) = 1. Then G is a monolithic primitive group because M is saturated. Assume that N = Soc(G) is abelian. Then N < G because G is non-cyclic, and so N is a p-group for some prime p ∈ π(M). Hence G ∈ Sp M = M by Step 8. This contradiction implies that N is non-abelian. Suppose that N < G. Let M1 and M2 be two different maximal subgroups of G containing N . Then Mi ∈ M and Mi is F-subnormal in G, i = 1, 2, as N = GF (Lemma 6.1.7 (1)). Applying Theorem 6.3.3 (2), G ∈ F. Since G ∈ Eπi0 , it follows that G ∈ M, contrary to the assumption that G is M-critical. This contradiction proves that G/N has a unique maximal subgroup and so G/N is cyclic of prime power order. Applying Lemma 6.3.12, M is a lattice formation. Conversely, assume that F = M × G for subgroup-closed saturated formations M and G satisfying Statements 1 to 4. By Example 6.3.13, Lemma 6.3.11, and Lemma 6.3.12, F is a lattice formation. Corollary 6.3.16. Let F be a saturated formation of soluble groups of characteristic π. Then F is a lattice formation if and only if there exists a partition {πi : i ∈ I} of π such that F = Xi∈I Sπi . Corollary 6.3.16 holds not only for subgroup-closed saturated formations but also for Sn -closed saturated ones. This was proved in [VKS93]. Lockett [Loc71] described the F-injectors of soluble π-groups, here F is a lattice formation of soluble groups of characteristic π. It turns out that if G is a soluble π-group, the F-injectors of G are exactly the F-maximal subgroups of G containing GF , that is, F is a dominant Fitting class in Sπ . Theorem 6.3.17 ([Loc71]). Let π be a non-empy set of primes and let G be a soluble π-group. Assume that {πi : i ∈ I} is a partition of π and F = Xi∈I Sπi . For each i ∈ I, let Vi be a Hall πi -subgroup of Ci = CG Xj=i Oπj (G) . Then:

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1. [Vi , Vj ] = 1 if i = j, 2. the subgroup Vi : i ∈ I = Xi∈I Vi is an F-subgroup of G containing GF = Xi∈I Oπi (G). Let I(G) be the set of all such subgroups Vi : i ∈ I obtained from the various choices of Vi ∈ Hallπi (Ci ). Then 3. if W is an F-subgroup of G containing GF , then W ≤ V for some V ∈ I(G). 4. I(G) = InjF (G). Proof. It is clear that F(G) is contained in GF = Xi∈I Oπi (G) and CG F(G) is contained in F(G) because G is soluble. 1. Take i, j ∈ I, i = j. Then [Vi , Cj ] ≤ Ci ∩ Cj ≤ CG (GF ) ≤ CG F(G) ≤ F(G) ≤ GF . Therefore Cj normalises Vi GF = Vi × Xi=j Oπj (G). Since Vi = Oπi (Vi GF ), it follows that Cj normalises Vi . In particular, Vj normalises Vi . By a similar argument Vi normalises Vj . Hence [Vi , Vj ] ≤ Vi ∩ Vj = 1. 2. We deduce at once from Statement 1 that Vi : i ∈ I is the direct product of its Hall πi -subgroups and also that GF ≤ Vi : i ∈ I ∈ F. 3. Let i ∈ I. Since W ∈ F, the Hall πi -subgroup Wi of W centralises Oπj (W ), which contains Oπj (G) by assumption, i = j. Therefore Wi is contained in a Hall πi -subgroup, Vi say, of CG Xj=i Oπi (G) . Hence W = Xi∈I Wi ≤ Xi∈I Vi ∈ I(G). 4. It is enough to prove that I(G) is a conjugacy class of G. of I(G). For each Let V = Xi∈I Vi and V¯ = Xi∈I V¯i be two typical elements ∈ C X O (G) such that V¯i = Vixi . i ∈ I, there exists an element x i G j=i πj Let x = i∈I xi , where the product may be taken in any order. If i = j, the element xj normalises each conjugate of Vi , and therefore V x = Xi∈I Vix = Xi∈I Vixi = V¯ . The next result, due to A. Ballester-Bolinches, K. Doerk, and M. D. PérezRamos [BBDPR92], shows that these injectors have a good behaviour with respect to F-subnormal subgroups. Theorem 6.3.18. Let F be a lattice formation of soluble groups of characteristic π. If G is a soluble π-group and V is an F-injector of G and H is an F-subnormal subgroup of G, then V ∩ H is an F-injector of H. Proof. Assume that the result is not true and let G be a counterexample of minimal order. Clearly we may suppose that H is an F-normal maximal subgroup of G. Hence G/ CoreG (H) is a πi -group for some member πi of {πi : i ∈ I}, where {πi : i ∈ I} is the partition of π such that F = Xi∈I Sπi . Write π = πi . Then CoreG (H) contains every Hall π -subgroup of G. Note that HF is contained in GF because HF is an F-subnormal F-subgroup of G (Lemma 6.1.6 (1) and Theorem 6.3.3 (3)). Let V be an F-injector of G such that V ∩ H is not an F-injector of H. Since HF is contained in V ∩ H, it follows that V ∩ H is not F-maximal in H. Let R be an F-maximal subgroup of H containing V ∩ H. It is clear that R is an F-injector of H. Since the


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Hall π -subgroup Vπ of V is contained in the Hall π -subgroup Rπ of R and Rπ is contained in CoreG (H), it follows that Vπ = Rπ as V ∩ CoreG (H) is result, an F-injector of CoreG (H). On the other hand, according to Lockett’s the Hall π-subgroup Vπ of V is a Hall π-subgroup of C = CG Xj=i Oπj (G) and the Hall π-subgroup Rπ of R is a Hall π-subgroup of CH Xj=i Oπj (H) . Moreover Vπ ∩ H ≤ Rπ . Since G/ CoreG (H) is a π-group, it follows that Xj=i Oπj (G) = Xj=i Oπj (H) and so there exists an element g ∈ C such that Vπg ∩ H = Rπ . If C = G, then Vπ is a Hall π-subgroup of G. Thus G = CoreG (H)Vπ and Vπg ∩ H = Vπh ∩ H for some h ∈ CoreG (H). This implies that |Vπ ∩ H| = |Rπ | and Rπ = Vπ ∩ H, contrary to our supposition. Consequently C is a proper subgroup of G. Since C is normal in G, it follows that V ∩ C is an F-injector of C. Moreover H ∩ C is F-subnormal in C by Lemma 6.1.7 (2). The minimality of G yields V ∩ H ∩ C is an F-injector of H ∩ C. In particular V ∩ H ∩ C = R ∩ C. Since Rπ is a Hall π-subgroup of R ∩ C, we have that Vπ ∩ H = Rπ . This contradiction proves the result. The following result is a characterisation of saturated lattice formations of soluble groups by means of properties of Fitting type. Most of the work is already contained in the above theorem. Theorem 6.3.19. Let F be a saturated formation of soluble groups of characteristic π. The following statements are pairwise equivalent: 1. F is a lattice formation. 2. F is a Fitting class satisfying that if G is a soluble π-group, V is an F-injector of G and H is an F-subnormal subgroup of G, then V ∩ H is an F-injector of H. 3. F is a Fitting class and if H is an F-subnormal F-subgroup of a soluble π-group G, then H, H g ∈ F for every g ∈ G. Proof. Applying Theorem 6.3.3 (3) and Theorem 6.3.18, we have that 1 implies 2. Assume that Statement 2 holds. Let H be an F-subnormal F-subgroup of a soluble π-group G. If g ∈ G, then H g is contained in every F-injector of G. Therefore H, H g ∈ F. Suppose, arguing by contradiction, that the statement 3 is true but F is not a lattice formation. On this supposition, by Theorem 6.3.3 (3), there exists a group G of minimal order having an F-subnormal F-subgroup 1 = H which is not contained in the F-radical of G. If N is a minimal normal subgroup of G, then HN/N is contained in the F-radical K/N of G/N by minimality of G. Since H is F-subnormal in K by Lemma 6.1.7 (2), and KF is contained in GF , it follows that K = G. Hence G/N ∈ F for every minimal normal subgroup of G. Thus G is a monolithic primitive group. Let A = GF be the unique minimal normal subgroup of G. By Lemma 6.1.7 (2) and Proposition 6.1.11, T = HA = HF ∗ (T ) is an F-group. Note that T, T g is F-subnormal in G for all g ∈ G because A is contained in T (Lemma 6.1.7 (1)). Applying

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Statement 3, it follows that T G ∈ F. Since T G is normal in G and F is a Fitting class, T G is contained in the F-radical of G. In particular, H is contained in GF , which contradicts our assumption. Hence F is a lattice formation. Remark 6.3.20. In [BBMPPR00], it is proved that an Sn -closed saturated formation of soluble groups of full characteristic satisfying Statement 3 of Theorem 6.3.19 is actually subgroup-closed. Therefore Theorem 6.3.19 hold not only for subgroup-closed saturated formations of soluble groups, but also for Sn -closed ones. We round this section off with a characterisation of lattice formations of soluble groups. It is not always true in general that a lattice formation of soluble groups is saturated. It is enough to consider the formation of all abelian groups. In the sequel we shall take a closer look at this family of formations, following ideas of A. F. Vasil’ev and S. F. Kamornikov [VK02]. Therefore until further notice we make the following general assumption. Hypothesis 6.3.21. F is not only a subgroup-closed formation but also soluble. Let ZF be the class of all groups G such that every subgroup of G is F-subnormal in G. The basic properties of F-subnormal subgroups imply that ZF is an homomorph containing F. The formation of all abelian groups shows that the inclusion could be proper. Moreover it is rather easy to see that ZF = F if F is saturated. We gather together in a convenient “portmanteau” lemma some relevant properties of ZF , when F is a lattice formation. Lemma 6.3.22. Let F be a lattice formation. Then: 1. ZF is a subgroup-closed formation of soluble groups. 2. π(F) = π(ZF ) and ZF contains all nilpotent π(F)-groups. 3. ZF is a Fitting class. Proof. 1. First we prove that ZF is a soluble class. Suppose, by way of contradiction, that ZF is not contained in S. Then ZF \ S is not empty. Let G be a group of minimal order in ZF \ S. By familiar reasoning, G is a nonabelian simple group such that every subgroup of G is F-subnormal in G. Let M be a maximal subgroup of G. Then 1 = M and GF is contained in M because M is F-subnormal in G. But then GF = 1 because G is simple. This means that G ∈ F and so G is soluble, contrary to supposition. Hence ZF is composed of soluble groups. It is clear that ZF is a homomorph. Let N1 and N2 be minimal normal subgroups of a group G such that N1 ∩N2 = 1 and G/Ni ∈ ZF for i = 1, 2. Let P be a Sylow subgroup of G. Then our assumption implies that P Ni /Ni is


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F-subnormal in G/Ni for i = 1, 2. By Lemma 6.1.7 (3), P N1 ∩ P N2 = P is F-subnormal in G. Moreover since Ni /Ni is F-subnormal in G/Ni , it follows that Ni is F-subnormal in G for i = 1, 2 by Lemma 6.1.6 (2). Hence 1 = N1 ∩ N2 is F-subnormal in G. By Lemma 6.1.7 (2), 1 is F-subnormal in P . Therefore every subgroup of P is F-subnormal in G by Lemma 6.1.9 (3) and Lemma 6.1.6 (2). Since every subgroup of G is generated by its subgroups of prime power order, and F is a lattice formation, it follows that G ∈ ZF . Applying [DH92, II, 2.6], ZF is R0 -closed and so ZF is a formation. Let G ∈ ZF and let H be a subgroup of G. Since every subgroup of H is F-subnormal in G, it follows by Lemma 6.1.7 (2) that H ∈ ZF . Hence ZF is subgroup-closed. 2. It is clear that π(F) ⊆ π(ZF ) because F ⊆ ZF . Let p ∈ π(ZF ) and let G be a group in ZF such that p divides |G|. Then Cp ∈ ZF because ZF is subgroup-closed. Hence 1 is F-subnormal in Cp and so Cp ∈ F. This shows that p ∈ π(F). If P is a p-group for some prime p ∈ π(F) = π(ZF ), then Cp ∈ ZF because ZF is subgroup-closed. By Lemma 6.1.9 (4), every subgroup of P is F-subnormal in P . Hence P ∈ ZF . This implies that every nilpotent π(F)group is a ZF -group. 3. It is clear that only the N0 -closure of ZF needs checking. Let A and B be normal subgroups of a group G such that G = AB and A and B belong to ZF . We prove that G ∈ ZF by induction on the order of G. If G is nilpotent, then G ∈ ZF because π(G) ⊆ π(ZF ) and ZF contains all nilpotent π(ZF )-groups. Hence we may suppose that G is not nilpotent. Let P be a Sylow subgroup of G. Then P is the product of P ∩ A and P ∩ B, which are obviously normal subgroups of P . Moreover, P ∩ A and P ∩B are ZF -subgroups of P because ZF is subgroup-closed. Since P is a proper subgroup of G, the induction hypothesis leads to the conclusion that P ∈ ZF . Furthermore, G is soluble because ZF is composed of soluble groups. Therefore G ∈ K E(F) as π(F) = π(ZF ) = char ZF . This implies that A and B are F-subnormal in G by Proposition 6.1.10. Since A and B belong to ZF , it follows that P ∩ A and P ∩ B are F-subnormal subgroups of G by Lemma 6.1.6 (1). Therefore P = (P ∩ A)(P ∩ B) is F-subnormal in G because F is a lattice formation. Hence every subgroup of P is F-subnormal in G Lemma 6.1.6 (1). Since every subgroup of G is generated by its subgroups of prime power order, it follows that G ∈ ZF . Consequently ZF is N0 -closed and so ZF is a Fitting class. Combining Theorem 2.5.2 and Lemma 6.3.22, we have: Proposition 6.3.23. Let F be a lattice formation. Then ZF is a saturated formation. Theorem 6.3.24. Let F be a lattice formation. Then ZF is a lattice formation. Proof. By Theorem 6.3.3 (3), it is sufficient to prove that every ZF -subnormal ZF -subgroup H of a group G is contained in the ZF -radical GZF of G.

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Suppose, by way of contradiction, that there exists a group G of minimal order having a ZF -subnormal ZF -subgroup H such that H is not contained in GZF . Among the ZF -subnormal ZF -subgroups of G that are not contained in GZF , let H be one of maximal order. Let N be a minimal normal subgroup of G. Then HN/N is a ZF -subnormal ZF -subgroup of G/N by Lemma 6.1.6 (3). The choice of G implies that HN/N ≤ (G/N )ZF = L/N . Assume that L is a proper subgroup of G. The minimality of G forces the conclusion that H is contained in LZF as H is ZF -subnormal ZF -subgroup of L. Since L is normal in G, it follows that LZF is contained in GZF . This contradiction shows that L = G and so G/N ∈ ZF for every minimal normal subgroup of G. Consequently G is a monolithic primitive group and N = GZF is the unique minimal normal subgroup of G. Moreover CG (N ) ≤ N . By Lemma 6.1.7 (2) and Proposition 6.1.11, HN = HF ∗ (HN ) is an ZF -group. Since HN is ZF -subnormal in G by Lemma 6.1.7 (1), it follows that N ≤ H by the choice of the pair (G, H). Therefore H = N (H ∩M ), where M is a core-free maximal subgroup of G complementing N in G. On the other hand, H/N is F-subnormal in G/N . Hence H is F-subnormal in G by Lemma 6.1.6 (2). Since H ∈ ZF , it follows that H ∩M = X is also F-subnormal in G Lemma 6.1.6 (1). Consequently X M , the normal closure of X in M , is F-subnormal in G because F is a lattice formation. Furthermore X M ∈ ZF because M ∈ ZF and ZF is subgroup-closed. Since X M is ZF -subnormal in G, it follows that X M is ZF -subnormal in N X M by Lemma 6.1.7 (2), and N X M belongs to ZF by Proposition 6.1.11. Since N X M is a normal subgroup of G and ZF is a Fitting class, it follows that H ≤ N X M ≤ GZF . This contradiction proves the theorem. We are now in a position to state and prove Vasil’ev and Kamornikov’s characterisation of lattice formations of soluble groups. Theorem 6.3.25 ([VK02]). Let F be a formation of soluble groups. The following statements are pairwise equivalent: 1. The set of all K-F-subnormal subgroups is a sublattice of the subgroup lattice of every group. 2. The set of all F-subnormal subgroups is a sublattice of the subgroup lattice of every group. 3. There exists a partition {πi : i ∈ I} of the set π(F) such that F = Xi∈I Fπi , where Fπi = F ∩ Sπi . Moreover, Fπi = Sπi for all i ∈ I such that |πi | > 1. Proof. Of the three statements in the theorem, it follows that 1 implies 2. Assume that Statement 2 holds. Then the preceding results show that ZF is a saturated lattice formation of soluble groups. Hence, by Theorem 6.3.15, there exists a partition {πi : i ∈ I} of π := char ZF = π(ZF ) = π(F) such that ZF = Xi∈I Sπi . Hence F = Xi∈I Fπi , where Fπi = F ∩ Sπi for all i ∈ I. Let G be a soluble primitive πi -group for some i ∈ I. Then G ∈ ZF and so every subgroup of G is F-subnormal in G. In particular, G ∈ F and


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thus G ∈ Fπi . Suppose, in addition, that |πi | ≥ 2. If A is a πi -group such that |π(A)| ≥ 2, then by a theorem of Hawkes [Haw75, Theorem 1], A is isomorphic to a subgroup of a multiprimitive π(A)-group G, that is, every epimorphic image of G is primitive. Since G is primitive and G is a πi -group, it follows that G ∈ Fπi . Hence A ∈ Fπi because Fπi is subgroup-closed. If A is a πi -group, then A is isomorphic to a subgroup of a πi -group B such that |π(B)| ≥ 2 as |πi | ≥ 2. Hence A ∈ Fπi . Consequently Fπi = Sπi for all i ∈ I such that |πi | ≥ 2 and Statement 3 is true. To complete the proof we now show that 3 implies 1. Suppose that F is a formation such that F = Xi∈I Fπi for a partition {πi : i ∈ I} of π = π(F). Assume, in addition, that Fπi = Sπi if |πi | ≥ 2. Consider the subgroup-closed formation H = Xi∈I Sπi . By Lemma 6.3.11, H is a saturated lattice formation and char H = π. We aim to show that SnK-H (G) = SnK-F (G) for every group G. Assume, arguing by contradiction, there exists a group G of minimal order such that SnK-H (G) = SnK-F (G). Clearly SnK-F (G) ⊆ SnK-H (G) because F ⊆ H. Hence there exists a subgroup H ∈ SnK-H (G) \ SnK-F (G). Then H is a proper subgroup of G and thus there exists a subgroup M of G such that either M is normal in G or M is an H-normal maximal subgroup of G. Since H is KH- subnormal in M, it follows that H is K-F-subnormal in M by minimality of G. If M were normal in G, we would have that H would be K-F-subnormal in G. This would contradict our choice of H. Hence M is an H-normal maximal subgroup of G and so GH is contained in M . Then G/ CoreG (M ) is a πi group for some πi ⊆ π as G/ CoreG (M ) is a soluble primitive H-group. Note that |πi | > 1 because M is not normal in G. Therefore G/ CoreG (M ) ∈ Fπi . This means that M is F-normal in G and H is K-F-normal in G, contrary to our initial supposition. Therefore SnK-F (X) = SnK-H (X) for all groups X. Applying Corollary 6.3.16 and Theorem 6.3.9, the set SnK-F (X) is a sublattice of the subgroup lattice of X for all groups X. Example 6.3.26. Let F be the formation of all abelian groups. Then F is a lattice formation of soluble groups such that ZF = N, the class of all nilpotent groups. It is clear that Fp = Sp for all p ∈ π(F) = P. In [VK01], A. F. Vasil’ev and S. F. Kamornikov consider w-inherited subgroup functors f enjoying the following property: If G is a group and H, K ∈ f(G), then H ∩ K ∈ f(G) and H, K ∈ f(G). They called them subgroup NTL-functors. The techniques employed in this section allow them to prove the following nice result in the universe of all soluble groups: Theorem 6.3.27. Let f be a subgroup NTL-functor. Then: 1. The class χf = {G | f(G) = S(G)} is a subgroup-closed saturated formation,

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2. there exists a partition {πi | i ∈ I} of π(χf ) such that χf = Xi∈I Sπi , 3. For every group G, f(G) = Snχf (G). Consequently the subgroup NTL-functors in the soluble universe are exactly f = SnF , for some subgroup-closed saturated lattice formation F. The authors also consider the problem in the general finite universe. The best they were able to prove is the following: Proposition 6.3.28. Let f be a subgroup NTL-functor. Then: 1. The class χf = {G | f(G) = S(G)} is a subgroup-closed solubly saturated formation, 2. For every group G, Sn χf (G) is contained in f(G). Consequently the following question remains open. Open question 6.3.29 ([VK01]). Let f be a subgroup NTL-functor. Is there a solubly saturated formation F such that f = SnF ?. The reader is referred to [KS03] for more information about subgroup functors and classes of groups. Postscript Lattice formations have been also involved in the study of F-normality associated with subgroup-closed saturated formations F in the soluble universe. As it is known, this notion was primarily associated with maximal subgroups. A first attempt to give a definition valid for arbitrary subgroups was made by A. Ballester-Bolinches, K. Doerk, and M. D. Pérez-Ramos in [BBDPR95]. In the case F = N, the class of all nilpotent groups, the F-normality coincides with the classical normality and, for a general subgroup-closed saturated formation F, the F-subnormality turns out to be associated naturally with the F-normality in the obvious way. However, the results concerning lattice properties of F-normal subgroups differ from the corresponding ones for F-subnormal subgroups. More recently, M. Arroyo-Jord´ a and M. D. Pérez-Ramos [AJPR01] study an alternative definition of F-normality, the F-Dnormality. It was suggested by K. Doerk. This new definition satisfies all the desired properties. Moreover, in this case, lattice formations turn out to be the subgroup-closed saturated formations for which the set of all F-Dnormal subgroups is a sublattice of the subgroup lattice in every soluble group. The same authors [AJPR04a], [AJPR04b], studied Fitting classes with stronger closure properties involving F-subnormal subgroups, for a lattice formation F of full characteristic.

6.4 F-subnormal subgroups and F-critical groups

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Definition 6.3.30. 1. Let F be a lattice formation containing the class N of nilpotent groups. A class X of groups is said to be an F-Fitting class if: a) for every G ∈ X and every F-subnormal subgroup H of G we have H ∈ X; and b) for G = H, K with H, K F-subnormal in G, if H, K ∈ X, then G ∈ X. 2. A subgroup of a group G is said to be an (X, F)-injector if, for every F-subnormal subgroup K of G, V ∩ K is X-maximal in K. Every F-Fitting class is also a Fitting class. They proved in [AJPR04b] the following nice result (see Theorem 2.4.26): Theorem 6.3.31. Let F be a lattice formation containing N, and X an F-Fitting class. Then for every group G, a subgroup V of G is an (X, F)-injector if and only if it is an X-injector.

6.4 F-subnormal subgroups and F-critical groups We saw in Section 6.3 that if F is a saturated formation, then F is a lattice formation if and only if F contains all groups generated by two F-subnormal F-subgroups (Theorem 6.3.3 (2)). As a consequence, a saturated lattice formation F enjoys the following property: If A and B are F-subnormal F-subgroups of a group G and G = AB, then G ∈ F. (6.1) It turns out that Condition (6.1) is not sufficient for a subgroup-closed saturated formation to be a lattice formation: the formation of all p-nilpotent groups, p a prime, satisfies Condition (6.1), but it is not a lattice formation (see Example 6.3.1). Moreover, the formation of all groups with nilpotent length at most two does not satisfy Condition (6.1). Consequently the question of determining the subgroup-closed saturated formations which are closed under taking products of F-subnormal subgroups arises (see [MK99, Problem 14.99]). This problem has already been settled and solved in the soluble universe by A. Ballester-Bolinches in [BB92] for subgroup-closed saturated formations of full characteristic (see also [Sem92]). The first result of this section puts a rich source of subgroup-closed saturated formations satisfying Condition 6.1 at our disposal. Proposition 6.4.1. Let F be a saturated formation. Suppose that, for every p ∈ π = char F, there exists a set of primes π(p) with p ∈ π(p) such that F is locally defined by the formation function f given by f (p) = Eπ(p) if p ∈ π and f (q) = ∅ if q ∈ / π. Then F is closed under taking products of F-subnormal subgroups. Proof. Assume that the result is false and derive a contradiction. Then there exists a group G of minimal order with two F-subnormal F-subgroups A and B

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such that G = AB and G ∈ / F. If N is a minimal normal subgroup of G, then it is clear that G/N is the product of the F-subnormal F-subgroups AN/N and BN/N by Lemma 6.1.6 (3). The choice of G implies that G/N ∈ F. Therefore G is in the boundary of F and so G is a monolithic primitive group. Then N = GF is the unique minimal normal subgroup of G and CG (N ) ≤ N . By Lemma 6.1.7 (2) and Proposition 6.1.11, AN = AF ∗ (AN ) is an F-group. Analogously BN ∈ F. Since CG (N ) ⊆ N , it follows that AN and BN belong to Eπ(p) for all p ∈ π(N ). Therefore G ∈ Eπ(p) for each prime p dividing |N |. This implies that G ∈ F, contrary to our supposition. Consequently F is closed under taking products of F-subnormal products. Note that the above result also holds if we replace Eπ(p) by Sπ(p) , for all p ∈ char F. Unfortunately, the converse of Proposition 6.4.1 is not true in general, as the following example shows. Example 6.4.2. Let S be a non-abelian simple group, and consider the saturated formation H = G : S ∈ / Q(G) . Let F be the largest subgroup-closed formation contained in H. By Theorem 3.1.42, F = G : S(G) ⊆ H is saturated. In addition, F cannot be locally defined by a formation function as in Proposition 6.4.1. We assert that F is closed under taking products of F-subnormal subgroups. Suppose, for a contradiction, that this is not true and let G be a counterexample of least order. Then G has two proper F-subnormal F- subgroups A and B such that G = AB and G ∈ / F. Let N be a minimal normal subgroup of G. Since G/N is a product of the F-subnormal F-subgroups AN/N and BN/N , the choice of G implies that G/N ∈ F. Therefore G is in the boundary of F and so G is a monolithic primitive group. In particular, / H. Then N = GF is the unique minimal normal subgroup of G. Assume G ∈ G ∈ b(H) = (S). Hence G is non-abelian and simple. This implies that N = G and therefore G = A = B, contrary to supposition. Consequently G ∈ H. Since G∈ / F, it follows that S(G) is not contained in H. Among the proper subgroups X of G not belonging to H, we choose H of minimal order. Then every proper subgroup of H belongs to H. Applying [DH92, III, 2.2(c)], there exists a normal subgroup K of H such that H/K ∈ b(H). Hence H/K is a non-abelian simple group. Since H/H ∩ N belongs to F, it follows that H = (H ∩ N )K. It H ∩ N were a proper subgroup of H, we would have H ∩ N ∈ F and so H/K ∈ F ⊆ H, contrary to supposition. Hence H ∩ N = H and H is a subgroup of N . By Lemma 6.1.7 (2) and Proposition 6.1.11, AN = HF ∗ (AN ) is an F-group. Consequently N ∈ F and so H ∈ F. This final contradiction proves that F is closed under taking F-subnormal subgroups. At the time of writing no useful characterisation of subgroup-closed saturated formations satisfying (6.1) is known. The picture improves, however, if attention is confined just to subgroup-closed saturated formations of soluble groups. The following result supports this view.


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Theorem 6.4.3. Let F be a saturated formation of soluble groups of characteristic π. The following statements are equivalent: 1. For each prime p ∈ π, there exists a set of primes π(p), with p ∈ π(p), such that F is locally defined by the formation function f given by f (p) = Sπ(p) and f (q) = ∅ if q ∈ / π. 2. F satisfies Condition (6.1). Proof. It follows at once from Proposition 6.4.1 that 1 implies 2. 2 implies 1. We are assuming in this chapter that F is subgroup-closed. Hence, for every p ∈ π, F (p) is a subgroup-closed formation by Proposition 3.1.40. Therefore F (p) is contained in F ∩ Eπ(p) ⊆ Sπ(p) , where π(p) = char F (p). Suppose, for a contradiction, that the inclusion is proper and choose a group G of minimal order in (F ∩ Sπ(p) ) \ F (p). Then every proper subgroup of G belongs to F (p) and G is a soluble monolithic group. Assume that G contains two inconjugate maximal subgroups, L and M say. Then G = M L by [DH92, A, 16.2]. Moreover M and L belong to F (p). Let W be a faithful G-module over GF(p) and denote by Z = [W ]G the corresponding semidirect product. Then Z F is contained in W and therefore W M and W L are two F-subnormal subgroups of Z by Lemma 6.1.7 (1). Moreover W M an W L belong to Sp F (p) = F (p). Hence W M and W L are F-groups. Since Z = (W M )(W L) and F satisfies (6.1), it follows that Z ∈ F. This implies that G ∈ F (p), which is clearly not the case. Hence G has a single conjugacy class of maximal subgroups. This implies that G is a cyclic group whose order is a power of a prime, q say. Moreover, q ∈ π(p). On the other hand, it is rather easy to see that F is clearly a Fitting class as F is closed under taking products of F- subnormal subgroups. Hence F (p) is also a Fitting class by Proposition 3.1.40. Since q ∈ π(p), it follows that F (p) contains Sq by [DH92, IX, 1.9]. Hence G ∈ F (p) and we have reached a contradiction. Consequently F (p) = F∩Sπ(p) for all p ∈ π. It remains to prove that F = LF(f ), where f is the formation / π. To this end function defined by f (p) = Sπ(p) , p ∈ π, and f (q) = ∅ if q ∈ assume, by way of contradiction, that M = LF(f ) is not contained in F and let G be a group of minimal order in M \ F. Then G is a soluble primitive group and N = Soc(G) = GF is the unique minimal normal subgroup of G. Let q be the prime dividing |N |. Then q ∈ π and G/N ∈ f (q) = Sπ(q) . Hence G/N ∈ F ∩ Sπ(q) = F (q). By Remark 3.1.7 (2), G ∈ F. It follows that our supposition is wrong and hence M is contained in F. Since F is obviously contained in M, we have F = M and the proof of the theorem is complete. From now on we focus our attention on formations whose associated critical groups have special properties. In order to carry out our task we shall need some definitions. Recall that if Z be a class of groups, a group G is called S-critical for Z, or simply Z-critical, if G is not in Z but all proper subgroups of G are in Z. Following [DH92, VII, 6.1], we denote CritS (Z) the class of all Z-critical groups. The motivation for investigating such minimal classes is that detailed

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knowledge of groups that just fail to have a group theoretic property is likely to give some insight into just what makes a group have the property. The minimal classes have been investigated for a number of classes of groups. For instance, O. J. Schmidt (see [Hup67, III, 5.2]) studied the N-critical groups. These groups are also called Schmidt groups. They have a very restricted structure and they are useful in proving a known result of H. Wielandt about groups with nilpotent Hall π-subgroups (see [Hup67, III, 5.8]). K. Doerk [Doe66] studied the critical groups with respect to the class of all supersoluble groups and R. W. Carter, B. Fischer, and T. O. Hawkes (see [DH92, VII, Section 6]) used a method of extreme classes to study the soluble F-critical groups in the case when F is the formation of all soluble groups with nilpotent length less than or equal to r. K. Doerk and T. O. Hawkes [DH92, VII, 6.18] gave a complete description of a soluble group G which is not in F but all maximal subgroups are in F, where F is an arbitrary (not necessarily subgroup-closed) saturated formation of soluble groups (note that such a group G is F-critical if F is a subgroup-closed formation). This result was extended by A. BallesterBolinches and M. C. Pedraza-Aguilera to the general universe of all finite groups in [BBPA96]. The reader is referred to [Rob02], [BBERR05], and [BBERss] for further information about critical groups associated with some interesting classes of groups. A useful property for a formation F in this connection is that of having F-critical groups with a well-known structure. For instance, if F is either a soluble saturated lattice formation or the formation of all p-nilpotent groups for a prime p, then every F-critical group is either a Schmidt group or a cyclic group of prime order. Therefore a subgroup-closed class Z is contained in F if and only if F contains every Schmidt group and every cyclic group of prime order in Z. This raises the following question. Which are the saturated formations F such that every F-critical group is either a Schmidt group or a cyclic group of prime order? This question was proposed by L. A. Shemetkov in [MK92, Problem 9.74]. Hence we shall say that a formation F has the Shemetkov property or F is a ˇ S-formation if every F-critical group is a Schmidt group or a cyclic group of prime order. ˇ The first investigation of S-formations was taken up by V. N. Semenchuk and A. F. Vasil’ev [SV84] in the soluble realm. A. Ballester-Bolinches and M. D. Pérez-Ramos [BBPR95] determined necessary and sufficient conditions for ˇ a subgroup-closed saturated formation to be a S-formation. This result can be ˇ used to give examples of subgroup-closed saturated S-formations of different nature. On the other hand, L. A. Shemetkov [She92, Problem 10.22] proposes the following question: ˇ Let F be a subgroup-closed S-formation. Is F saturated?


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A. N. Skiba [Ski90] answered this question affirmatively in the soluble universe. However his result does not remain true in the general case as A. Ballester-Bolinches and M. D. Pérez-Ramos showed in [BBPR96b]. In this ˇ paper, they gave a criterion for a subgroup-closed S-formation to be saturated from which Skiba’s result emerges. An alternative approach to Shemetkov’s question is due to S. F. Kamornikov [Kam94]. There he proved that a ˇ subgroup-closed S-formation is a Baer-local formation. We shall begin our treatment of this material with a general result concerning formations F whose F-critical groups have the composition factors of their F-residual in a class of simple groups X. We shall then specialise X to J and in this class aim to give a detailed account of the present state of knowledge. Theorem 6.4.4 ([BB05]). Let ∅ = X be a class of simple groups satisfying π(X) = char X. Denote Y = X ∩ P, the abelian groups in X. For a formation F, the following statements are equivalent: 1. Every F-critical group G such that GF is contained in OX (G) is either a Schmidt group or a cyclic group of prime order. 2. Every F-critical group G whose F-residual is contained in OX (G) is soluble, F is a Y-local formation, and for each prime p ∈ Y ∩ char F there exists a set of primes π(p) with p ∈ π(p) such that F is Y-locally defined by the Y-formation function f given by ⎧ ∼ ⎪ ⎨Eπ(p) if S = Cp , p ∈ Y ∩ char F, f (S) = ∅ if S ∼ = Cp , p ∈ Y \ char F, ⎪ ⎩ F if S ∈ X ∪ (X \ Y). Proof. 1 implies 2. It is clear that every F-critical group G with GF ≤ OX (G) is soluble by Statement 1. The next stage of the proof is to show that F is a Y-local formation. Applying Lemma 3.1.21, it is enough to prove that F is (Cp )-local for all primes p ∈ Y. Let p be a prime in Y. By Theorem 3.1.11, the smallest (Cp )-local formation F1 containing F is (Cp )-locally defined by the (Cp )-local formation function f given by f (p) = Q R0 A/ CA (H/K) : A ∈ F and H/K is an abelian p-chief factor of A , and f (S) = Q R0 A/L : A/L is monolithic and if Soc(A/L) ∈ E(S) if S ∈ (Cp ) . Denote π(p) = π Sp f (p) , and consider the (Cp )-local formation M = LF(Cp ) (g), where g(p) = Eπ(p) , g(S) = F

if S ∈ (Cp ) .

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It is clear that F is contained in M. Assume that M is not contained in F and derive a contradiction. Let G be a group of minimal order in the non-empty class M \ F. Since G is monolithic and G ∈ / F, it follows that N := Soc(G) is a p-group. Then p ∈ char F. Moreover there exists a subgroup H of G such that H ∈ CritS (F). By Statement 1, H is a Schmidt group as π(G) ⊆ char F. Suppose that H is a proper subgroup of G. Since HN/N ∈ F and H ∈ / F, it follows that H ∩ N = 1. In particular, |H| = pa q b for some prime q = p and one of the non-trivial Sylow subgroups of H is normal in H by [Hup67, III, 5.2]. Assume that a Sylow q-subgroup Q of H is normal in H. Then H/Q ∈ F because p ∈ char F and so M contains Sp . Since H/(H ∩ N ) belongs to F, it follows that H ∈ R0 F = F. This would contradict the choice of G. Therefore H has a normal Sylow p-subgroup and a Sylow q-subgroup of H is cyclic by [Hup67, III, 5.2]. Suppose that q does not belong to π(p) . Then H ∩ N is a Sylow p- subgroup of H. Assume not, and let P be a Sylow p-subgroup of H containing H ∩ N . Since [P, Q]=P (see the proof of [Hup67, III, 5.2(c)]), it follows that Q(H ∩ N )/(H ∩ N ) is not contained in Op ,p (H/H ∩ N ). This implies that q ∈ π(p), contrary to our supposition. Hence H ∩ N is a Sylow p-subgroup of G. Let C = CG (N ). If H is a subgroup of C, then H is nilpotent. This is not possible. Hence H is not contained in C. Since H ∩ N is contained in C, it follows that q divides |G/C|. Denote A = [N ](G/C). By Corollary 2.2.5, A ∈ M. Hence q ∈ π(p). This contradiction proves that q ∈ π(p). The definition of π(p) implies the existence of a group B ∈ F and an abelian p-chief factor L/M of B satisfying that q divides B CB (L/M ) . By Corollary 2.2.5, C = [L/M ] B CB (L/M ) ∈ F. Denote V = L/M and B∗ = B CB (L/M ), and E = g CB (L/M ) for some element g ∈ B such that o g CB (L/M ) = q. It is clear that V is a faithful and irreducible B ∗ -module over GF(p). Moreover V , regarded as E-module, is completely reducible by Maschke’s theorem [DH92, B, 4.5]. Since V is faithful for B ∗ and E is a cyclic group of order q, we can find an irreducible E-submodule W of V such that W is a faithful E-module. Let F = W E be the corresponding semidirect product. Then F is isomorphic to E(q|p), the unique Schmidt primitive group in defined (H) = 1, [DH92, B, 12.5]. Then F ∈ F because F is subgroup-closed. If Φ O p then H/Φ Op (H) is isomorphic to E(q|p) constructed above. This implies that H/Φ Op (H) ∈ F ⊆ M and so H ∈ M because M is (Cp )-saturated by Theorem 3.2.14. The minimality of G implies that H ∈ F, and this contradicts the fact that H ∈ CritS (F). Thus our supposition is false and Φ Op (H) = 1. But then H is isomorphic to E(q|p) ∈ F and we have reached the contradiction that H ∈ F. Consequently H = G and G is a Schmidt group with a normal Sylow p- subgroup, P say. Let Q be a Sylow q-subgroup of G. Since N is the unique minimal normal subgroup of G and Φ(Q) is normal in G, we have that Φ(Q) = 1 and Q is a cyclic group of order q. Note that Φ(P ) is elementary abelian by [DH92, VII, 6.18] and G/Φ(P ) ∼ = E(q|p) and Φ(P ) = Φ(G). Hence G is


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an epimorphic image of the maximal Frattini extension E of E(q|p) with p-elementary abelian kernel (see [DH92, Appendix β]). Denote by A the kernel of the above extension. Then E/A is isomorphic to E(q|p) and A ≤ Φ(E). Moreover CE Soc(A) is a p-group by [GS78, Theorem 1]. Let Q∗ be a Sylow / F and A is q- subgroup of E. If AQ∗ belongs to F, then E is F-critical because G ∈ contained in each maximal subgroup of E (note that every Sylow subgroup of ∗ is E belongs to F). This implies that E is a Schmidt group. In particular, AQ ∗ ∗ / F. nilpotent and then 1 = Q ≤ CE Soc(G) . This contradiction yields AQ ∈ In this case, we can find a subgroup J of AQ∗ such that J ∈ CritS (F). By Statement 1, J should be a Schmidt group with an elementary abelian Sylow p-subgroup. This implies that J is isomorphic to E(q|p) ∈ F, and we have a contradiction. Therefore F = M is a Y-local formation, and we have completed the proof of the implication. 2 implies 1. Suppose that F is a Y-local formation and there exists a set of primes π(p) with p ∈ π(p), for each p ∈ π = char F, such that F is Y-locally defined by the Y-formation function f given by f (p) = Eπ(p) if p ∈ π ∩ Y, f (q) = ∅, if p ∈ Y \ π, and f (E) = F for every simple group E ∈ X ∪ (X \ Y). We shall prove that every group in CritS (F) whose F-residual is an X-group is a Schmidt group or a cyclic group of prime order. Let G be a group in CritS (F) such that GF ≤ OX (G). By Condition 2, G is soluble. We prove by induction on |G| that G is a Schmidt group or a cyclic group of prime order. If G is a p-group for some prime p and G has not order p, then p ∈ π ∩ Y and so G is an Eπ(p) -group. In particular, G is an F-group. This contradicts our choice of G. Hence G is cyclic group of prime order. Assume that G has not prime power order and there exists a minimal normal subgroup B of G such that G/B is not an F-group. Then B has to be contained in Φ(G) because G is F-critical. Therefore G/B ∈ CritS (F). By induction, G/B is either a Schmidt group or a cyclic group of prime order. If G/B is a cyclic group of prime order, then so is G. This contradiction shows that G/B is a Schmidt group. Let p be the prime dividing |B|. Then G is a {p, q}-group, for some prime q = p and either G has a normal Sylow p- subgroup or G has a normal Sylow q-subgroup. Suppose that G has a normal Sylow p-subgroup, P say. Then GF is a p- group and so p ∈ π ∩ Y. Since G is not nilpotent because it is F-critical, then there exists a q-element g ∈ G such that g does not centralise P . Let us choose g of minimal order. Then every proper subgroup of N = g centralises P . Consequently P N is a Schmidt group. Suppose that P N is a proper subgroup of G. Then P N ∈ F. Hence P N/ Op ,p (P N ) belongs to Eπ(p) . It follows that q ∈ Eπ(p) , G ∈ f (p) = Eπ(p) and G is an F-group. This contradiction yields G = P N and G is a Schmidt group. If G has a normal Sylow q-subgroup, a similar argument can be used to conclude that G is a Schmidt group. Consequently we may assume that G/B ∈ F for every minimal normal subgroup B of G. Then N = GF = Soc(G) is a minimal normal subgroup of G and it is a p-group for some prime p ∈ π ∩ Y. If N is contained in

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Φ(G), then Op ,p (G/N ) = Op ,p (G)/N = Op (G)/N and so G/ Op (G) ∈ Eπ(p) . Hence G ∈ f (p) = Eπ(p) and G is an F-group. This contradiction implies that Φ(G) = 1 and G is a monolithic primitive group. If N = G, then G is a cyclic group of prime order. Thus we may assume that N is a proper subgroup of G. Since G ∈ / F, it follows that G ∈ / f (p) = Eπ(p) . Hence there exists an element g ∈ G whose order is a prime q ∈ / π(p). Denote A = g. If N A were a proper subgroup of G, then N A ∈ F. Hence N A/ Op ,p (N A) ∼ = A belongs to Eπ(p) . This contradiction yields G = AN and then every maximal subgroup of G is nilpotent. Consequently G is a Schmidt group and the Statement 1 of the theorem is now clear. If X = J, the class of all simple groups, then Y is the class of all abelian simple groups. Therefore we have: Corollary 6.4.5. Let F be a formation. The following statements are equivalent: 1. Every F-critical group is either a Schmidt group or a cyclic group of prime order. 2. Every F-critical group is soluble, F is solubly saturated and, for each prime p ∈ char F, there exists a set of primes π(p) with p ∈ π(p) such that F is P-locally defined by the P-formation function f given by ⎧ ∼ ⎪ ⎨Eπ(p) if S = Cp , p ∈ char F, f (S) := ∅ / char F, if S ∼ = Cp , p ∈ ⎪ ⎩ F if S ∈ J \ P. ˇ In particular, every subgroup closed S-formation is solubly saturated (see [Kam94]). It is clear that every soluble formation is saturated if and only if it is solubly saturated. Hence combining Theorem 6.4.4 and Theorem 6.4.3 we have ˇ Corollary 6.4.6. Let F be a soluble S-formation. Then F is saturated and it is closed under taking products of F-subnormal subgroups. The saturated formation S of all soluble groups shows that the converse of the above result does not hold. ˇ There also exist non-saturated S-formations. Example 6.4.7 ([BBPR96b]). Let F = (G : every {3, 5}-subgroup of G is nilpotent). By [DH92, VII, 6.5], F is a subgroup-closed formation. Let G be an F-critical group. Then G has a {3, 5}-subgroup H such that H is not nilpotent. The choice of G implies that H = G. Especially, G is a {3, 5}-group which is not nilpotent but all its subgroups are nilpotent. Therefore G is a Schmidt group.


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Take G = Alt(5), the alternating group of degree 5. Then G ∈ F. Let E be the maximal Frattini extension of G with 5-elementary abelian kernel. Then E/Φ(E) is isomorphic to G and CG Φ(E) = O5 (G) = 1 by [GS78, Proposition 5]. If F were saturated, it would be true that E ∈ F. But this is not true because Φ(E)P , for a Sylow 3-subgroup P of E, is not nilpotent inasmuch as P does not centralise Φ(E). ˇ The following result provides a criterion for a S-formation to be saturated. ˇ Theorem 6.4.8 ([BBPR96b]). Let F be a S-formation. The following statements are equivalent: 1. F is a saturated formation. 2. Let G be a primitive group of type 2 such that G ∈ F. If p is a prime dividing |Soc(G)| and V is an irreducible and faithful G-module over GF(p), then every Schmidt subgroup isomorphic to E(q|p) of [V ]G belongs to F. Proof. If F is a saturated formation, then the statement 2 is always true: Let G ∈ F be a primitive group of type 2; then G ∈ F (p) for every prime p ∈ π Soc(G) , where F is the canonical local definition of F. The semidirect product [V ]G ∈ Sp F (p) = F (p) ⊆ F, for each irreducible and faithful G-module V over GF(p) and p dividing the order of Soc(G). Now the result is clear because F is subgroup-closed. To complete the proof we now show that 2 implies 1. By Corollary 6.4.5, F is solubly saturated and, for each prime p ∈ char F, there exists a set of primes π(p) with p ∈ π(p) such that F = LFP (f ), where f is the P-formation function given by ⎧ ∼ ⎪ ⎨Eπ(p) if S = Cp , p ∈ char F, f (S) := ∅ / char F, if S ∼ = Cp , p ∈ ⎪ ⎩ F if S ∈ J \ P. Applying Theorem 3.4.5, the formation H = LF(f ) is the largest saturated formation contained in F. Suppose, by way of contradiction, that the class F \ H is non empty, and let G be a group of minimal order in this class. Then G group of type 2 and, since G ∈ / H, there exists a prime is a primitive p ∈ π Soc(G) ⊆ char F such that G ∈ / f (p) = Eπ(p) . Consequently there exists an element g ∈ G of order q, for some prime q ∈ / π(p). Furthermore, by [DH92, B, 10.9], G has an irreducible and faithful module V over GF(p). Let X = [V ]G be the corresponding semidirect product. Denote A = g and consider the subgroup V A of X. V , regarded as an A-module, is semisimple by [DH92, B, 4.5]. Moreover, since V is faithful and A is a cyclic group of order q, we can find an irreducible A-submodule W of V such that W is a faithful A-module. Let B = W A be the corresponding semidirect product. It is clear that B is a Schmidt group which is isomorphic to E(q|p). By Condition 2, B ∈ F. It yields B/ CB (W ) ∈ f (p). Hence q ∈ π(p). This contradiction shows that H = F and F is saturated.

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ˇ Remark 6.4.9. Let F be a saturated S-formation. According to Corollary 6.4.5, F = LFP (f ), where f is the P-formation function given by f (p) = Eπ(p) , p ∈ π(p), if p ∈ char F, f (p) = ∅ if p ∈ / char F and f (S) = F if S ∈ J \ P. By Theorem 3.1.17, the canonical P-local definition of F, F say, is given by ⎧ ⎪ ⎨F ∩ Eπ(p) if p ∈ char F, F (S) = ∅ if p ∈ / char F, ⎪ ⎩ F if S ∈ J \ P. Furthermore, by Corollary 3.1.18, the canonical local definition of F is F (p) = F ∩ Eπ(p) if p ∈ char F and F (p) = ∅ otherwise. Using familiar arguments it can be proved that F = LF(f ). Unfortunately, not every saturated formation which is locally defined as ˇ above is a S-formation. Example 6.4.10 ([BBPR95]). Consider F = LF(f ) which is locally defined by the formation function given by f (2) = f (3) = E{2,3} , f (5) = E{2,5} , and f (q) = ∅ if q = 2, 3, 5. Then F is subgroup-closed and char F = {2, 3, 5}; Alt(5) is F-critical but it is neither a Schmidt group nor a cyclic group of prime order. For saturated formations of soluble groups, the following characterisation holds. Theorem 6.4.11. Let F be a saturated formation of soluble groups. Then every soluble F-critical group is either a Schmidt group or a cyclic group of prime order if and only if F satisfies the following condition: there exists a formation function f , defined by f (p) = Sπ(p) for a set of primes π(p) such that p ∈ π(p) if p ∈ char F and f (p) = ∅ otherwise, such that F = LF(f ). Proof. Assume that every soluble F-critical group is either a Schmidt group or a cyclic group of prime order. Let F be the canonical local definition of F = LF(F ). Since F is subgroup-closed, it follows that F (p) is subgroupclosed for each p ∈ π = char F by Proposition 3.1.40. Hence F (p) is contained in F ∩ Sπ(p) , where π(p) = char F (p) for every p ∈ π. Assume that there exists a prime p ∈ π such that F (p) = Sπ(p) ∩ F and let G be a group of minimal order in the non-empty class (F ∩ Sπ(p) ) \ F (p). Then 1 = Soc(G) is the unique minimal normal subgroup of G which is not a p-group. By [DH92, B, 10.9] there exists an irreducible and faithful G-module V over GF(p). Let X = [V ]G be the corresponding semidirect product. It is clear that X is a primitive group and V = Soc(X) is the unique minimal normal subgroup of X. Since G ∈ / F (p), we have that X is not an F-group. Let M be a maximal subgroup of X. If CoreX (M ) = 1, then M is isomorphic to G. Hence M ∈ F. Assume that CoreX (M ) = 1. Then V ≤ M and M ∩ G is a maximal subgroup of G. In this case M ∩ G ∈ F (p) and so M ∈ Sp F (p) = F (p) ⊆ F. Hence X is


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an F-critical soluble group. By hypothesis, X is a Schmidt group (clearly X cannot be a cyclic group of prime order). In particular G is a nilpotent group. Assume that Soc(G) is a proper subgroup of G. Let A be a maximal subgroup of G containing Soc(G). Then V A is a maximal subgroup of X and so V A is nilpotent. Let q be the prime dividing |Soc(G)|. Then the Sylow q-subgroup Aq of A is non-trivial and Aq ≤ CV A (V ) = V . This contradiction yields Soc(G) = G and hence G is a cyclic group of order q ∈ π(p) = char F (p). This means that G ∈ F (p), contrary to our supposition. Consequently, for each prime p ∈ π, we have that F (p) = F ∩ Sπ(p) , where π(p) = char F (p). We are now close to completing the proof of the implication. Let f be the formation function given by f (p) = Sπ(p) if p ∈ π and f (q) = ∅, if q = π and F (q) = ∅. It is clear that F is contained in LF(f ). Assume that the equality is not true and take a group G ∈ LF(f ) \ F of minimal order. Since LF(f ) is composed of soluble groups, it follows that G is a soluble primitive group. Let p be the prime dividing |Soc(G)|. Then G/N ∈ Sπ(p) ∩ F = F (p). Consequently G ∈ Sp F (p) ⊆ F, and we have reached a contradiction. Hence F = LF(f ). Suppose now that there exists a formation function f , defined by f (p) = Sπ(p) for a set of primes π(p) such that p ∈ π(p) if p ∈ char F and f (p) = ∅ otherwise, such that F = LF(f ). Let G a soluble F-critical group. Assume that Φ(G) = 1. Then G is a primitive group. Let p be the prime dividing Soc(G). If q = p is a prime dividing the order of G, and g ∈ G is an element of G of order q, then g does not centralise N . Denote A = g. If N A were a proper subgroup of G, then N A ∈ F. Hence N A/ Op ,p (N A) ∼ = A belongs to Sπ(p) and q ∈ π(p). Since G does not belong to Sπ(p) , it follows that G = AN for some subgroup A of G of prime order. This means that G is a Schmidt group. Hence, in this case, G is either a Schmidt group or a cyclic group of prime order. Assume that Φ(G) = 1. The group G∗ = G/Φ(G) is an F-critical group and Φ(G∗ ) = 1. The above argument implies that G∗ is either a Schmidt group or a cyclic group of prime order. Consequently, G is a Schmidt group and the other implication of the theorem is now clear. We now present a set of necessary and sufficient conditions for a saturated ˇ formation to be a S-formation. Theorem 6.4.12 ([BBPR95]). Let F be a saturated formation. Then F is a ˇ S-formation if and only if F satisfies the following two conditions: 1. There exists a formation function f , defined by f (p) = Eπ(p) for a set of primes π(p) such that p ∈ π(p) if p ∈ char F and f (p) = ∅ otherwise, such that F = LF(f ); this formation function f satisfies the following property: If G ∈ CritS (F) ∩ b(F) and G isan almost simple group such that G∈ / f (p) for some prime p ∈ π Soc(G) , then G ∈ / f (q) for each prime q ∈ π Soc(G) . (6.2) 2. CritS (F) ∩ b(F) does not contain non-abelian simple groups.

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ˇ Proof. Denote π := char F. If F is a S-formation, every group G ∈ CritS (F) ∩ b (F) has abelian socle. Bearing in mind Remark 6.4.9, only the sufficiency of the conditions is in doubt. Assume that there exists a set of primes π(p) with p ∈ π(p), for each p ∈ π, such that F is locally defined by the formation function f given by / π. Then F = LFP (fˆ), where fˆ is the f (p) = Eπ(p) if p ∈ π, and f (q) = ∅ if q ∈ P-formation function defined by fˆ(p) = f (p) for all p ∈ P and fˆ(S) = F for all S ∈ J \ P (see Corollary 3.1.13 and Corollary 3.1.18). By Corollary 6.4.5, it will be sufficient to show that every F-critical group is soluble to conclude ˇ that F is a S-formation. Suppose that CritS (F) \ S is not empty and derive a contradiction. Let G be a group of minimal order in CritS (F) \ S. Then Φ(G) = 1 and G is a monolithic primitive group in b(F). Let N = Soc(G) be the unique minimal normal subgroup of G. If N = G, then G is simple. Since this contradicts 2, we must have N < G, so that N ∈ F. Assume that N is non-abelian. Then CG (N ) = 1 and π(N ) ⊆ π(p), for every p ∈ π(N ). Now since G ∈ / F, there exists a prime q ∈ π(G) such that q ∈ / π(p) for some prime p ∈ π(N ); in particular, q ∈ / π(N ). Let g be an element of G of order q. Denote A = g. The group A operates by conjugation on N and (|N |, |A|) = 1. By [DH92, I, 1.3], there exists an A-invariant Sylow p- subgroup Np of N. Since G is not soluble, it follows that Np A is a proper subgroup of G. Hence Np A ∈ F. Since Np is normal in Np A and q ∈ / π(p), it follows that A ≤ CG (Np ). On the other hand, N = N1 × · · · × Nr is a direct product of non-abelian simple groups Ni , 1 ≤ i ≤ r, which are pairwise isomorphic. Since Np = (N1 )p × · · · × (Nr )p for some Sylow p-subgroup (Ni )p of Ni , 1 ≤ i ≤ r, and (Ni )p ≤ Ni ∩ Nig , it follows that Ni = Nig , 1 ≤ i ≤ r. Hence A normalises Ni for all i ∈ {1, . . . , r}. Suppose that Ni A = G for every i ∈ {1, . . . , r}. Then Ni A ∈ F. Consequently A ≤ CG (Ni ) for every i ∈ {1, . . . , r} because q ∈ / π(p). This implies that A ≤ CG (N ) = 1, which is impossible. We conclude for this contradiction that G = Ni A for some i ∈ {1, . . . , r} and N = Ni is a non-abelian simple group. In particular, G is an almost simple group. We may apply now Condition 1 and deduce that G∈ / f (r) for each r ∈ π(N ). But the above argument shows that A centralises a Sylow r-subgroup of N for each r ∈ π(N ). Hence A ≤ CG (N ) = 1, and again we have a contradiction. Therefore N must be abelian. Let p be the prime dividing |N |. Then p ∈ π and G ∈ / f (p) because G ∈ / F. Let g be an element of G whose order is a prime q ∈ / π(p). Denote again A = g. If N A = G, then G is soluble. This contradiction implies that N A is a proper subgroup of G. But in this case q ∈ π(p) because N is self-centralising in G, contrary to our initial supposition that q ∈ / π(p). Thus we are forced to the conclusion that every F-critical group is soluble ˇ and F is a S-formation. Remark 6.4.13. None of the conditions 1 and 2 can be dispensed with in Theorem 6.4.12 (see [BBPR95, Examples]).


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With the help of the preceding theorem we can now give examples of ˇ subgroup-closed saturated S-formations of different nature. The simplest example is the formation F of the p-nilpotent groups, p a prime number. It is clear that F = LF(f ), where f (p) = Sp and f (q) = E for every prime q = p. Hence F belongs to the family of saturated formations described in Theorem 6.4.12. Let G ∈ CritS (F) ∩ b(F). Then G is not a p -group. Thus p ∈ π(G). Since G is not p-nilpotent, we can apply the p-nilpotence criterion of Frobenius [Hup67, IV, 5.8] to conclude that G = NG (P ) for some p-subgroup 1 = P of G. Hence Soc(G) is abelian and F satisfies Conditions 1 and 2 of Theorem 6.4.12. Conˇ sequently F is a S-formation. This is a classical result due to Itˆ o ([Hup67, IV, 5.4]). Less trivial is the following result. Theorem 6.4.14 ([BBPR95]). Let {πi : i ∈ I} be a family of pairwise disjoint sets of primes and put π = {πi : i ∈ I}. Let F be the saturated formation locally defined by the formation function f given by f (p) = Eπi if / π. Then F is a subgroup-closed saturated p ∈ πi , i ∈ I, and f (q) = ∅ if q ∈ ˇ S-formation. Proof. By Proposition [DH92, IV, 3.14], F is a subgroup-closed saturated formation. It is clear that π = char F. Note that a group G belongs to F if and only if G has a normal Hall πi -subgroup for every i ∈ I. We claim that F satisfies Conditions 1 and 2 of Theorem 6.4.12. On one hand, the formation function defined above satisfies Condition 1. On the other hand, assume that G is a non-abelian simple group in CritS (F) ∩ b(F) and derive a contradiction. Then 2 ∈ char F, by the Odd Order Theorem [FT63], and so there exists an element i ∈ I such that 2 ∈ πi . Denote π1 = π \ πi and π2 = πi . If X is a group in F, we denote by X1 the normal Hall π1 subgroup of X. The normal Hall π2 -subgroup of X is denoted by X2 . We reach a contradiction after the following steps: Step 1. Let M be a maximal subgroup of G such that M1 = 1 and M2 = 1. Then Sylp (M ) ⊆ Sylp (G), for every prime p dividing the order of M . Let p be a prime dividing |M | and let Mp ∈ Sylp (M ). There exists a Sylow p-subgroup Gp of G such that Mp ⊆ Gp . Assume, arguing by contradiction, that Mp is a proper subgroup of Gp . Then Mp is a proper subgroup of Tp = NGp (Mp ). Suppose that p ∈ π2 (similar arguments can be used if p ∈ π1 ). In this case, we have that M1 ≤ NG (Mp ) and so M1 , Tp ≤ NG (Mp ), which is a proper subgroup of G because G is a non-abelian simple group. Let L be a maximal subgroup of G such that NG (Mp ) ≤ L. Then L = L1 × L2 because L ∈ F. Furthermore, L1 = 1 and L2 = 1 as p ∈ π(L) and M1 ≤ L1 . Hence M2 , L2 ≤ NG (M1 ) = M and so Mp is a Sylow p-subgroup of L2 . But then Tp is a Sylow p-subgroup of L2 containing properly a Sylow p-subgroup of L2 . This contradiction yields Mp = Gp and Mp is a Sylow p-subgroup of G. Step 2. Let p be a prime in π1 and let 1 = P be a p-subgroup of G. Then NG (P ) is of Glauberman type with respect to the prime p (cf. [Gor80, 4.1, page 281]).

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It is clear that NG (P ) is a proper subgroup of G. Hence NG (P ) ∈ F and NG (P ) = NG (P )1 × NG (P )2 . In particular, NG (P ) is p-soluble. Suppose that p = 3, then SL(2, 3) is not involved in G because the Sylow 2-subgroup of SL(2, 3) is not centralised by a Sylow 3-subgroup of SL(2, 3). Hence NG (P) is strongly p-soluble in the sense of [Gor80, page 234]. Moreover, Op NG (P ) = 1 and p is an odd prime. Therefore we can apply [Gor80, pages 268–269] to conclude that NG (P ) is p-constrained and p-stable. By [Gor80, Theorem 8.2.11, page 279] we have that NG (P ) is of Glauberman type with respect to the prime p. Step 3. Let p be a prime in π1 and let 1 = P be a Sylow p-subgroup of G. Then P ≤ N , where N = NG ZJ(P ) and ZJ(P ) is the centre of the Thompson subgroup of P . By Step 2, the normaliser of every nonidentity p-subgroup of G is of Glauberman type with respect to the prime p. Applying [Gor80, Theorem 8.4.3, page 282], we conclude that P ∩ G = P ∩ G = P ∩ N . Step 4. Let M be a maximal subgroup of G. Then M is either a π1 -group or a π2 -group. Since G is F-critical, we have that every maximal subgroup of G belongs to F. Assume that the above statement is not true. Then the set Σ := {M : M is a maximal subgroup of G, M1 = 1, and M2 = 1} is non-empty. We define a binary relation R in Σ by M R L if and only if M2 ≤ L2 . Clearly R is reflexive and transitive. Moreover, if M R L and L R M , then M2 = L2 and so M = NG (M2 ) = NG (L2 ) = L. Hence (Σ, R) is a partially ordered set. Let M be a maximal element of (Σ, R). Since M1 = 1 and M1 is soluble, by the Feit-Thompson theorem, we have that (M1 ) is a proper subgroup of M1 . Let p be a prime dividing |M1 : (M1 ) | and let P be a Sylow p-subgroup of M . Then, by Step 1, P is a Sylow p-subgroup of G and moreover P ≤ N , where N = NG ZJ(P ) , by Step 3. Clearly N is a proper subgroup of G and M2 ≤ N . Let L be a maximal subgroup of G containing N . Then M2 ≤ L2 . Moreover L1 = 1 because p ∈ π1 . Therefore L ∈ Σ and M R L. By the maximality of M , we have that L = M . In particular, P is contained in (M1 ) because N ≤ (M1 ) × (M2 ) . Hence |M1 : (M1 ) | is a p -number, contrary to our supposition. Step 5. G has a maximal subgroup of odd order. If every maximal subgroup of G were of even order, then G would be an Eπ2 -group by Step 4. This would imply that G ∈ F, and we would have a contradiction. Hence we conclude that G has a maximal subgroup of odd order. Applying [LS91, Theorem 2], we have that G is one of the following groups: Alt(p), p a prime number, p ≡ 3 (mod 4) and p = 7, 11, 23; L2 (q), q ≡ 3 (mod 4), Lεp (q), ε = ±1, p odd prime, and G = U3 (3) or U5 (2); M23 , Th, F2 , or F1 . In the remaining steps we rule out the above possibilities for the nonabelian simple group G.


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Step 6. G is not of the type Alt(p), p a prime number, p ≡ 3 (mod 4) and p = 7, 11, 23. Suppose that G = Alt(p) for some prime p, p ≡ 3 (mod 4). It is clear that Alt(p − 1) is a maximal subgroup of G, Alt(p − 1) ∈ F and Alt(p − 1) ∈ Eπ2 . Let P be a Sylow p-subgroup of G and let M be a maximal subgroup of G such that NG (P ) ≤ M . By Step 4, M is either a π1 -group or a π2 -group. If M ∈ Eπ2 , then G ∈ Eπ2 and so G ∈ F. This contradiction yields M ∈ Eπ1 . Hence M = P = NG (P ) and we have a contradiction. Step 7. G = L2 (q), q ≡ 3 (mod 4). Assume that G = L2 (q), for some q ≡ 3 (mod 4). Then by [LS91, Theorem 2], if M is a maximal subgroup of odd order, then M is isomorphic to a semidirect product of an elementary abelian group of order q and a cyclic group of order (q − 1)/2. On the other hand, by the theorem of Dickson [Hup67, II, 8.27], G has a subgroup H which is isomorphic to the dihedral group of order 2 (q − 1)/2 . Then H ∈ Eπ2 by Step 4, and therefore M ∈ Eπ2 . It means that G ∈ Eπ2 . This contradiction confirms Step 7. Step 8. G = Lεp (q), ε ∈ {±1}, p odd prime. Assume that G = Lεp (q) for some odd prime p, ε ∈ {±1} and G = U3 (3) or of G of odd U5 (2). Again, by [LS91, Theorem 2], if M is a maximal subgroup order, then the order of M is p (q p − ε)/(q − ε)(q − ε, p) . From Tables 3.5A and 3.5B and the corresponding results of Chapter 4 of [KL90], it follows that there exists a proper subgroup M of G of even order such that p ∈ π(M ). Hence M ∈ Eπ2 and then G ∈ Eπ2 . This contradiction proves Step 8. Step 9. G is not of type M23 , Th, F2 , or F1 . Using the Atlas [CCN+ 85] as reference for the list of maximal subgroups of G, we see that in this case G should be a π2 -group. This final contradiction proves the theorem. We now turn our attention to an application of Theorem 6.4.12 leading to ˇ a characterisation of the subgroup-closed saturated S-formations. Theorem 6.4.15 ([BBPR96b]). Let F = LF(F ) be a subgroup-closed saturated formation. Denote π = char F and π(p) = char F (p), for every p ∈ π. Any two of the following statements are equivalent: ˇ 1. F is a S-formation. 2. A π-group G belongs to F if and only if NG (Q)/ CG (Q) belongs to Eπ(p) for each p-subgroup Q of G and each prime p ∈ π. 3. A π-group G belongs to F if and only if NG (Q) ∈ Eπ\{p} Eπ(p) for each non-trivial p-subgroup Q of G and each prime p ∈ π. ˇ Proof. 1 implies 2. Assume that F is a S-formation. According to Theorem 6.4.12, we have that F (p) = Eπ(p) ∩ F, for every p ∈ π. Let G be a π-group in F. Suppose that a prime p ∈ π is fixed and let Q be a p-subgroup of G. Then NG(Q) ∈ F because F is subgroup-closed. In particular NG (Q)/ Op NG (Q) ∈ F (p) ⊆ Eπ(p) . Since Q is a normal p-subgroup of NG (Q), it follows that Op NG (Q) ≤ CG (Q). This means that

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NG (Q)/ CG (Q) ∈ Eπ(p) . Conversely, assume that G is a π-group such that NG (Q)/ CG (Q) belongs to Eπ(p) for each p-subgroup Q of G and each p ∈ π, but G is not an F-group. If we choose G of minimal order among the groups X ∈ / F satisfying the above property, we have that G is an F-critical group ˇ because this property holds in every subgroup of G. Since F is a S-formation, it follows that G is a Schmidt group. In particular π(G) = {p, q} for two distinct primes p and q in π and G has a normal Sylow p-subgroup, P say. By hypothesis, we have that G/ CG (P ) ∈ Eπ(p) . If q were not in π(p), it would be true that Q ≤ CG (P ). This is not possible. Hence q ∈ π(p) and then Q ∈ Eπ(p) ∩ F = F (p). Therefore G ∈ Sp F (p) = F (p) ⊆ F. This contradiction yields G ∈ F. 2 implies 1. We see that F satisfies the Statements 1 and 2 of Theorem 6.4.12. (a) For each prime p ∈ π, we have that F (p) = Eπ(p) ∩ F. Let p be a prime in π. Since F (p) is subgroup-closed by Proposition 3.1.40, it follows that F (p) is contained in Eπ(p) ∩ F. Assume, by way of contradiction, that F (p) = Eπ(p) ∩ F and let G be a group of minimal order in the non-empty class (Eπ(p) ∩ F) \ F (p). Then 1 = Soc(G) is the unique minimal normal subgroup of G and it is not a p-group. By [DH92, B, 10.9], there exists an irreducible and faithful G-module over GF(p). Let X = [V ]G be the corresponding semidirect product. X is a primitive group and X ∈ / F because G∈ / F (p). Let q be a prime in π and let Q be a non-trivial q-subgroup of G. Suppose that p = q. Then NX (Q) is a proper subgroup of G because V is the unique minimal normal subgroup of X. Let L be a maximal subgroup of G containing NX (Q). If CoreX (L) = 1, then L is isomorphic to G and if CoreG (L) = 1, then V ≤ L and L = V (G ∩ L). Since G ∩ L ∈ F (p), it follows that L ∈ Sp F (p) = F (p) ⊆ F. In both cases, L ∈ F. Therefore NX (Q) ∈ F and so NX (Q)/ CX (Q) ∈ Eπ(q) . Now, if p = q and NX (Q) is a proper subgroup of G, we can argue as above to conclude that NX (Q)/ CX (Q) ∈ Eπ(p) . If Q is a normal subgroup of X, then V = Q and X/ CX (Q) ∈ Eπ(p) because it is isomorphic to G. Since X is a π-group, we can apply Statement 2 to conclude that X ∈ F. This contradiction shows that F (p) = Eπ(p) ∩ F. (b) CritS (F) ∩ b(F) does not contain primitive groups of type 2. Assume that G is a primitive group of type 2 in CritS (F) ∩ b(F). Since G is F-critical, it follows that G is a π-group. On the other hand, applying Statement 2, we can determine a prime p ∈ π and a p-subgroup Q of G such / Eπ(p) . Then Q is non-trivial. Suppose that NG (Q) is that NG (Q)/ CG (Q) ∈ a proper subgroup of G. Then NG (Q) ∈ F ⊆ Eπ\{p} Eπ(p) . This means that NG (Q)/ CG (Q) ∈ Eπ(p) , contrary to our supposition. Hence Q is a normal subgroup of G. But then Soc(G) ≤ Q and Soc(G) is abelian. This contradiction confirms Statement b. From Statements a and b we deduce that F enjoys the properties given in ˇ Theorem 6.4.12. This means that F is a S-formation. Assume now that G is a π-group. Let p be a prime in π and let Q be a nontrivial p-subgroup of G. If NG (Q) ∈ Eπ\{p} Eπ(p) , then NG (Q)/ CG (Q) ∈ Eπ(p) .


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This elementary remark proves that 2 implies 3. Now, if Statement 3 holds, we can repeat the arguments used in the proof of 2 implies 1 to conclude that ˇ F is a S-formation. Consequently 3 implies 1 and the circle of implications is complete. Illustration 6.4.16. Let F be the saturated formation of p-nilpotent groups, p a prime number. It is clear that F = LF(F ), where F (p) = Sp and F (q) = F ˇ for every prime q = p. We have seen above that F is a S-formation. Therefore F satisfies Condition 2 of Theorem 6.4.15. Hence a group G is p-nilpotent if and only if NG (Q)/ CG (Q) is a p-group for every p-subgroup Q of G. The statement 3 of this theorem says that a group G is p-nilpotent if and only if NG (Q) is p-nilpotent for every p-subgroup Q of G. These statements are two equivalent forms of the well known p-nilpotence criterion due to Frobenius. The next topic we broach concerns the relation between the F-residual of a group and the subgroup generated by the F-residuals of some of its F-critical subgroups. The springboard for these results was a theorem of Berkovich [Ber99] stating that the nilpotent residual of a group G is the subgroup generated by the nilpotent residuals of the subgroups A of G such that A/Φ(A) is a Schmidt group. Berkovich’s result is a particular case of a more general theorem as we shall see below. Let F be a formation. Denote by BF the class of all groups G such that G/Φ(G) is an F-critical group. Note that if F = N, the class of all nilpotent groups, BN is the class of all groups such that G/Φ(G) is a Schmidt group (see [Ber99]). Let G be a group and let T(G) = AF : A ≤ G; A ∈ BF if BF ∩ S(G) = ∅; otherwise, we let T(G) = 1. Theorem 6.4.17 ([ABB02]). Let F be a saturated formation, and let G be a group. Then T(G) = GF . Proof. Clearly X F ≤ GF for every subgroup X of G because F is subgroupclosed. Hence T = T(G) is contained in GF . Assume, arguing by contradiction, that G/T ∈ / F. Then G/T has an F- critical subgroup, A/T say. Choose now a minimal supplement A0 of T in A. Then A0 ∩ T is contained in Φ(A0 ). Since A/T is isomorphic to A0 /A0 ∩ T , it follows that A0 /A0 ∩ T is F-critical. Therefore the factor group (A0 /A0 ∩ T ) Φ(A0 /A0 ∩ T ) is also F-critical because F is saturated. It means that F A0 ∈ BF and so AF 0 is contained in T . Hence A0 ≤ A0 ∩ T ≤ Φ(A0 ). It follows that A0 /Φ(A0 ) ∈ F. Now since F is saturated, we conclude that A0 ∈ F. This contradiction completes the proof. We continue the section with an application of Theorem 6.4.17 leading ˇ to a characterisation of the S-formations in the soluble universe among the subgroup-closed saturated formations. It rests on the following result.

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Theorem 6.4.18 ([ABB02]). Let F be a saturated formation of soluble groups of full characteristic such that every soluble group in CritS (F) is a Schmidt group. If A is a group in BF , then AN = AF . Proof. We shall argue by induction on |A|. Firstly, if Φ(A) = 1, then A is an F-critical group. Assume that A is not soluble. Then A is a non-abelian simple group. In this case AF = A = AN . Thus we may suppose that A is soluble and then A is a Schmidt group. Hence there exists a normal abelian Sylow p-subgroup of A, P say, for some prime p. It is rather clear that P coincides with both the nilpotent residual and the F-residual of A. Hence we can assume that Φ(A) = 1. Let N be a minimal normal subgroup of A contained in Φ(A). Then A/N ∈ BF . Hence the induction hypothesis implies that (A/N )F = (A/N )N . This yields AF N = AN N . If AN ∩ N = 1, then AN = AN ∩ AF N = AF (AN ∩ N ) = AF . Thus we can suppose that N is contained in AN and AN = AF N . If N is contained in AF , then AN = AF and the theorem is true. Consequently we shall assume that N ∩ AF = 1, and hence Φ(A) ∩ AF = 1. Note that we can suppose that A/Φ(A) is an extension of a p-group by a q-group for some primes p and q. Since this class is a saturated formation, we have that A is also an extension of a p-group by a q-group. Therefore AN is a p-group, and consequently N is a p-group, too. Let us have a look now at the structure of the F-group A/AF . Given ¯ the corresponding subgroup HAF /AF of a subgroup H of A, denote by H F ¯ By Theorem 6.4.11, we have that the class F is locally defined A/A = A. by a formation function f given by f (r) = Sπ(r) , where π(r) is a set of ¯ is a minimal primes such that r ∈ π(r), for all primes r. Now note that N ¯ ) ∈ Sπ(p) . We can conclude that ¯ Therefore, A/ ¯ CA¯ (N normal subgroup of A. ¯ CA¯ (N ¯ ) ∈ Sπ(p) . If q ∈ π(p), then A ∈ Sπ(p) = f (p) and so A/ CA (N ) ∼ = A/ A ∈ F, against the supposition that A/Φ(A) is F-critical. Therefore q ∈ / π(p) and we have that A/ CA (N ) is a p-group. Since the normal Sylow p-subgroup of A centralises N , it follows that N is central in G. On the other hand, A/Φ(A) is an F-critical group with trivial Frattini subgroup. Since AF ∩ Φ(A) = 1 and F AF Φ(A)/Φ(A) = A/Φ(A) , it follows that AF is abelian. But the equality AN = AF × N yields that AN is complemented in A by a Carter subgroup of A by Theorem 4.2.17. We can conclude that there exists a Carter subgroup C of A such that A = AN C and AN ∩ C = 1. Now N is central in A. Hence N ≤ NG (C) = C. Consequently N ≤ AN ∩ C = 1, contrary to supposition. This final contradiction proves the result. Theorem 6.4.19 ([ABB02]). Let F be a saturated Fitting formation of soluble groups of full characteristic. The following statements are equivalent: 1. Every soluble group in CritS (F) is a Schmidt group. 2. GF = AN : A ≤ G; A ∈ BF for every group G. Proof. By Theorem 6.4.17, we have that GF = AF : A ≤ G, A ∈ BF . Hence if every soluble group in CritS (F) is a Schmidt group, we apply Theorem 6.4.18


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to conclude that GF = AN : A ≤ G; A ∈ BF for every group G. Therefore 1 implies 2. Therefore, only the sufficiency of the Condition 2 is in doubt. To prove that every soluble group in CritS (F) is a Schmidt group, we shall use Theorem 6.4.11. Write F = LF(F ), where F denotes the canonical local definition of F. Consider any prime p. We prove that F (p) = Sπ(p) ∩ F, where π(p) = char F (p). Since F is a subgroup-closed Fitting formation, we have that F( p) is subgroup-closed Fitting formation by Proposition 3.1.40. Since F is integrated, we have that F (p) ⊆ Sπ( p) ∩ F. Assume that Sπ(p) ∩ F = F ( p) and take a group G in (Sπ( p) ∩ F) \ F (p) of minimal order. By familiar reasoning, 1 = Soc(G) is the unique minimal normal subgroup of G. Moreover, Soc(G) cannot be a p-group since, being F full, it holds that F ( p) = Sp F (p). Note that, in fact, Op (G) = 1. By [DH92, B, 10.9], there exists an irreducible and faithful G-module V over GF(p). Consider now the corresponding semidirect product X = [V ]G. Note that if X ∈ F, then X/ CX (V ) ∈ F ( p) and thus X/V ∈ F ( p). This is impossible because G ∼ / F = X/V . Therefore X ∈ and X is in fact an F-critical group. We are ready at this point to reach our final contradiction. Since X F = N A : A ≤ X; A ∈ BF , and X ∈ BF , we have that X F = X N = V . Therefore G ∼ = X/V = X/X N is nilpotent. Then G is a q-group for some prime q ∈ char F (p). Since F (p) is a Fitting class of soluble groups, it follows that Sq is contained in F ( p) by [DH92, IX, 1.9] and then G ∈ F (p). This contradiction yields F (p) = Sπ(p) ∩ F. It follows then that F = LF(f ), where f is the formation function defined by f ( p) = Sπ( p) if p ∈ char F and f ( p) = ∅ otherwise. Applying Theorem 6.4.11, every soluble group in CritS (F) is a Schmidt group. ˇ Remark 6.4.20. The formation in the above theorem is not a S-formation in general (see Example 6.4.10). ˇ We close our extended treatment of S-formations with a survey describing another context where this family of saturated formations appears. In [Keg65] O. H. Kegel introduced the notion of a triple factorisation. This is a factorisation of a group G involving three subgroups A, B, and C of the type G = AB = AC = BC. The evidence is that the existence of a triple factorisation can have greater consequences for the group structure than does a single factorisation. For example, Kegel shows that a group which has a triple factorisation by nilpotent groups is nilpotent. Consequently, it seems natural to wonder which are the saturated formations F which are closed under taking triple factorisations. The first contribution to the solution of this problem was made by Vasil’ev [Vas87, Vas92] in the soluble universe. The following three results are proved in that universe. Theorem 6.4.21 (Vasil’ev). Let F be an Sn -closed saturated formation. Then the following statements are equivalent:

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ˇ 1. F is a S-formation. 2. (Kegel’s property) F contains each group G = AB = AC = BC where A, B, C ∈ F. 3. F contains each group having three pairwise non-conjugate maximal subgroups belonging to F. The above result has been improved in [BBPAMP00]. Theorem 6.4.22. Let F be an Sn -closed saturated formation of full characteristic. The following statements are equivalent. ˇ 1. F is a S-formation. 2. F satisfies the property: If G is a group of the form G = AB = AC = BC, where A and B are F-subgroups of G and C is an F-subnormal F-subgroup of G, then G is an F-group. In the study of factorised groups, the case of a triply factorised group G = AB = AC = BC where C is a normal subgroup of G is of particular interest. For instance, the factoriser of a normal subgroup of a factorised group always has this form. Hence the following characterisation of the above formations is also of interest ([BBPAMP00]). Theorem 6.4.23. Let F be an Sn -closed saturated formation of full characteristic. The following statements are equivalent: ˇ 1. F is a S-formation. 2. F satisfies the property: If G is a group of the form G = AB = AC = BC, where A and B are F-subgroups of G and C is a normal subgroup of G, then GF = C F . Bearing in mind the above results, a natural question arises: Let F be a Fitting formation, non-necessarily subgroup-closed, with the Kegel property. Is F saturated? This question, proposed by Vasil’ev in the Kourovka Notebook [MK99] for formations of soluble groups, was partially answered in [BBE05]. Theorem 6.4.24. Let F be a Fitting formation with the following property: for every prime p ∈ char F, whenever G is a primitive F-group whose socle is a p-group, all groups E(q|p) are in F for all primes q = p such that q divides |G/ Soc(G)|. (6.3) ˇ Then F satisfies the Kegel property if and only if F is a subgroup-closed Sformation. Note that if F is saturated, then F satisfies (6.3). Let F be a Fitting formation. If for some primes p, q, the group E(q|p) ∈ F, then Sp (Cq ) ⊆ F by [DH92, XI, 2.5]. Since Sp (Cq ) ⊆ F ∩ N2 and F ∩ N2 is a Fitting formation of metanilpotent groups, it follows that Sp Sq ⊆ F∩N2 ⊆ F by [DH92, XI, 2.4]. Hence E(q|p) ∈ F if and only if Sp Sq ⊆ F.

6.5 Wielandt operators

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6.5 Wielandt operators One of the significant properties of subnormal subgroups is that the nilpotent residual of the subgroup generated by two subnormal subgroups of a group is the subgroup generated by the nilpotent residuals of the subgroups. This is a consequence of an elegant theory of operators created by H. Wielandt for proving results on permutability of subnormal subgroups. For a group G and the lattice Sn (G) of all subnormal subgroups of G, a map ω : Sn (G) −→ Sn (G) is called a Wielandt operator in G if, for any H, K ∈ Sn (G), the following conditions are satisfied: H, Kω = H ω , K ω ,

(6.4)

if H K, then H ω K.

(6.5)

Here, of course, H ω denotes the image of H under the map ω. Note that Condition 6.5 implies that H ω is a normal subgroup of H. The importance of the theory of operators is suggested by the following result of H. Wielandt. Theorem 6.5.1 ([Wie57]). Let ϕ and ψ be two Wielandt operators in a group G. Assume that two subnormal subgroups H and K of G are permutable if H = H ϕ = H ψ . Then Aϕ B ψ = B ψ Aϕ for any pair (A, B) of subnormal subgroups of G. It is a consequence of the above result that each new operator leads to the discovering of a new case of permutability of subnormal subgroups and gives new insights on the construction of subnormal subgroup generation. Wielandt’s theory of operators is clearly of interest in relation to the theory of classes of groups and may repay further study. Suppose that a Wielandt operator ω is defined in all groups G. If ω satisfies (X ω )α = (X α )ω for any homomorphism α of a group X, then the class F := (X : X ω = 1) is a Fitting formation and Gω is the F-residual of G for every group G. Conversely if F is a Fitting formation, then the map δ : Sn (G) −→ δ F Sn (G), H = H for all H ∈ Sn (G), defines a Wielandt operator in every group G, permuting with all homomorphisms provided that δ satisfies Condition 6.4. Consequently, the problem of finding Wielandt operators which are permutable with homomorphisms is reduced to the description of Fitting formations F satisfying the following property: If U and V are subnormal subgroups of a group G, then U, V F = (6.6) U F , V F . Let us state this property in a formal definition. Definition 6.5.2. Let F be a formation. We say that F satisfies the Wielandt property for residuals if whenever U and V are subnormal subgroups of U, V in a group G, then U, V F = U F , V F .

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The formations appearing in this section are not subgroup-closed in general. Not all formations have the Wielandt property for residuals. For instance, let F be the saturated formation composed of all groups with no epimorphic image isomorphic to Alt(5). Then if G is the symmetric group of degree 5, it follows that GF = 1 = Alt(5), 1F = Alt(5). In fact, we have: Proposition 6.5.3. Let F be a formation. If F satisfies the Wielandt property for residuals, then F is a Fitting formation. Proof. Let G be a group in F, and N a subnormal subgroup of G. Then N F ≤ N F , GF = N, GF = GF = 1. Hence N ∈ F and F is Sn -closed. Suppose that G = N1 N2 for normal subgroups N1 and N2 such that Ni ∈ F, i = 1, 2. Then GF = N1F N2F = 1. This means that G ∈ F and F is N0 -closed. Consequently, F is a Fitting class. The validity of the converse is not known at the time of writing and seems to be quite difficult. Our aim in the first part of this section is to show that many of the known Fitting formations have the Wielandt property for residuals. The procedure we describe here is based on the papers [KS95] and [BBCE01]. The basic strategy is the following: first we prove a reduction theorem for a minimal counterexample. This allows us to reduce the problem in many cases to considering a restricted class of groups in the boundary of the formation. As an application, we deduce that many known Fitting formations have the Wielandt property for residuals. The main obstacle in giving the complete answer for the problem is in understanding the restriction of an irreducible module to a subnormal subgroup. Although a certain amount of information can be derived from repeated application of the Clifford theorems, the closed relation between the components of the restriction is lost. In particular, for a subnormal subgroup, it is difficult to find the relationship between the kernels of the action of the subnormal subgroup on each component of the restriction. We begin by describing two ways to obtain new formations with the Wielandt property from some old ones. Proposition 6.5.4. If F1 , F2 , and Fi , i ∈ I, are formations satisfying the Wielandt property for residuals, then 1. the formation F1 ◦ F2 satisfies the Wielandt property, and 2. the formation i∈I Fi satisfies the Wielandt property. Proof. 1. We have that X F1 ◦F2 = (X F2 )F1 by Proposition 2.2.11 (4) for any group X. Let G be a group and U and V subgroups of G such that U and V are subnormal subgroups of H = U, V . Then H F1 ◦F2 = (H F2 )F1 = U F2 , V F2 F1 = U F1 ◦F2 , V F1 ◦F2 .


287

2. We have that X i∈I Fi = i∈I X Fi for any group X, where in the product only a finite set of residuals appear since X is finite. Consider a group G and U and V subgroups of G suchthat U and V are subnormal subgroups of H = U, V . Then H i∈I Fi = i∈I H Fi = i∈I U Fi , V Fi = i∈I U Fi , i∈I V Fi = U i∈I Fi , V i∈I Fi . Note that if F is a Fitting formation, then U F is contained in GF for every subnormal subgroup U of G. Therefore it is always true that U F , V F is contained in U, V F provided that U and V are subnormal in U, V . If U and V permute, the equality holds as the next result shows. Proposition 6.5.5. Let F be a Fitting formation. If U and V are subgroups of G such that U V = V U and U and V are subnormal in U V , then (U V )F = U FV F. Proof. Assume that the result is false and let G be a counterexample of least order. Let U and V be subnormal subgroups of U V = V U such that |U | + |V | is maximal doing false the result. Clearly U and V are proper subgroups of G and G = U V . Let N be a proper normal subgroup of G such that U ≤ N . Then N = U (V ∩ N ). The minimality of G yields N F = U F (V ∩ N )F . If U is a proper subgroup of N , then GF = N F V F by the maximality of the pair (U, V ). Hence GF = U F V F . This contradiction shows that U and V are maximal normal subgroups of G. Thus U F and V F are normal in G. Assume that one of them, U F say, is not trivial, and let N be a minimal normal subgroup of G such that N ≤ U F . It follows that G/N is a group generated by the subnormal subgroups U N/N and V N/N of G/N . Then, by minimality of G, we have that GF = U F (V F N ) = U F V F , contrary to our initial supposition. Hence U F = 1 = V F or, equivalently, U and V are in F. Since F is a Fitting class, we deduce that G ∈ F, i.e. GF = 1. This final contradiction proves the proposition. Corollary 6.5.6. Let F be a Fitting formation. If U and V are subgroups of a group G such that U and V are subnormal in U, V and U ∈ F, then U, V F = V F . Proof. Since U is a subnormal subgroup of U, V and U ∈ F, we have that U is contained in the F-radical U, V F of U, V . Hence U, V = U, V F V . By Proposition 6.5.5, we deduce that U, V F = (U, V F )F V F = V F . A well-known result of H. Wielandt (see [Wie94b]) asserts that the Fitting subgroup of a group G normalises the nilpotent residual of each subnormal subgroup of G. The next corollary extends this result to an arbitrary Fitting formation. Corollary 6.5.7. Let F be a Fitting formation. If U and V are subgroups of a group G such that U and V are subnormal in U, V , it follows that UF normalises V F . In particular, GF normalises the F-residual H F of each subnormal subgroup H of G.

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Proof. Consider the subgroup K = UF , V generated by its subnormal subgroups UF and V . Then K F = V F by Corollary 6.5.6 and K F is normal in K. Hence UF normalises V F . Let F be a Fitting formation. Given a group X, we denote by W(X, F) the set of all pairs (A, B) such that A and B are subnormal subgroups of A, B ≤ X and AF , B F < A, BF . Let B(F) denote the class of all groups X such that W(X, F) = ∅. If F does not satisfy the Wielandt property for residuals, then the class B(F) is non-empty. In the following we analyse the structure of a group G of minimal order in B(F). We consider a pair (U, V ) in W(G, F) such that |U |+|V | is maximal. Denote H = U F , V F and A = U ∩V . By Proposition 6.5.5, U and V do not permute. In particular, neither U nor V F by Corollary 6.5.6. V is normal in G. Moreover, U F = 1 = Statement 6.5.8. G = U, V . Moreover, U F = V F . Proof. By minimality of G, it is clear that G = U, V and 1 = GF . If N = U F = V F , then N is normal in G. The minimal choice of G implies that GF /N = U F /N, V F /N = 1. Then N = GF . This contradiction yields U F = V F. Statement 6.5.9. GF = HN for every minimal normal subgroup N of G. In particular, H is core-free in G. Moreover, H is normal in GF . Proof. Let N be a minimal normal subgroup of G. We consider G/N = U N/N, V N/N . By minimality of G, we deduce that GF N = HN . If N is not contained in GF , then N ∩GF = 1. This means that GF N = GF ×N . Since H ≤ GF , it follows that H ∩ N = 1. But GF N = HN implies that |GF | = |H| and then GF = H, contrary to our supposition. Hence Soc(G) ≤ GF and GF = HN for any minimal normal subgroup N of G. By [DH92, A, 14.3], Soc(G) normalises H because H is subnormal in G. Hence H is normal in GF . Assume that H is not core-free in G. Then H contains a minimal normal subgroup of G, N say. Hence GF = GF N = HN = H, against to our choice of G. Therefore H is core-free in G. Statement 6.5.10. If Soc(G) is non-abelian, then Soc(G) is a minimal normal subgroup of G and G is in the boundary of F. In this case, GF is the unique minimal normal subgroup of G. Proof. First, note that for every minimal normal subgroup N of G, since H ∩ N is normal in N , we have that N = (H ∩ N ) × N ∗ and GF = H × N ∗ with N ∗ = 1. This implies that H centralises N ∗ . If there exist two minimal ∗ ), for i = 1, normal subgroups N1 and N2 of G, then GF = H ×Ni∗ ≤ CG (N3−i ∗ F 2. Therefore Ni ≤ Z(G ) and both N1 and N2 are abelian. In other words, if Soc(G) is not a minimal normal subgroup of G, then Soc(G) is abelian.


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Assume that N = Soc(G) is non-abelian. Then N is a minimal normal subgroup of G and CG (N ) = 1. It is clear that N is a direct product of copies of a non-abelian simple group, E say. This means that N ∈ D0 (1, E) = X. Then GF /H ∈ X and (GF )X ≤ H. Since (GF )X is normal in G, it follows that (GF )X = 1 by Statement 6.5.9. Hence GF ∈ X. Assume that N is a proper subgroup of GF . Then there exists a copy of E centralising N . This is a contradiction. Hence N = GF . In particular, G is in the boundary of F. Statement 6.5.11. If Soc(G) is abelian, then GF is an elementary abelian p-group for some prime p. Proof. Let N be a minimal normal subgroup of G. By Statement 6.5.9, GF = HN . Since Soc(G) is abelian, N is an elementary abelian p-group for some prime p. In particular, Op (GF ) and (GF ) are normal subgroups of G contained in H. Since H is core-free in G, Op (GF ) = (GF ) = 1, and GF is an abelian p-group. If, on the other hand, Φ(GF ) = 1, then we can take N to be contained in Φ(GF ). In this case, GF = HN = H. This contradiction leads to Φ(GF ) = 1, and GF is an elementary abelian p-group. Statement 6.5.12. H = U F V F . Furthermore, U F and V F are proper subgroups of U and V , respectively. Proof. Whether or not Soc(G) is abelian, every subnormal subgroup of GF is a normal subgroup of GF . In particular, U F and V F are normal in GF . Therefore H = U F V F . Assume that U F = U . Then U normalises V F . This would imply that V F is normal in G, contrary to Statement 6.5.9. Therefore U F < U and V F < V . Statement 6.5.13. A = GF and GF is contained in A. Moreover, 1. A is a maximal normal subgroup of U and V , and G/A is a q-group for some prime q ∈ char F; 2. if Soc(G) is a p-group, then p ∈ char F; and 3. GN = Oq (G) is contained in A. Proof. Let M be a proper subnormal subgroup of G such that U ≤ M and consider the subgroup Y = U, M ∩V . The lattice properties of the subnormal subgroups imply that Y is subnormal in G. Furthermore Y is contained in M . Assume that U is a proper subgroup of Y . Then, by maximality of the pair (U, V ), we have that GF = Y F , V F . By minimality of G, it follows that Y F = U F , (M ∩ V )F . Therefore GF = U F , (M ∩ V )F , V F = U F , V F = H and we have reached a contradiction. Hence U = Y and so M ∩ V ≤ U . In particular M ∩ V = U ∩ V = A. The arguments for a proper subnormal subgroup of G containing V are analogous. Let M be a maximal normal subgroup of G such that U ≤ M . By the foregoing arguments, we have that M ∩ V = A. Therefore, A is a normal

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∼ V M/M = G/M subgroup of V . Moreover, V /A = V /(V ∩ U ) = V /(M ∩ V ) = is a simple group (note that V is not contained in M ). Analogously, we deduce that A is normal in U and that U/A is a simple group. This implies that A is a normal subgroup of G and that A is a maximal normal subgroup of U and V . Since AF is a normal subgroup of G contained in H, it follows that AF = 1 by Statement 6.5.9. This means that A ∈ F and A is contained in GF . Since GF is normal in G, we have that (U GF )F = U F by Corollary 6.5.6. Therefore U GF is a proper subnormal subgroup of G. Assume that U < U GF . Since G = U GF , V , we deduce that GF = (U GF )F , V F = U F , V F by maximality of the pair (U, V ), contrary to supposition. Hence GF is a subgroup of U . Analogously, GF is contained in V , and we have the equality. Assume that U/A is a non-abelian simple group. Then U/A normalises V /A by Theorem 2.2.19, and V is normal in G. This contradiction yields that U/A is a cyclic group of prime order, q say. The same argument for V proves that V /A is a cyclic group of prime order, r say. If r = q, then [U/A, V /A] ≤ [Oq (G/A), Or (G/A)] = 1. Then G/A = U/A × V /A is abelian, and U and V are normal subgroups of G. This possibility cannot happen. Therefore, r = q and G/A is a group generated by two subnormal q-subgroups, U/A and V /A, and so G/A is a q-group. Suppose that A = 1. Then G is a q- group. If q ∈ / char F, then U F = U and V F = V , contrary to Statement 6.5.12. Therefore we must have q ∈ char F and so G ∈ F by [DH92, IX, 1.9]. This contradiction yields A = 1 and then A contains a minimal normal subgroup of G. On the other hand, we can assume that either Soc(G) is an elementary abelian p-group for some prime p, or Soc(G) is a non-abelian minimal normal subgroup of G by Statements 6.5.10 and 6.5.11. In both cases, we have that Soc(G) and GF are subgroups of A and Statement 2 holds. Since GF ≤ A, we have that the q-group G/A belongs to F. Therefore q ∈ char F and Statement 1 holds. Now we prove that if M is a maximal normal subgroup of G, then A is contained in M . Assume that there exists a maximal normal subgroup M of G such that A is not contained in M . Then G = AM , U = A(U ∩ M ), and V = A(V ∩M ). The subnormal subgroup T = U ∩M, V ∩M is a supplement of A in G and A∩M ≤ T ≤ M . Hence M = G∩M = T A∩M = T (A∩M ) = T . By minimality of G, M F = (U ∩ M )F , (V ∩ M )F . On the other hand, since G = M U , by Proposition 6.5.5, it follows that GF = M F U F = (U ∩M )F , (V ∩ M )F , U F = (V ∩ M )F , U F = U F , V F , contrary to our initial supposition. Therefore, every maximal normal subgroup of G contains A. Clearly, GN is contained in A. Since every maximal subgroup of G/GN is normal in G/GN , it follows that A/GN ≤ Φ(G/GN ). Since G/A is a q-group, we deduce that G/GN is a q-group. Then GN = Oq (G) and 3 holds. Next, we assume that Soc(G) is abelian. This implies that GF is an elementary abelian p-group for some prime p ∈ char F by Statements 6.5.11 and 6.5.13. Denote B = GF . Then B is a G-module over the field GF(p).


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Statement 6.5.14. B is a completely reducible A-module over GF(p). Proof. We denote by J(BA ) the intersection of all maximal A-submodules of BA . Since A is normal in G, the action of G permutes these maximal submodules, and thus J(BA ) is a normal subgroup of G. Suppose that J(BA ) = 0, and let N be a minimal normal subgroup of G such that N ≤ J(BA ). By Statement 6.5.9, we have, in additive notation, that B = H + N . Since B, H, and N are A-submodules and N ≤ J(BA ), we have that B = H by Nakayama’s lemma ([HB82a, VII, 6.4]). This is a contradiction. Therefore, J(BA ) = 0, and B is a completely reducible A-module over GF(p) by [HB82a, VII, 1.6]. It is clear that H is an A-submodule of B. Statement 6.5.15. Let Z be an arbitrary irreducible A-submodule of H. Then if Z1 is an irreducible A-submodule of B, then there exists g ∈ G such that Z1 is A-isomorphic to Z g . Proof. Let Z be an irreducible A-submodule of H and consider the normal closure Z G = g∈G Z g . Then Z G A is a completely reducible A-module and is a direct sum of its irreducible submodules which are isomorphic to some conjugate of Z. Let N be a minimal normal subgroup of G such that N ≤ Z G . Hence NA is again a completely reducible A-module and is a direct sum of its irreducible submodules, which are isomorphic to some conjugate of Z. On the other hand, B = H + N by Statement 6.5.9. Therefore, every A-composition factor of B/H is isomorphic to a conjugate of Z. Let Z1 be an irreducible A-submodule of B. The normal closure N1 = Z1G is not contained in H, and every A-composition factor of N1 is isomorphic to a conjugate of Z1 . Again by Statement 6.5.9, B = H + N1 and so every Acomposition factor of B/H is isomorphic to a conjugate of Z1 . This implies that Z1 is A-isomorphic to a conjugate of Z. The following lemma is needed in the proof of our next statement. Lemma 6.5.16. Let K be a field of characteristic p, and let G be a group with a normal subgroup N such that G/N is a p-group. If W is an irreducible KN module, then the induced KG-module W G has all of its composition factors isomorphic. Proof. Let T be the inertia subgroup of W in G. First note that (W T )N = g W g, where g runs over a transversal of N in T . This is a particular case of Mackey’s theorem ([DH92, B, 6.21]). Since T is the inertia subgroup of W in G, we have that W g ∼ = W for all g ∈ T . Therefore (W T )N is homogeneous, and all of its composition factors are isomorphic to W . In particular, if U/V is a composition factor of W T , then (U/V )N is homogeneous, and all its composition factors are isomorphic to W . If U/V is a composition factor of W T , then the G-module (U/V )G ∼ = G U /V G is irreducible by [DH92, B, 7.4]. It is thus sufficient to prove that all composition factors of W T are isomorphic. Let U/V be a composition factor

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of W T . Then, by [DH92, B, 8.3], (U/V )N is an irreducible N -module. Hence (U/V )N is isomorphic to W . By [DH92, B, 5.17], all composition factors of W T are isomorphic. Statement 6.5.17. If p = q, then all composition factors of B are isomorphic. Proof. Suppose that p = q. Let Z be an irreducible A-submodule of B. By Lemma 6.5.16, the induced module Z G has all its composition factors isomorphic. Let M be a composition factor of B, and let Z1 be an irreducible A-submodule of MA . By Statement 6.5.15, Z1 is A-isomorphic to Z g for some −1 g ∈ G. Then Z1g is an irreducible A-submodule of M which is isomorphic to Z. In other words, MA has an irreducible submodule isomorphic to Z, that is, 0 = HomKA (Z, MA ). By Nakayama’s reciprocity theorem [DH92, B, 6.5], it follows that 0 = HomKG (Z G , M ). Therefore a composition factor of Z G is isomorphic to M , and then all composition factors of Z G are isomorphic to M . Statement 6.5.18. If p = q, then B = Soc(G). Proof. Let Z be an irreducible A-submodule of BA . Since p = q, it follows that Z G is a completely reducible G-module by [HB82a, VII, 9.4]. Denote by α the inclusion of Z in BA . Applying [HB82a, VII, 4.4], there exists a KG-homomorphism α : Z G −→ B such that (z ⊗ g)α = z g for all g ∈ G and all z ∈ Z. Hence Im(α ) = Z G , the normal closure of Z in G. Therefore Z G is a completely reducible G-module and Z G ≤ Soc(G). In particular, Z is contained in Soc(G). Since, by Statement 6.5.14, BA is a completely reducible A-module, it follows that B is contained in Soc(G) and the equality holds by Statement 6.5.9. The most important examples of Fitting formations are as follows: 1. The solubly saturated Fitting formations (see Chapter 3). 2. The Fitting formations constructed by Fitting families of modules ([DH92, Chapter IX, 2, Construction F]). Fix a prime r. Let K be an extension field of GF(r). For any r-soluble group G, denote TK (G) the class of all irreducible KG-modules V such that V is a composition factor of the module W K = W ⊗ K, where W is an r-chief factor of G. Suppose that, for every group G, a class of irreducible KG-modules M(G) is defined. Then the class M := G M(G) is called a Fitting family if it satisfies the four properties listed in Definition 2.5.5. Applying Theorem 2.5.6, the class T(1, M) = G : G is r-soluble and TK (G) ⊆ M(G) is a Fitting formation provided that M is a Fitting family. In both cases, we have a way to distinguish between the abelian r-chief factors of any group X in the following sense:


293

1. If F is a solubly saturated formation defined by the canonical P-local formation function F , then an abelian r-chief factor M of X can be F-central if X/ CX (M ) ∈ F (r) or F-eccentric otherwise. 2. If F = T(1, M) is a Fitting formation constructed by a Fitting family of modules M, then an abelian r-chief factor M of X can be such that all composition factors of M K are in M(X) or not. Let X be an arbitrary group, and let M be an X-module over GF(r). Denote by Irr(M ) the class of all irreducible X-modules occurring as composition factors of M . Suppose that F is either a solubly saturated formation or a Fitting formation defined by a Fitting family of modules, and let ModF (U ) denote the class of all irreducible U -modules occurring as 1. F-central chief factors of U below B, if F is a solubly saturated Fitting formation, or 2. abelian chief factors M of U below B such that every composition factor of M K is in M(U ), if F = T(1, M) is a Fitting formation constructed by a Fitting family of modules M. Analogously, let ModF (V ) denote the corresponding set for V . Statement 6.5.19. If F is either a solubly saturated Fitting formation or a Fitting formation defined by a Fitting family of modules, then G is in the boundary of F. Proof. Assume first that p = q. In this case, all composition factors of B are isomorphic G-modules by Statement 6.5.17. We consider a G-composition series of B, 0 = B0 ≤ B1 ≤ · · · ≤ Br = B say. The composition factor B1 is a minimal normal subgroup of G and so B = HB1 by Statement 6.5.9. Since H is core-free in G, it follows that B1 = U F . Moreover, B1 is a completely reducible U -module by Clifford’s theorems [DH92, B, 7.3]. It then decomposes as B1 = B1 ⊕ B1∗ , where B1 = B1 ∩ U F . Since B1 is not contained in U F , we have that B1∗ = 0. Let M be a U -composition factor of B1∗ . Then M is isomorphic to a U -composition factor of B1 /B1 , which is a section of U/U F ∈ F. This implies that M ∈ ModF (U ) and Irr(B1∗ ) is contained in ModF (U ). Assume now that B1 = 0 and let M be an irreducible U -submodule of /F B1 . Then U F = M ⊕ M1 , for some U -submodule M1 of U F . Since U/M1 ∈ and U/U F ∈ F, it follows that M is not in ModF (U ). Consequently Irr(B1 ) ∩ ModF (U ) = ∅ if B1 = 0. The arguments for V are completely analogous. Hence B1 , considered as V -module, decomposes as B1 = B1 ⊕ B1∗∗ where B1 ≤ B1 ∩ V F and B1∗∗ is a non-trivial V -submodule of B1 such that Irr(B1∗∗ ) is contained in ModF (V ). Moreover, Irr(B1 ) ∩ ModF (V ) = ∅ provided that B1 = 0. Let B = U F + Br−1 . Assume that B /Br−1 = 0 and let M/Br−1 be an irreducible U -submodule of B /Br−1 . Since B /Br−1 is completely reducible as U -module, it follows that B /Br−1 = M/Br−1 ⊕ M1 /Br−1 . Note that

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M1 = M1 ∩(U F +Br−1 ) = (M1 ∩U F )+Br−1 and U F +M1 = U F +Br−1 = B . Therefore U F /(M1 ∩ U F ) is U -isomorphic to (U F + M1 )/M1 ∼ = B /M1 and B /M1 is U -isomorphic to M/Br−1 . Since M/Br−1 = 0, we have that U F is not contained in M1 and so U/(M1 ∩ U F ) is not in F. Hence M/Br−1 ∈ / ModF (U ). Consequently Irr(B /Br−1 ) ∩ ModF (U ) = ∅. The same argument holds for V , that is, if B = V F + Br−1 , then Irr(B /Br−1 ) ∩ ModF (V ) = ∅ provided that B /Br−1 = 0. Suppose that r ≥ 2. Then the composition factors B1 and B/Br−1 are different. Furthermore B = B + B . Assume that B /Br−1 = 0. Then B = B and Irr(B/Br−1 ) ∩ ModF (V ) = ∅. Let ϕ : B/Br−1 −→ B1 be a G-isomorphism. Since ϕ is a V -isomorphism, it follows that Irr(B1 ) ∩ ModF (V ) = ∅. This is a contradiction because B1∗∗ is a non-trivial V -submodule of B1 such that Irr(B1∗∗ ) ⊆ ModF (V ). Consequently B /Br−1 = 0 and B /Br−1 = 0. Moreover ϕ(B /Br−1 ) is contained in B1 and ϕ(B /Br−1 ) is contained in B1 . Hence B1 = ϕ(B/Br−1 ) = ϕ(B /Br−1 + B /Br−1 ) = ϕ(B /Br−1 )+ϕ(B /Br−1 ) = B1 +B1 ≤ (B1 ∩U F )+(B1 ∩V F ) ≤ H. This is a contradiction. Therefore r = 1 and B is an irreducible G-module. Consider now the case where p = q. Then, by Statement 6.5.18, B is a completely reducible G-module. By Clifford’s theorem, U F is a completely reducible U -module. If M is an irreducible U -submodule of U F , then there exists a U -submodule M0 of U F such that U F = M ⊕ M0 . Since U/M0 is not in F, we have that M ∈ / ModF (U ). That is, Irr(U F ) ∩ ModF (U ) = ∅. On the other hand, since U/U F ∈ F, it follows that M ∈ ModF (U ), for every chief factor M of U between U F and U . That is, Irr(B/U F ) is contained in ModF (U ). With a similar argument, we deduce the corresponding result for V . Hence Irr(BU /U F ) ∩ Irr(U F ) = ∅ = Irr(V F ) ∩ Irr(BV /V F ). Now suppose that B = N1 × · · · × Nr , where Ni is a minimal normal subgroup of G, i = 1, . . . , r, and r ≥ 2. Each Ni can be decomposed as Ni = Ni∗ ⊕ of Ni ∩ U F in Ni as U -modules. Then (Ni ∩ U F ), where Ni∗ is a complement ∗ ∗ F BU = (N1 ⊕ · · · ⊕ Nr ) ⊕ (N1 ∩ U ) ⊕ · · · ⊕ (Nr ∩ U F ) . Denote B ∗ = N1∗ ⊕ · · · ⊕ Nr∗ . Then U F ∩ B ∗ = 0 because Irr(U F ) ∩ Irr(BU /U F ) = ∅. Hence U F = (N1 ∩ U F ) ⊕ · · · ⊕ (Nr ∩ U F ). The same arguments hold for V : V F = decomposition as vector spaces, (N1 ∩V F )⊕· · ·⊕(Nr ∩V F ). Comparing the two F F F F = N + (N ∩U )⊕· · ·⊕(N ∩U ) + (N ∩V B = N1 +H =N1 +U F +V 1 1 r 1 )⊕ F F F F F · · ·⊕(Nr ∩V ) = N1 ⊕ (N2 ∩U )+(N2 ∩V ) ⊕· · ·⊕ (Nr ∩U )+(Nr ∩V ) = N1 ⊕ · · · ⊕ Nr , we deduce that Ni = (Ni ∩ U F ) + (Ni ∩ V F ) for each i ≥ 2. Therefore Ni ≤ U F + V F = H for each i ≥ 2. This contradiction leads to r = 1 and B is a minimal normal subgroup of G. Consequently, in both cases, we have that G is a monolithic group in the boundary of F. For convenience, we incorporate the class of all groups satisfying the above statements in a formal definition.


295

Definition 6.5.20. Let F be a Fitting formation. Define b3 (F) as the class of all triples (G, U, V ) such that 1. G ∈ b(F) and U and V are subnormal subgroups of G; 2. G = U, V ; 3. U ∩ V = GF = 1; 4. U/GF and V /GF are cyclic groups of order q, a prime. Note that if (G, U, V ) ∈ b3 (F), then GF is contained in GF and G/GF is a q-group, q ∈ char F. The above statements lead to the following result. Theorem 6.5.21. Let F be a Fitting formation. Suppose that either 1. F is a solubly saturated Fitting formation, or 2. F = T(1, M) is a Fitting formation defined by a Fitting family of modules M constructed over an extension field K of GF(r). Then the following statements are equivalent: 1. F satisfies the Wielandt property for residuals. 2. For every triple (G, U, V ) ∈ b3 (F), we have that GF = U F , V F . Applying Theorem 6.5.21, a large number of Fitting formations satisfying the Wielandt property for residuals appear. Corollary 6.5.22. Let F be a Fitting formation. Then F satisfies the Fitting property for residuals provided that one of the following conditions hold: 1. Sp F = F, for all primes p ∈ char F. 2. FSp = F, for all primes p ∈ char F. 3. F is solubly saturated, and its boundary is composed of non-abelian simple groups. 4. char F = ∅. 5. F = E X for some class X of simple groups. 6. F = D0 (1, X1 ), where X1 is a class of non-abelian simple groups. Let p be a prime, and let Mp be the class of all groups whose abelian p-chief factors are central. It is rather clear that Mp is a Fitting formation. Moreover, Mp is solubly saturated by Lemma 3.2.15 and Mp ∩S is the class of all soluble p-nilpotent groups. The Mp -radical of a group G is the intersection of the centralisers of the abelian p-chief factors of G. This subgroup also appears when a P-local definition of a solubly saturated formation is considered (see Section 3.2). Corollary 6.5.23. Let p be a prime. Then Mp satisfies the Wielandt property for residuals.

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Proof. Applying Theorem 6.5.21, we need only consider triples in b3 (Mp ). Suppose that (G, U, V ) is a triple in b3 (Mp ). Then G = U, V is a monolithic group in b(Mp ), U and V are subnormal subgroups of G, U ∩ V is the Mp -radical of G and G/(U ∩ V ) is a q-group for some prime q ∈ char Mp = P. Denote by N the Mp -residual of G. Then N is an abelian p-group contained in U ∩ V = A. Since N is a completely reducible A-module, it follows that A ≤ CG (N ). Consequently G = QA = Q CG (N ) for every Sylow q-subgroup Q of G. Let B = N Q. We have that B is soluble and N is a minimal normal subgroup of Q. It is clear that Q does not centralise N because G ∈ / Mp . This implies that B Mp = N . On the other hand, U = A(Q ∩ U ) and V = A(Q ∩ V ). Hence G = AQ ∩ U, Q ∩ V and Q = Q ∩ U, Q ∩ V . It means that B = N Q ∩ U, Q ∩ V = N (Q ∩ U ), N (Q ∩ V ) = U ∩ B, V ∩ B. Note that B Mp = B F , where F is the saturated formation of all p -nilpotent groups. Combining Proposition 6.5.4 (1) and Corollary 6.5.22 (1), it follows that F satisfies the Wielandt property for residuals. Therefore B Mp = (U ∩ B)Mp , (V ∩ B)Mp ≤ U Mp , V Mp and so N = U Mp , V Mp . Let F be a solubly saturated formation. Then, applying Theorem 3.2.14, there exists a Baer function f such that F = LFP (f ). Denote Supp(f ) = {p ∈P : f (p) = ∅} ∪ {S ∈ J \ P : f (S) = ∅}. Then it rather clear that F = p∈Supp(f ) Mp ◦ f (p) ∩ S∈Supp(f )\P E (S) ◦ f (S) by Remarks 3.1.2 and Remark 3.1.9. Therefore, applying Proposition 6.5.4 and Corollary 6.5.23, we have: Theorem 6.5.24 ([KS95]). Let F be a solubly saturated formation and let f be a Baer function P-locally defining F. If for all S ∈ Supp(f ), f (S) satisfies the Wielandt property for residuals, then F satisfies the Wielandt property for residuals. Corollary 6.5.25. Let F be a saturated formation locally defined by a formation function f . If for all primes p, the formations f (p) satisfy the Wielandt property for residuals, then F satisfies the Wielandt property for residuals. Proof. Set

f (p) g(J) =

when J ∼ = Cp , p ∈ P and p||J| f (p) when J ∈ J \ P,

then it is clear that F = LFP (g). Applying Proposition 6.5.4 (2), g(J) satisfies the Wielandt property for residuals if J ∈ J\P. By Theorem 6.5.24, F satisfies the Wielandt property for residuals. Corollary 6.5.26. Any soluble subgroup-closed Fitting formation satisfies the Wielandt property for residuals. Proof. Any soluble subgroup-closed Fitting formation F is a primitive saturated formation. Therefore, F has a local definition f such that f (p) satisfies the Wielandt property for residuals for all prime numbers p (see [DH92, page 497]).


297

Example 6.5.27. (see Example 2.2.17) Let Q be the Fitting formation of all quasinilpotent groups. Then Q is a solubly saturated formation P-locally defined by the P-local formation function f given by (1) when S ∼ = Cp , and f (S) = D0 (1, S) when S ∈ J \ P. Since f (S) satisfies the Wielandt property for residuals for all S, it follows that Q satisfies the Wielandt property for residuals by Theorem 6.5.24. In the next examples, we work in the universe of all soluble groups. Let Xi be Fitting formations, i = 1, 2. For every group G, denote by M(G) the class of all irreducible KG-modules V such that V = U ⊗ W with U πspecial, W π -special, and G/ Ker(G on U ) ∈ X1 and G/ Ker(G on W ) ∈ X2 . Applying Theorem 2.5.10, M = M(K, P, X1 , X2 ) = G M(G) is a Fitting family. Let T(1, M) = T(1, r, P, X1 , X2 )be the Fitting formation defined by M. Theorem 6.5.28. Let π be a set of primes and consider the partition P = {π, π } of the set of all prime numbers. The Fitting formation F = T(1, M) = T(1, r, P, X1 , X2 ) satisfies the Wielandt property for residuals in the following case: X1 = Sρ and X2 = Sσ for some sets of primes ρ and σ (not both empty). The following result is used in the proof of Theorem 6.5.28. It can be proved by using similar arguments to those used in the proof of [HB82a, VII, 9.13]. Lemma 6.5.29. Let N be a normal subgroup of G, and let V1 and V2 be two KG-modules such that 1. (V1 )N is absolutely irreducible, and 2. V2 is absolutely irreducible and (V2 )N is homogeneous, and all of its constituents are isomorphic to (V1 )N . Write (V2 )N ∼ = s(V1 )N . Then there exists an irreducible K(G/N )-module W with dim W = s such that V2 ∼ = V1 ⊗ W . Proof (of Theorem 6.5.28). We use only the restriction on the Xi at one point, and so have written the proof as far as possible to be independent of that hypothesis. Applying Theorem 6.5.21, we need only consider groups in b3 (F). Hence we suppose that G is in the boundary of F and, moreover, that U and V are subnormal subgroups of G satisfying G = U, V , A = U ∩ V = GF = 1, and U/A and V /A are of prime order q, q ∈ char F. Note that G/A is a q-group and Oq (G) = GN = Oq (U ) = Oq (V ) = Oq (A). Furthermore, G has a unique minimal normal subgroup B = GF which is a p-group for some prime p. First, we observe that p = r (the characteristic of K), since otherwise all r-chief factors would come from G/B ∈ F, and so G would be in F. We are working with a field K which is algebraically closed. However, when dealing with dimensions of KX-modules for a subgroup X of G, we

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can assume that K is a splitting field for G and all its subgroups. In fact, by Brauer’s theorem [HB82a, VII, 2.6], we can assume that K is a finite Galois extension of k = GF(p). We are interested in the behaviour of the irreducible components of B K . By [HB82a, VII, 1.15], the KG-module B K is completely reducible. Let N be an irreducible component of B K . Applying [HB82a, VII, 1.18 (b)], every irreducible KG-submodule of B K is G-isomorphic to N η for some η ∈ G(K/k). We collect some properties we need. First, if L is a normal subgroup of G, then a KL-module Q is π-special if and only if all of its G-conjugates are π-special, and L/ Ker(L on Q) ∈ Xi if and only if the same is true for all of the G-conjugates of Q. Further, Q is π-special if and only if all of its Galois conjugates are special and L/ Ker(L on Q) ∈ Xi if and only if the same is true for all of the Galois conjugates of Q. Clearly we may assume that q ∈ π. Suppose, by way of contradiction, that B = U F , V F . Then CoreG (U F , V F ) = 1. We have that BU is completely reducible as U -module and so B = U F ⊕ B0 , with B = B0 = 0. It follows that (U F )K can contain no components in M(U ) and (B0 )K must have all its components in M(U ). Let N be an irreducible component of B K . If no component of (BU )K is in M(U ), then no component of (BU )K is in M(U ) and thus B0 = 0. This is a contradiction. If every component of NU is in M(U ), then every component of (BU )K is in M(U ). This implies that U F = 1 (or, equivalently, B = B0 ). It is also a contradiction. Hence, if we denote by D(NU ) the sum of all irreducible KU -submodules of NU which do not lie in M(U ), then 0 = D(NU ) = N . In particular, NU is not a homogeneous module. Similar remarks apply to V . Furthermore, NA = N1 ⊕ · · · ⊕ Nt , where the Ni are irreducible KAmodules, all conjugate by elements of G. Since A ∈ F, for each i we have that Ni = Zi ⊗ Xi , where Zi is a π-special irreducible KA-module with A/ Ker(A on Zi ) ∈ X1 and Xi is a π -special irreducible KA-module with A/ Ker(A on Xi ) ∈ X2 . Note that since all Ni are G-conjugates, so are the Zi g g and the Xi , because if Nig ∼ = N1 for some G ∈ G, then Zi ⊗Xi ∼ = (Zi ⊗Xi )g = g ∼ g ∼ g ∼ Ni = N1 = Z1 ⊗ X1 , and thus Zi = Z1 and Xi = X1 , by [CK87, 2.4]. We break the proof into a number of cases. Case 1. Suppose that all of the Zi , as well as all of the Xi , are isomorphic. This is equivalent to saying that NA is homogeneous. If p = q, then NA is irreducible by [DH92, B, 8.3]. This implies that NU is irreducible and either N ∈ M(U ) or N ∈ / M(U ). This contradiction yields p = q. Since NU is a completely irreducible U -module, we can write NU = L1 ⊕ · · · ⊕ Lu , where Li are irreducible KU -modules. Analogously, NV = P1 ⊕ · · · ⊕ Pv , where Pi are irreducible KV -modules. If Lj is an irreducible component of NU such that Ni is a component of (Lj )A , then (Lj )A ∼ = tj Ni for some tj . Since q divides |K| − 1 by [HB82a, VII, 2.6], we have that tj is either 1 or q by [DH92, B, 8.5]. Analogously, if Pk is an irreducible component of NV such that Ni is a component of (Pk )A , then either (Pk )A = Ni or (Pk )A ∼ = qNi . We have that


299

NA = N1 ⊕ · · · ⊕ Nr , and each irreducible component Ni is Ni = Z ⊗ X, with Z a π-special KA-module and X a π -special KA-module. Applying [CK87, 2.3], there is a unique π -special KU -module Y contained in X U such that X = YA . Moreover, Z U is completely reducible by [HB82a, VII, 9.4]. Let W be an irreducible component of Z U . By the Nakayama’s reciprocity theorem, 0 = HomU (Z U , W ) ∼ = HomA (Z, WA ) ([DH92, B, 6.5]). Therefore Z is an irreducible component of WA . Since Z is π-special, then so is W by [CK87, 2.3]. It is clear that the inertia subgroup of Z in U is the whole U . Then WA is homogeneous, i.e. WA ∼ = tZ. Again, by [DH92, B, 8.5], either t = 1 or t = q. Assume that t = q. Therefore, we have that dim W = dim Z U = q dim Z. This implies that W ∼ = Z U and Z U is a π-special KU -module. Let L be any irreducible KU -module such that Z ⊗ X is a component of LA . It follows that Z U ⊗ Y is irreducible by [CK87, 2.4]. By [HB82a, VII, 4.5 (a)], we have that theorem (Z ⊗ X)U = (Z ⊗ YA )U ∼ = Z U ⊗ Y . Applying Nakayama’s reciprocity ([DH92, B, 6.5]), it follows that 0 = HomA (Z ⊗X, LA ) ∼ = HomU (Z ⊗X)U , L . Consequently Z U ⊗Y ∼ = L. This implies that Li ∼ = Z U ⊗Y for all i ∈ {1, . . . , u} and NU is homogeneous, contrary to 0 = D(NU ) = N . Hence t = 1, and W has the same dimension as Z. Consequently, W ⊗ Y is an irreducible KU module with (W ⊗ Y )A = Z ⊗ X. For any irreducible component Lj of NU , it follows from Lemma 6.5.29 that Lj = (W ⊗ Y ) ⊗ Jj , where Jj is an irreducible K(U/A)-module (regarded as KU -module) and dim Jj = 1 or q. Since U/A is cyclic, it follows that dim Jj = 1 by [DH92, B, 9.2]. Hence (Lj )A = Ni . Arguing with V , we have that if NV = P1 ⊕· · ·⊕Pv , with the Pi irreducible V -modules, and Pk is an irreducible component of NV such that Ni is a component of (Pk )A , then (Pk )A = Ni is irreducible. It implies that Ni is in fact U -module and V -module. Therefore Ni = N is an irreducible G-module. This is a contradiction. Case 2. Suppose that not all of the Xi are isomorphic. We let T denote the inertia subgroup of X1 and note that A ≤ T = G. Since G/A is a q-group generated by U/A and V/A, we have that there is a maximal normal subgroup M of G satisfying T ≤ M and so either U or V is not contained in M . We may suppose that U is not contained in M . Recall that all Xi are isomorphic to G-conjugates of X1 , and so the inertia subgroups are conjugate in G. It then follows that U is not contained in the inertia subgroup of any Xi . Now let L be a component of NU and suppose that N1 is a component of LA . If L is π-factorable, then L = D ⊗ E with D π-special and E π special. Note that LA = DA ⊗ EA ; if DA = D1 ⊕ · · · ⊕ Dm with all Di irreducible A-modules, then Di is π-special for all i ∈ {1, . . . , m} by [CK87, 2.2]. Suppose that EA is irreducible. Then LA = (D1 ⊗EA )⊕· · ·⊕(Dm ⊗EA ). Therefore EA is isomorphic to X1 by [CK87, 2.4], and then U is contained in the inertia subgroup of X1 , contrary to supposition. Hence we cannot have EA irreducible. By Clifford’s theorem, since the inertia subgroup of X1 in U is A, we have that E is the direct sum of q = |U/A| irreducible modules conjugate to X1 . But then the dimension of E is not a π -number. This contradiction

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6 F-subnormality

yields that L cannot be a π-factorable module. It follows that no component of NU can be π-factorable, and so no component of NU can be in M(U ), i.e. NU = D(NU ), and we have reached a contradiction. Case 3. Suppose that all of the Xi are isomorphic. By Case 1, we may assume that not all the Zi are isomorphic and let T be the inertia subgroup of Z1 . As before, it follows that we may suppose that U is not contained in the inertia subgroup of any Zi . Now let L be any irreducible KU -module such that Z1 ⊗X1 is a component of LA . We then have that X1 has a unique extension to a π -special KU module, (X1 )∗ say by [CK87, 2.3]. Also, since Z1 is not U -invariant, we have that (Z1 )U is irreducible by [DH92, B, 7.8] and π-special by [CK87, 2.3]. It by [CK87, 2.4]. By [HB82a, VII, 4.5], follows that (Z1 )U ⊗ (X1 )∗ is irreducible U U ∼ we have that (Z1 ⊗ X1 ) = Z1 ⊗ (X1 )∗ A = Z1U ⊗ (X1 )∗ . Now 0 = U ∼ HomA (Z1 ⊗ X1 , LA ) = HomU (Z1 ⊗ X1 ) , L by the Nakayama’s reciprocity theorem ([DH92, B, 6.5]). Then L is isomorphic to (Z1 )U ⊗ (X1 )∗ . It follows that if NU = L1 ⊕ · · · ⊕ LU with the Li irreducible, then Li = (Zi )∗ ⊕(Xi )∗ with (Zi )∗ π-special and (Xi )∗ π -special, 1 ≤ i ≤ u.In each case the π -special factor is isomorphic to (X1 )∗ , and thus if U/ Ker U on(X1 )∗ is not in X2 , then no component of NU is in MU , i.e. NU = D(NU ). This contradiction proves that U/ Ker U on(X1 )∗ ∈ X2 . Some of the Lj is in M(U ). Suppose Li ∈ M(U ). Then the group U/ Ker U on(Zi )∗ ∈ X1 . On the other hand, since A ∈ F, the group A/ Ker(A on Zj ) belongs to X1 for all j. Recall that all Zj are conjugate and then so are the Ker(A on Zj ). Since (Zj )∗ = (Zj )U , we have that Ker U on(Zj )∗ = CoreU Ker(A on Zj ) . Thus A/ Ker U on(Zj )∗ ∈ X1 . At this point, we must invoke the special form of Xi , i = 1, 2. Since Ker U on(Zi )∗ is contained in A, we must U/ Ker U on(Zi )∗ ∈ Sρ and have Q ∈ ρ. Then U/ Ker U on(Zj )∗ is a ρ-group and hence is in X1 for all j. Thus Lj ∈ M(U ) for all j. In other words, D(NU ) = 0. This final contradiction proves GF = U F , V F and then F has the Wielandt property for residuals.

Examples 6.5.30. 1. Set P = π = {p}, π = {p} , p a prime, X1 = (1) and X2 = S, then T(1, p, P, X1 , X2 ) = T(1, Mp ) are the Fitting classes introduced by T. O. Hawkes in [Haw70]. Applying Theorem 6.5.28, T(1, Mp ) satisfies the Wielandt property for residuals. 2. The Fitting formations studied by K. L. Haberl and H. Heineken in [HH84] can be seen as Fitting formations constructed by the Cossey-Kanes method with X1 = S and X2 = (1). Hence, by Theorem 6.5.28, these classes also satisfy the Wielandt property for residuals. Let F be a Fitting formation satisfying the Wielandt property for residuals. In general, the F-residual of a group generated by two F-subnormal subgroups is not the subgroup generated by their F-residuals, as the following example shows.


301

Example 6.5.31. Let F be the saturated Fitting formation of all groups of nilpotent length at most 2. Then F is a subgroup-closed formation of soluble groups. Applying Corollary 6.5.26, F has the Wielandt property for residuals. Let G be the symmetric group of degree 4. If A is the alternating group of degree 4 and B is a Sylow 2-subgroup of G, then A and B are both F- subnormal in G = A, B, A and B belong to F, but G ∈ / F. From this example the following problem arises: Find a precise description of those formations F for which the F-residual of a group generated by two F-subnormal subgroups is the subgroup generated by their F-residuals. We will be mainly concerned with this problem from now on. Our treatment of the question closely follows the approaches developed in the papers of S. F. Kamornikov [Kam96] and A. Ballester-Bolinches, M. C. PedrazaAguilera, and M. D. Pérez-Ramos [BBPAPR96], and A. Ballester-Bolinches [BB05]. For the purposes of this discussion, let F be a fixed, but arbitrary, formation. Definition 6.5.32. 1. We say that F has the generalised Wielandt property for residuals, F is a GWP-formation for short, if F enjoys the following property: If G is a group generated by two F-subnormal subgroups A and B, then GF = AF , B F . 2. F satisfies the Kegel-Wielandt property for residuals, F is a KW-formation for short, if F has the following property: Let G = A, B be a group generated by two K-F-subnormal subgroups A and B. Then GF = AF , B F . Obviously, every KW-formation is a GWP-formation. We show in the following that that the converse holds for saturated formations, and the soluble GWP-formations are exactly the soluble subgroup-closed saturated lattice formations. We need a couple of preliminary results. To be a subgroup-closed Fitting formation is a necessary condition for a formation F to have the generalised Wielandt property for residuals. Lemma 6.5.33. If F is a GWP-formation, then F is a subgroup-closed Fitting formation. Proof. First suppose, by way of contradiction, that F is not subgroup-closed. Let G be an F-group of minimal order having a subgroup not in F and, among subgroups of G not in F, let M be one of maximal order. Then M is a maximal subgroup of G. Since GF = 1, it follows that M is F-subnormal in G. Since F is a GWP-formation, we have that M F = M F , 1 = M F , GF = GF = 1. This contradicts the choice of G. Consequently F is subgroup-closed. In

302

6 F-subnormality

particular, F is Sn -closed. To complete the proof we now show that F is N0 closed. Suppose that this is not true and derive a contradiction. Let G be a group of minimal order having two normal subgroups N1 and N2 such that G = N1 N2 and Ni ∈ F for i = 1, 2. If N is a minimal normal subgroup of G, it follows that G/N ∈ F. Therefore G is in the boundary of F and N = GF is the unique minimal normal subgroup of G. It is clear that Ni = 1, i = 1, 2. Hence N is contained in N1 ∩ N2 and thus Ni is F-subnormal in G, i = 1, 2 by Lemma 6.1.7 (1). Since F is a GWP-formation, it follows that GF = N1F N2F = 1, contrary to supposition. Therefore F is N0 -closed. The proof of the lemma is now complete. The following result is another step to attain our objectives. Theorem 6.5.34. Let F be a GWP-formation. Then F is a lattice formation. Proof. By Lemma 6.5.33, F is subgroup-closed. Hence the intersection of F- subnormal subgroups of a group is F-subnormal by Lemma 6.1.7 (3). Suppose that F is not a lattice formation and derive a contradiction. By this supposition, there exists a group G of minimal order such that SnF (G) is not a sublattice of the subgroup lattice of G. In particular, G has two F-subnormal subgroups A and B such that A, B is not F-subnormal in G. Let N be a minimal normal subgroup of G. Then AN/N and BN/N are F-subnormal in G/N by Lemma 6.1.6 (3). Hence AN/N, BN/N = A, BN/N is F-subnormal in G/N by minimality of G. Therefore X = A, BN is F-subnormal in G by Lemma 6.1.6 (2). Since A and B are F-subnormal in X by Lemma 6.1.7 (2), it follows that A, B is F-subnormal in X provided that X is a proper subgroup of G. This would imply the F-subnormality of A, B in G by Lemma 6.1.6 (1). Consequently G = A, BN for every minimal normal subgroup N of G. Hence either G = A, B or CoreG (A, B) = 1. If G = A, B, then A, B is F-subnormal in G, contrary to supposition. Hence CoreG (A, B) = 1. On the other hand, AF and B F are subnormal subgroups of G by Lemma 6.1.9 (1). Hence AF , B F is subnormal in G and so N normalises AF , B F ([DH92, A, 14.3 and 14.4]). Since F is a GWPformation, we have that AF , B F = A, BF . This implies that A, BF is normal in G. Hence A, BF ≤ CoreG (A, B) = 1 and A, B is an F-group. Let us consider the subgroup AN . Clearly N is not contained in A. If N F = N , then no simple component of N belongs to F and thus (AN )F = N . This contradicts the fact that A is F-subnormal in AN (Lemma 6.1.6 (2)). Therefore N ∈ F. This implies that G ∈ E K(F) and so N is F-subnormal in G by Proposition 6.1.10. Then N is an F-subnormal subgroup of AN by Lemma 6.1.6 (2). In particular, (AN )F = AF N F = 1 = (BN )F because the property of F. Since G = AN, BN and F is a GWP-formation, it follows that GF = (AN )F , (BN )F = 1. This final contradiction proves that F is a lattice formation. A challenging unsolved problem in the theory of formations is whether the converse of Theorem 6.5.34 is true. The chance of finding the answer


303

seems remote. With our present knowledge even the saturated case remains unanswered. We shall prove a result that provides a test for a subgroup-closed saturated lattice formation to be a GWP-formation in terms of its boundary. This allows us to present the complete answer to the problem in the soluble universe and give interesting examples. As in the case of groups generated by subnormal subgroups, we thought it would be desirable to collect the arguments common to our next results. Let F be a subgroup-closed Fitting formation. Given a group Z, we denote by R(Z, F) the set of all pairs (H, K) such that H and K are F-subnormal subgroups of H, K and H F , K F < H, KF . Let W(F) denote the class of all groups Z such that R(Z, F) = ∅. If F is not a GWP-formation, then the class W(F) is not empty. In the following we analyse the structure of a group G of minimal order in W(F). Then G has two F-subnormal subgroups A and B such that A, BF = AF , B F . Choose A and B with |A| + |B| maximal. Arguing as in the subnormal case, we have: Result 6.5.35. G = A, B, and Result 6.5.36. Soc(G) ≤ GF and GF = AF , B F N for any minimal normal subgroup of G. In particular, CoreG (AF , B F ) = 1. Result 6.5.37. AF , B F is normal in GF . Proof. Applying Lemma 6.1.9 (1), AF and B F are subnormal subgroups of G. Hence Soc(G) ≤ NG (AF , B F ) by [DH92, A, 14.3 and 14.4]. This implies that AF , B F is normal in GF . Result 6.5.38. GF ∈ Q R0 (N ) for any minimal normal subgroup N of G. Proof. Let N be and AF , B F normal subgroup Result 6.5.36, we

a minimal normal subgroup of G. Then GF = AF , B F N GF by Results 6.5.36 and 6.5.37. Hence (GF )Q R0 (N ) is a of G contained in AF , B F . Since CoreG (AF , B F ) = 1 by have that (GF )Q R0 (N ) = 1 and GF ∈ Q R0 (N ).

Result 6.5.39. N ∈ F for any minimal normal subgroup N of G. Proof. Since N F is normal in G, we have that either N F = 1 or N F = N . Assume that N F = N . By Lemma 6.1.6 (2), A is F-subnormal in AN . Hence AF is normal in AN by Lemma 6.1.9 (1) and [DH92, A, 14.3]. This implies that AN/AF N ∈ F and so (AN )F = AF N . Hence AN = A(AN )F . The F-subnormality of A in AN yields AN = A. Since F is subgroup-closed, it follows that N AF /AF ∈ F, whence N/(N ∩ AF ) ∈ F. Therefore N = N ∩ AF and GF = AF , B F , contrary to our initial supposition. Consequently N ∈ F.

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6 F-subnormality

Result 6.5.40. GF ∈ F. Proof. Let N be a minimal normal subgroup of G contained in GF . Then N ∈ F by Result 6.5.39 and GF ∈ Q R0 (N ) by Result 6.5.38. Hence GF ∈ F. Result 6.5.41. GF is contained in A ∩ B. In particular, AF B F is a subgroup of G. Proof. Clearly AGF is a proper subgroup of G. Hence AGF is contained in a maximal F-normal subgroup of G. The minimality of G yields (AGF )F = AF . Assume that A is a proper subgroup of AGF . Since AGF is F-subnormal in G, it follows that GF = (AGF )F , B F = AF , B F by the choice of the pair (A, B), contrary to our initial supposition. Hence A = AGF and GF is contained in A. Analogously GF is contained in B. With the same arguments to those used in Statement 6.5.10, we have: Result 6.5.42. If GF is non-abelian, then G is in the boundary of F. Suppose now that there exists a family of subgroup-closed formations {Fi }i∈I such that π(Fi ) ∩ π(Fj ) = ∅, i = j, and F = Xi∈I Fi . Result 6.5.43. There exist i, j ∈ I such that G/GF ∈ Fi and GF ∈ Fj . Moreover if either G ∈ / b(F) or GF is non-abelian, then i = j. Proof. By Result 6.5.40, we have that GF ∈ F and, by Result 6.5.38, GF is a direct product of copies of a simple group. Hence there exists j ∈ I such that GF ∈ Fj . On the other hand, G/GF = Xi1 /GF × · · · × Xit /GF , where Xik /GF ∈ Fik is a Hall π(Fik )-subgroup of G/GF , 1 ≤ k ≤ t, for some set {i1 , . . . , it } ⊆ I. Let k ∈ {1, . . . , t}. Then Xik /GF = (A ∩ Xik )/GF , (B ∩ Xik )/GF = A ∩ Xik , B ∩ Xik /GF and Xik = A ∩ Xik , B ∩ Xik . Applying Lemma 6.1.7 (2), A ∩ Xik and B ∩ Xik are F-subnormal subgroups of Xik . Assume that Xik is a proper subgroup of G for all k ∈ {1, . . . , t}. Then XiFk = (A∩Xik )F , (B ∩Xik )F by the minimal choice of G, leading to XiFk = 1. This is due to the fact that XiFk is a normal subgroup of G contained in AF , B F and CoreG (AF , B F ) = 1 by Result 6.5.36. Hence G ∈ N0 F = F, contrary to hypothesis. Therefore there exists an index i = ik ∈ {i1 , . . . , it } such that Xi = G. This means that G/GF ∈ Fi . Suppose that i = j. Then GF is a Hall π(Fj )-subgroup of G and there exists a Hall π(Fi )-subgroup C of G such that G = GF C and GF ∩ C = 1. It follows that A/AF = GF /AF × (C ∩ A)AF /AF and so A normalises AF B F . Analogously B normalises AF B F . It implies that AF B F = 1 by Result 6.5.36 and GF is a minimal normal subgroup of G. Hence G ∈ b(F). If GF is non-abelian, CG (GF ) = 1. Since A = GF × (C ∩ A) and B = F G × (C ∩ B), it follows that A = B = GF . Then, by Results 6.5.35 and 6.5.41, A = B = G, and this contradicts our initial hypothesis. Consequently if either G ∈ / b(F) or GF is non-abelian, we have that i = j.


305

Assume now that F is a saturated GWP-formation. Then F is a lattice formation by Theorem 6.5.34 and, since F is saturated, it follows that F is a K-lattice formation by Theorem 6.3.9. Suppose that F is not a KW-formation. Then there exists a group G and a pair (A, B) of K-F-subnormal subgroups of G such that G = A, B and GF = AF , B F . Let us take (A, B) satisfying |A| + |B| maximal. Then, as in the above reductions, G enjoys the properties stated in Results 6.5.35, 6.5.36, 6.5.37, and 6.5.38. G also has the following property. Result 6.5.44. GF ∈ F. Proof. Consider the subgroup M = A, B F . Suppose that M = G. Then G = AGF . Since by Lemma 6.1.9 (1), AF is subnormal in G and GF is a direct product of isomorphic simple groups by Result 6.5.38, it follows that GF normalises AF and so AF is a normal subgroup of G. By Result 6.5.36, we have that A ∈ F. By virtue of Lemma 6.3.8, it follows that A ≤ GF . If GF ∩ GF = 1, then GF ∈ F and the result follows. Hence GF ∩ GF = 1 and G = GF × GF . By Result 6.5.36, we have that Soc(G) ≤ GF . It implies that GF = 1 and so A = 1 and G = B, giving a contradiction. Therefore we may assume that M is a proper subgroup of G. The choice of G, Lemma 6.1.6 (1) and Lemma 6.1.7 (2) imply that M F = A, B F F = AF , (B F )F . Arguing in a similar way with B, we have AF , BF = (AF )F , B F . If either A < A, B F or B < AF , B, it follows that GF = A, B F F , AF , BF = AF , B F by the choice of G (note that F is subgroup-closed). This contradiction yields A = A, B F and B = AF , B. Then B F is contained in A and AF is a normal subgroup of GF . Hence (A ∩ GF )/AF is a K-F-subnormal F-subgroup of GF /AF by Lemma 6.1.7 (2) and Lemma 6.1.6 (3). Applying Lemma 6.3.8, (A ∩ GF )/AF is contained in (GF /AF )F . If AF = A ∩ GF , then (GF /AF )F = 1 and GF ∈ F because GF is a direct product of simple groups. Therefore we may assume AF = A ∩ GF . In this case, B F is contained in AF . Arguing in a similar way with B, we conclude that AF is contained in B F . Consequently AF = B F is a normal subgroup of G. By Result 6.5.36, A and B are F-groups. By Lemma 6.3.8, G ∈ F and GF = 1. This completes our preparations, and we can now deduce the main results. Theorem 6.5.45. Let F be a saturated formation. Then: F is a GWP-formation if and only if F is a KW-formation. Proof. Only the necessity of the condition is in doubt. Assume that F is a GWP-formation which is not a KW-formation. Then there exists a group G and a pair (A, B) of K-F-subnormal subgroups of G such that G = A, B and GF = AF , B F . If |A| + |B| maximal, then GF ∈ F by Result 6.5.44. Then SnF (G) = SnK-F (G) by Proposition 6.1.10. This contradiction yields that F is a KW-formation.

306

6 F-subnormality

Our next main result provides a test for a subgroup-closed saturated lattice formation to have the generalised Wielandt property for residuals in terms of its boundary. If F is a subgroup-closed Fitting formation, let bn (F) denote the class of all groups G ∈ b(F) such that Soc(G) is not abelian and G has the properties stated in Results 6.5.35–6.5.43. Theorem 6.5.46. Let F be a subgroup-closed saturated lattice formation. Then F is a GWP-formation if and only if the following condition is fulfilled by all groups G ∈ bn (F): If G = A, B with A and B F-subnormal subgroups of G, then GF = (6.7) AF , B F . Proof. It is clear that only the sufficiency of the condition is in doubt. Assume that Property (6.7) holds. We suppose that F is not a GWPformation and derive a contradiction. Since W(F) is not empty, a group G of minimal order in W(F) satisfies the properties stated in Results 6.5.35–6.5.43 for a pair of F-subnormal subgroups A and B of G with |A| + |B| maximal. Applying Theorem 6.3.15, F = M × H, where M is a subgroup-closed saturated Fitting formation such that Sπ(M) M = M and H = Xi∈I Sπi , with / bn (F), GF is πl ∩πk = ∅ for all k = l in I. Moreover π(M)∩π(H) = ∅. Since G ∈ an elementary abelian p-group for some prime p by Results 6.5.38 and 6.5.42. Therefore GF ∈ M or GF ∈ Sπi for some i ∈ I. In addition, by Result 6.5.43, G/GF ∈ M or G/GF ∈ Sπj for some j ∈ I. If GF ∈ M and G/GF ∈ M, then p ∈ π(M) and G ∈ Sπ(M) M = M ⊆ F, contradicting G ∈ W(F). Assume now that G/GF ∈ Sπj for some j ∈ I. Then GF is a Hall π(M)-subgroup of G and there exists a Hall πj -subgroup C of G such that G = GF C and GF ∩ C = 1. Then A/AF = GF /A × (C ∩ A)AF /AF . It follows that A normalises AF B F . Analogously B normalises AF B F . Consequently AF B F = 1 and A and B are F-groups. Since F is a lattice formation and A and B are F-subnormal in G, we have that G ∈ F by Theorem 6.3.3 (3). It contradicts our supposition. Suppose that GF ∈ Sπi . If G/GF ∈ M or G/GF ∈ Sπj for some j ∈ I, i = j, we can argue as above and obtain a contradiction. Hence G/GF ∈ Sπi and so G ∈ Sπi ⊆ F, contradicting G ∈ W(F). It follows that our supposition is wrong and hence F is a GWP-formation. If F is a soluble subgroup-closed saturated lattice formation, then bn (F) = ∅. Moreover, if F is a soluble GWP-formation, then F is a subgroup-closed Fitting formation by Lemma 6.5.33, and hence saturated by Theorem 2.5.2. Therefore we have: Corollary 6.5.47 (see [Kam96, BBPAPR96]). Let F be a soluble formation. Then F is a GWP-formation if and only if F is a subgroup-closed saturated lattice formation. Another interesting examples of GWP-formations follow from the following result.


307

Corollary 6.5.48 ([Kam96]). Let F be a saturated formation representable as F = M×H, where π(M)∩π(H) = ∅, M = M2 is a subgroup-closed saturated Fitting formation, H = Xi∈I Sπi , and moreover πl ∩ πk = ∅ for all k = l in I. Then F is a GWP-formation. Proof. Applying Theorem 6.3.15, F is a subgroup-closed saturated lattice formation. Hence, by Theorem 6.5.46, it is enough to check the property in groups in bn (F) generated by two F-subnormal subgroups. Let G be one of them. Then G ∈ b(F) and Soc(G) = GF is non-abelian. Moreover, by Result 6.5.43, GF ∈ M and G/GF ∈ M (note that H is a soluble formation). Hence G ∈ M2 = M ⊆ F. This contradiction proves that F is a GWP-formation. This completes our discussion about GWP-formations. We can turn this situation on its head and ask the following. Let F be a subgroup-closed formation and let G be a group generated by two F-subnormal subgroups A and B of G. When do we have GF = AF , B F ? The question is answered in [BBEPA02] for subgroup-closed saturated formations. It is proved there that if G is a group whose derived subgroup is nilpotent, then GF = AF , B F provided that A and B are F-subnormal in G = A, B. Furthermore the class NA of all groups whose derived subgroup is nilpotent is characterised as the largest subgroup-closed saturated formation enjoying that property. Let F be a GWP-formation. Then F has the following property: If A and B are K-F-subnormal (F-subnormal) subgroups of a group (6.8) G and G = AB, then GF = AF B F . In general, Property 6.8 does not characterise the GWP-formations as the class of all 2-nilpotent groups shows. Hence the question of how one subgroupclosed formation satisfying Property 6.8 can be characterised arises. This question is closely related to the characterisation of the subgroup-closed formations satisfying Property 6.1. The above question has a nice answer in the soluble universe for subgroupclosed saturated formations of full characteristic. Theorem 6.5.49 ([BBPAPR96]). Let F be a subgroup-closed saturated formation of soluble groups of full characteristic. The following statements are pairwise equivalent: 1. F satisfies the property: If A and B are two F-subnormal subgroups of a soluble group G and G = AB, then GF = AF B F . 2. For each prime number p, there exists a set of primes π(p) with p ∈ π(p) such that F is locally defined by the formation function f given by f (p) = Sπ(p) . These sets of primes satisfy the following property:

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6 F-subnormality

If q ∈ π(p), then π(q) ⊆ π(p) for every pair of primes p and q. Let F be a subgroup-closed saturated formation of full characteristic satisfying the conditions of the above theorem. Then a soluble group G is an F-group if and only if G has a normal π(p)-complement for every prime p, where π(p) is the set of primes such that p ∈ π(p).

7 Fitting classes and injectors

7.1 A non-injective Fitting class After B. Fischer, W. Gasch¨ utz, and B. Hartley’s result about the injective character of the Fitting classes of soluble groups (Theorem 2.4.26), and bearing in mind the extension of the projective theory to the general universe of finite groups, it seemed to be reasonable to think about the validity of Theorem 2.4.26 outside the soluble realm. It was conjectured then that if F is an arbitrary Fitting class and G is a finite group, then InjF (G) = ∅. In the eighties of the last century, a big effort of some mathematicians was addressed to find methods to obtain injectors for Fitting classes in all finite groups. These efforts were successful for a big number of Fitting classes and they will be presented in Section 7.2. In this atmosphere, the construction of E. Salomon [Sal] of an example of a non-injective Fitting class caused a deep shock. Salomon’s construction, never published, is based in a pull-back construction of induced extensions due to F. Gross and L. G. Kov´ acs (see Section 1.1). The aim of this section is to present the Salomon’s example in full detail. We begin with a quick insight to the group A = Aut Alt(6) . Let D denote the normal subgroup of inner automorphisms D ∼ = Alt(6) of A. It is wellknown that the quotient group A/D is isomorphic to an elementary abelian 2-group of order 4 and A does not split over D, i.e. there is no complement of D in A (see [Suz82]). If u is an involution of Sym(6), the symmetric group of degree 6, then u is a complement of Alt(6) in Sym(6) and the element u acts on Alt(6) as an outer automorphism. Likewise, Alt(6) ∼ PGL(2, 9) (see [Hup67, pages = = PSL(2, 9) but Sym(6) ∼ 183 and 184]). There exist elements of order 2 in PGL(2, 9) which are not 1 in PSL(2, 9) (for instance the coclass of the matrix in the quotient −1 group GL(2, 9)/ Z GL(2, 9) ∼ = PGL(2, 9)). If v is one of these involutions, 309

310


then v is a complement of PSL(2, 9) in PGL(2, 9) and the element v acts on Alt(6) ∼ = PSL(2, 9) as an outer automorphism. The subgroup B = Du ∼ = Sym(6) and the subgroup C = Dv ∼ = PGL(2, 9) are normal subgroups of A of index 2. Clearly A = BC and B ∩ C = D. Let S be a non-abelian simple group. If x is an involution in S, define the group homomorphism α1 : B −→ S

such that Ker(α1 ) = D, B α1 = x,

Put |S : Im(α1 )| = |S|/2 = n1 , and consider the right transversal T1 = {s1 = 1, s2 , . . . , sn1 }, of Im(α1 ) in S and the transitive action ρ1 : S −→ Sym(n1 ) on the set of indices I1 = {1, . . . , n1 }. For each i ∈ I1 and each s ∈ S, ρ1 si s = xi,s sj , for some xi,s ∈ Im(α1 ) and is = j. Write PS = S ρ1 ≤ Sym(n1 ) and consider the monomorphism (see Lemma 1.1.26) λ1 = λT1 : S −→ Im(α1 ) ρ1 PS , defined by sλ1 = (x1,s , . . . , xn1 ,s )sρ1 , for any x ∈ S, and the epimorphism α ¯ 1 : W1 = B ρ1 PS −→ Im(α1 ) ρ1 PS α¯ 1 α1 1 defined by (b1 , . . . , bn1 )τ = (bα 1 , . . . , bn1 )τ , for b1 , . . . , bn1 ∈ B and τ ∈ n1 ∼ n1 ¯ 1 ) = D = Alt(6) . PS . Write M1 = Ker(α Construct the induced extension G1 , defined by α1 (see Definition 1.1.27), σ

1 S −→ 1 Eλ1 : 1 −→ M1 −→ G1 −→

Recall that G1 = {w ∈ W1 : wα¯ 1 = sλ1

for some s ∈ S},

and σ1 : G1 −→ S

defined by wσ1 = s, where wα¯ 1 = sλ1 .

The following diagram is commutative: Eλ1 : 1

/ M1

E: 1

/ M1

id

/ G1 / W1

σ1

/1

/S λ1

α ¯1

/ Im(α1 ) ρ1 Ps

/1

7.1 A non-injective Fitting class

311

Then, applying Theorem 1.1.35, G1 splits over M1 , since B splits over D. For the group C we repeat the previous arguments to construct a similar group G2 . Let T be a non-abelian simple group. If y is an involution in T , define the group homomorphism α2 : C −→ T

such that Ker(α2 ) = D, C α2 = y.

Put |T : Im(α2 )| = |T |/2 = n2 , and consider the right transversal T2 = {t1 = 1, t2 , . . . , tn2 } of Im(α2 ) in T and the transitive action ρ2 : T −→ Sym(n2 ) on the set of indices I2 = {1, . . . , n2 }. For each i ∈ I2 and each t ∈ T , ρ2 ti t = yi,t tj , for some yi,t ∈ Im(α2 ) and it = j. With the obvious changes of notation, construct the induced extension defined by α2 as in Definition 1.1.27. Then, for G2 = {w ∈ W2 = C ρ2 PT : wα¯ 2 = tλ2 for some t ∈ T } and σ2 : G2 −→ T defined as above, we also have that the following diagram is commutative Eλ2 : 1

/ M2

E2 : 1

/ M2

id

/ G2 / W2

σ2

/1

/T λ2

α ¯2

/ Im(α2 ) ρ2 PT

/1

Then, again by Theorem 1.1.35, G2 splits over M2 since C splits over D. Finally, consider the homomorphism α : A −→ S × T such that bα = (bα1 , 1), cα = (1, cα2 ) for any b ∈ B, c ∈ C. Then, Ker(α) = D and Im(α) = |T | Im(α1 ) × Im(α2 ). Put |S × T : Im(α)| = |S| 2 2 = n1 n2 , and consider the right transversal of Im(α) in S × T T = T1 × T2 = {(s1 , t1 ) = (1, 1), (s1 , t2 ), . . . , (s1 , tn2 ), (s2 , t1 ), (s2 , t2 ), . . . , (sn1 , tn2 )}. The transitive action ρ : S × T −→ Sym(n1 n2 ) on the set of indices I = I1 × I2 = {(1, 1), . . . , (n1 , n2 )} (lexicographically ordered) gives P = (S × T )ρ = PS × PT . Consider the monomorphism λ = λT : S × T −→ Im(α) ρ P, defined by (s, t)λ = (x1,s , y1,t ), (x1,s , y2,t ), . . . , (xn1 ,s , yn2 ,t ) (s, t)ρ

312


for any s ∈ S, t ∈ T , the epimorphism α ¯ : W = A ρ P −→ Im(α) ρ P defined by α¯ α α (a(1,1) , a(1,2) , . . . , a(n1 ,n2 ) )τ = (aα (1,1) , a(1,2) , . . . , a(n1 ,n2 ) )τ for a(1,1) , a(1,2) , . . . , a(n1 ,n2 ) ∈ A and τ ∈ P , and write M = Ker(¯ α) = D = n1 n2 ∼ n1 n2 . D = Alt(6) Construct the induced extension defined by the homomorphism α : A −→ S × T: Eλ : 1

/M

E: 1

/M

id

/G /W

σ

/ S×T

/1

λ

α ¯

/ Im(α) p (PS × PT )

/1

Then, G = {w ∈ W = A ρ P : wα¯ = (s, t)λ

for some (s, t) ∈ S × T }

and σ : G −→ S × T defined by wσ = (s, t) such that wα¯ = (s, t)λ , for all w ∈ G. Now applying Theorem 1.1.35, the group G does not split over M , since A does not split over D. Every element w ∈ W can be written uniquely as w = (a(1,1) , . . . , a(n1 ,n2 ) )(τ1 , τ2 ) where a(1,1) , a(1,2) , . . . , a(n1 ,n2 ) ∈ A for all (i, j) ∈ I, τ1 ∈ PS and τ2 ∈ PT . If w ∈ G, and wα¯ = (s, t)λ , then α wα¯ = (aα (1,1) , . . . , a(n1 ,n2 ) )(τ1 , τ2 )

= wσλ = (x1,s , y1,t ), (x1,s , y2,t ), . . . , (xn1 ,s , yn2 ,t ) (s, t)ρ ρ1 and aα = τ1 and tρ2 = τ2 . (i,j) = (xi,s , yj,t ), for all (i, j) ∈ I, s

Proposition 7.1.1. The group W possesses subgroups W(1) and W(2) which are isomorphic to W1 and W2 , respectively. Proof. Let W(1) be the subset of all elements w in W such that 1. a(i,1) = a(i,2) = · · · = a(i,n2 ) , for all i = 1, . . . , n1 , 2. a(i,j) ∈ B, for all (i, j) ∈ I, and 3. τ2 = 1.

7.1 A non-injective Fitting class

313

Then W(1) is a subgroup of W and the map ψ1 : W1 −→ W(1) such that ψ1 is the element w ∈ W(1) such that (b1 , . . . , bn1 )τ 1. a(i,1) = a(i,2) = · · · = a(i,n2 ) = bi , for all i = 1, . . . , n1 , 2. τ1 = τ and τ2 = 1, is a group isomorphism. Put M(1) = M1ψ1 . A similar argument and construction holds for W2 .

Proposition 7.1.2. The group G possesses two subgroups which are isomorphic to G1 and G2 , respectively. Proof. Consider the subgroup G(1) = W(1) ∩ G and note that G(1) = {x ∈ W(1) : xα¯ = (s, 1)λ for some s ∈ S}. Note that the kernel of the group epimorphism σ(1) = σπ1 : G(1) −→ S, where π1 : S × T −→ S is the canonical projection, is M(1) = M1ψ1 , as in Proposition 7.1.1. Define the group homomorphism β1 = ι(1) ψ1−1 : G(1) −→ W1 , where ι(1) : G(1) −→ W(1) is the canonical inclusion and ψ1 as in Proposition 7.1.1. Consider an element x = (a(1,1) , . . . , a(n1 ,n2 ) )(τ1 , 1) ∈ G(1) . Then, if xα¯ = (s, 1)λ , we have that sρ1 = τ1 and aα (i,j) = (xi,s , 1) ∈ S ×1, for all i = 1, . . . , n1 , α1 i.e. a(i,j) ∈ B and a(i,j) = xi,s , for all i = 1, . . . , n1 . Observe that xα¯ = (s, 1)λ = (x1,s , 1), (x1,s , 1) . . . , (xn1 ,s , 1) (sρ1 , 1), and α¯ 1 −1 −1 = xβ1 α¯ 1 = xι(1) ψ1 α¯ 1 = xψ1 α¯ 1 = (a(1,1) , . . . , a(n1 ,1) )τ1 α1 α1 ρ1 = (a(1,1) , . . . , a(n1 ,1) )τ1 = (x1,s , . . . , xn1 ,s )s = = sλ1 = (s, 1)π1 λ1 = xσπ1 λ1 = xσ(1) λ1 . Then the diagram 1

/ M1

1

/ M1

id

is commutative.

/ G(1)

σ(1)

β1

/ W1

/1

/S λ1

α ¯1

/ Im(α1 ) ρ1 Ps

/1

314


By the universal property, Theorem 1.1.23 (2), we have that G(1) is isomorphic to G1 . Analogously we can proceed with G2 and it appears a subgroup G(2) in W(2) which is isomorphic to G2 . Let S and T be two non-abelian simple groups. Recall that the class F = composed by the trivial group and all groups which are direct products of the form D0 (S, T, 1)

S 1 × · · · × Sn × T1 × · · · × Tm , ∼ S, Tj ∼ where Si = = T , 1 ≤ i ≤ n, 1 ≤ j ≤ m, for some positive integers n and m, is a Fitting formation (see Lemma 2.2.3). Theorem 7.1.3. Let S and T be two non-abelian simple groups. Suppose that S and T satisfy the three following conditions: 1. no subgroup of S is isomorphic to T , 2. no subgroup of T is isomorphic to S, and 3. either S or T are isomorphic to no subgroup of a direct product of copies of the alternating group Alt(6) of degree 6. Consider the Fitting formation F = D0 (S, T, 1). Then the group G, constructed above, has no F-injectors. Proof. The group G possesses two subgroups, S˜ and T˜, which are isomorphic to S and T , respectively. Write G/M = (H1 /M ) × (H2 /M ), with H1 /M ∼ =S ∼ ˜ ˜ ˜ ˜ ˜ S/( S ∩ M ) = S, since S ∩ M = 1, by T . Observe that SM/M and H2 /M ∼ = = ˜ ) = 1, then the group G/H1 ∼ T would have condition 3. If (H1 /M ) ∩ (SM/M = a subgroup isomorphic to S, and this is not possible by Condition 2. Hence ˜ . A similar argument with T˜ and H2 leads to H2 = T˜M . Both H1 H1 = SM and H2 are maximal normal subgroups of G. ˜ ) = {U : U M = SM, ˜ U ∼ We observe that MaxF (SM = S}. If U ∈ ˜ ), then U ∩ M = 1 by condition 3. Since U ∈ F and U M ≤ SM ˜ , MaxF (SM ˜ . we have that U ∼ = S and U M = SM Similarly MaxF (T˜M ) = {V : V M = T˜M, V ∼ = T }. ˜ = R1 is Suppose that X is an F-injector of G. Then, the subgroup X ∩ SM ˜ . Hence R1 ∼ F-maximal in SM = S. Likewise, X ∩ T˜M = R2 ∼ = T . Hence R1 ×R2 is a normal subgroup of X and R1 ×R2 ∼ = S ×T . Moreover, (R1 ×R2 )∩M = 1. Since |G| = |M ||S × T | = |M ||R1 × R2 |, we conclude that R1 × R2 is a complement of M in G, i.e. G splits over M . But this is not true. Therefore the group G has no F-injectors and F is a non-injective Fitting class. Remark 7.1.4. The simple groups S = Alt(7) and T = PSL(2, 11) satisfy the above conditions 1, 2, and 3.

7.2 Injective Fitting classes

315

7.2 Injective Fitting classes We have proved in Corollary 2.4.28 that every Fitting class F is injective in the universe FS. In fact, in the attempt of investigating classes of groups, larger than the soluble one, in which there exist F-injectors for a particular Fitting class F, the first remarkable contribution comes from A. Mann in [Man71]. There, following some ideas due to B. Fischer and E. C. Dade (see [DH92, page 623]), it is proved that in every N-constrained group G, there exists a single conjugacy class of N-injectors and each N-injector is an N- maximal subgroup containing the Fitting subgroup. A group G is said to be N-constrained if CG F(G) ≤ F(G). It is well-known that every soluble group is N-constrained (see [DH92, A, 10.6]). In [BL79] D. Blessenohl and H. Laue proved that the class Q of all quasinilpotent groups is an injective Fitting class in E. In fact they prove something more (see [DH92, IX, 4.15]). Theorem 7.2.1 (D. Blessenohl and H. Laue). Every finite group G has a single conjugacy class of Q-injectors, and this consists of those Q-maximal subgroups of G containing F∗ (G). In the decade of the eighties of the last century there was a considerable amount of contributions to obtain more injective Fitting classes. P. F¨ orster proved the existence of a certain non-empty characteristic conjugacy class of N-injectors in every finite group in [F¨ or85a]. Later M. J. Iranzo and F. Pérez-Monasor obtained the existence of injectors in all finite groups with respect to various Fitting classes, including a new type of N-injectors. Their investigations, together with M. Torres, gave light to a “test” to prove the injectivity of a number of Fitting classes. Some of the most interesting results obtained from this test have been published recently by M. J. Iranzo, J. Lafuente, and F. Pérez-Monasor. Their achievements illuminate the validity of a L. A. Shemetkov conjecture saying that any Fitting class composed of soluble groups is injective. We present here some of the fruits of these investigations. Proposition 7.2.2. Let F be a Fitting class and G be a group. 1. A perfect comonolithic subnormal subgroup E of G is an F-component of G if and only of EGF /GF is a component of G/GF . 2. If E is an F-component of G, the F-maximal subgroups of E containing EF are F-injectors of E. Proof. 1. Let E be a perfect comonolithic subnormal subgroup of a group G. Suppose that E is an F-component of G. Then N(E) is a subnormal F-subgroup of G, i.e. N(E) ≤ GF . Therefore EGF /GF is isomorphic to a quotient group of E/ N(E), and then EGF /GF is a quasisimple subnormal subgroup of G/GF . Conversely, if EGF /GF is a component of G/GF , then E/(E ∩ GF ) is a quasisimple group. Since E is subnormal in G, EF = E ∩ GF

316


by Remark 2.4.4. If E ∈ F, then E is contained in GF , contrary to supposition. Hence EF ≤ Cosoc(E). Moreover, Cosoc(E)/EF = Z(E/EF ). Therefore N(E) = [E, Cosoc(E)] ≤ EF . Hence N(E) ∈ F. 2. Suppose E is an F-component of G and V is an F-maximal subgroup of E such that EF ≤ V . Since N(E) ≤ EF ≤ Cosoc(E) and Cosoc(E)/ N(E) is abelian, EF is the F-injector of Cosoc(G). Moreover, V ∩ Cosoc(E) is normal in Cosoc(E) and then is a subnormal F-subgroup of E. Hence V ∩ Cosoc(E) = EF and V is an F-injector of E. Proposition 7.2.3. Let K be a subnormal subgroup of a group G. If E is an F-component of G such that E is not contained in K, we have that [K, E] ≤ N(E). Proof. Denote M = Cosoc(E). By Theorem 2.2.19, the subgroup K normalises E. Therefore K normalises M . Clearly K is subnormal in KE and KM is normal in KE. Since K ∩ E is subnormal in the comonolithic group E and E ≤ K, we have that K ∩ E ≤ M . Therefore [K, E] ≤ [KM, E] ≤ KM ∩ E = M (K ∩ E) ≤ M. Hence [K, E, E] = [E, K, E] ≤ [M, E] = N(E) and the Three-Subgroups Lemma (see [KS04, 1.5.6]) yields that [E, K] = [E, E, K] ≤ N(E). Now we are ready to state and prove the result of Iranzo, Pérez-Monasor, and Torres. Theorem 7.2.4 ([IPMT90]). Let F be a Fitting class and G a group. Let {E1 , . . . , En } be a set of F-components of G which is invariant by conjugation of the elements of G. For each i = 1, . . . , n, let Ji be an F-injector of Ei . Consider the subgroup J = J1 , . . . , Jn . Then InjF NG (J) ⊆ InjF (G). Proof. Note that, by Proposition 7.2.2 (2) and Proposition 7.2.3, J is a normal product J = J1 · · · Jn , and therefore J ∈ F. Let H be an F-injector of NG (J). We have to prove that for any subnormal subgroup S of G, the subgroup H ∩ S is F-maximal in S. To do that we consider an F-subgroup K of S such that H ∩ S ≤ K and argue that H ∩ S = K. We may assume without loss of generality that the F-components E1 , . . . , Em are those contained in S, for m ≤ n, and the other ones are not in S. This implies that {E1 , . . . , Em } is a set of F-components of S which is invariant by conjugation of the elements of S. Observe that J ≤ NG (J)F ≤ H. Therefore, for any i = 1, . . . , m, we have that Ji ≤ J ∩ Ei ≤ H ∩ Ei ≤ H ∩ S ∩ Ei ≤ K ∩ Ei ∈ F,


317

since K ∩ Ei is subnormal in K. Therefore Ji = J ∩ Ei = H ∩ Ei = K ∩ Ei , since Ji ∈ MaxF (Ei ), i = 1, . . . , m. Observe that if x ∈ K, for every i ∈ {1, . . . , m}, there exists an index j ∈ {1, . . . , m} such that Jix = (J ∩ Ei )x = K ∩ Eix = K ∩ Ej = Jj . Choose now j ∈ {m+1, . . . , n}. Applying Proposition 7.2.3, it can be deduced that [Jj , S] ≤ [Ej , S] ≤ N(Ej ) ≤ Jj . This is to say that S normalises Jj for every j ∈ {m + 1, . . . , n}. Therefore K ≤ NS (J1 . . . Jm ) ≤ NS (J). Hence H ∩ S ≤ K ≤ NS (J) and then H ∩ S = H ∩ NS (J). The subgroup NS (J) is subnormal in NG (J). Since H ∈ InjF NG (J) , we have that H ∩ S ∈ MaxF (NS (J)). This implies that H ∩ S = K, as desired. Theorem 7.2.4 is a crucial result when proving the injectivity of a Fitting class by inductive arguments: with the above notation, if InjF NG (J) = ∅, then the group G possesses F-injectors. Equipped with this theorem we can obtain several results of M. J. Iranzo, J. Lafuente, and F. Pérez-Monasor in [ILPM03] and [ILPM04], which go much further on the theorems about the existence of injectors. Lemma 7.2.5 (see [ILPM03]). Let G be a group and m a preboundary of perfect groups. Set B = Fit Cosoc(Z) : Z ∈ m . 1. If X, Y ∈ bm (G), then a) Cosoc(X) = XB , [X, Y ] ≤ X ∩ Y and (XY )B = XB YB , b) X = Y if and only if XGB /GB = Y GB /GB . 2. Suppose that bm (G) = {X1 , . . . , Xn } = ∅ and write E = Em (G); then a) E = X1 . . . Xn and EB = (X1 )B . . . (Xn )B , b) E/EB ∼ = X1 /(X1 )B × · · · × Xn /(Xn )B is a direct product of nonabelian simple groups. Proof. 1a. By definition Cosoc(X) ∈ B. Assume that of B, we have that X ∈ B. Then X ∈ Sn Cosoc(Z) : Z ∈ m , by [DH92, XI, 4.14]. But this is not possible since m is subnormally independent. Therefore Cosoc(X) = XB . Trivially, if X = Y , then [X, Y ] ≤ X ∩ Y . Suppose that X = Y . Observe that, since m is subnormally independent, we have that X ≤ Y and Y ≤ X. By Theorem 2.2.19, Y normalises X and X normalises Y . Hence [X, Y ] ≤ X ∩ Y . If X = Y , then X ∩ Y ≤ Cosoc(X) ∩ Cosoc(Y ) = XB ∩ YB . Moreover, XYB ∩ Y XB = (X ∩ Y XB )YB = (X ∩ Y )XB YB = XB YB

318


and then XY /XB YB = XYB /XB YB × Y XB /XB YB is a direct product of non-abelian simple groups. Since (XY )B /XB YB ≤ Z(XY /XB YB ) by [DH92, IX, 1.1], we conclude that (XY )B = XB YB . 1b. Observe that XGB /GB ∼ = X/(X ∩ GB ) = X/XB is a non-abelian simple group. Suppose that X = Y and XGB /GB = Y GB /GB . Notice that [X, Y ] ≤ X ∩ Y ∈ B, and then, XGB /GB = (XGB /GB ) = [XGB /GB , Y GB /GB ] = [X, Y ]GB /GB = 1. This is a contradiction. Part 2 follows immediately from 1. Lemma 7.2.6 (M. J. Iranzo, J. Lafuente, and F. P´ erez-Monasor, un¯ published). Let F be a Fitting class and n a subclass of b(F). Then Fit(F, n) = F · Fit n = G ∈ E : G = GF En (G) . Proof. Let G be a group. If X ∈ bn (G), then clearly Cosoc(X) = XF . Write X = G ∈ E : G = GF En (G) and Y = Fit n. For each group G, the subgroup En (G) is in Fit n, i.e. En (G) ≤ GY . Therefore X ⊆ F · Fit n ⊆ Fit(F, n). Let us prove that X is a Fitting class. If G ∈ X, then G/GF ∼ = En (G)/ En (G)F is a direct product of non-abelian simple groups by Lemma 7.2.5 (2b). Let N be a normal subgroup of G. Then bn (N ) ⊆ bn (G). Thus, if bn (N ) = {X1 , . . . , Xr }, then N GF /GF = X1 GF /GF × · · · × Xr GF /GF and then N = N ∩N GF = N ∩X1 . . . Xr GF = N ∩En (N )GF = En (N )NF ∈ X. If N and M are normal subgroups of a group G = N M and N, M ∈ X, then G = N M = NF En (N )MF En (M ) ≤ GF En (G). Hence G ∈ X. Therefore X is a Fitting class. It is clear that F and n are contained in X. Hence X = Fit(F, n). Lemma 7.2.7. Let T be a Fitting class such that T = TS. Consider F = ¯ ¯ ⊆ b(F). Tb = Fit Cosoc(X) : X ∈ b(T) . Then b(T) = b(T) Proof. Let G be a group in b(T). Then G group is a comonolithic perfect and Cosoc(G) ∈ F. If G ∈ F, then G ∈ Sn Cosoc(X) : X ∈ b(T) by [DH92, XI, 4.14]. This is to say that there exists a group X ∈ b(T) such that G is a proper subnormal subgroup of X. In particular G ∈ T, and this contradicts ¯ our assumption. Hence G ∈ b(F). Theorem 7.2.8. Let T be a class of groups. The following statements are equivalent: 1. T is a Fitting class such that T = TS. 2. T = (G ∈ E : GX ∈ F) for a pair of Fitting classes X and F such that F = X ∩ FA. In this case, for each group G, we have GT = CG (GX /GF ).


319

Proof. 1 implies 2. Set m = b(T), and consider the Fitting classes F = Tb and ¯ ¯ X = Fit m. Clearly F ⊆ X ∩ T. Since T = TS, we have that m = b(T) ⊆ b(F), by the above lemma. Then we can apply Lemma 7.2.6 and conclude that X = Fit(F, m) = G ∈ E : G = GF Em (G) . If G ∈ X ∩ FA, then G/GF ∼ = Em (G)/ Em (G) ∩ GF and this group is abelian and a direct product of non-abelian simple groups, by Lemma 7.2.5 (2b). Hence G ∈ F, and then F = X ∩ FA. Set H = (G ∈ E : GX ∈ F). If a group G ∈ H \ T, there exists a subnormal subgroup N of G such that N ∈ m. Thus N ≤ GX ∈ F ⊆ T, and this is a contradiction. Hence H ⊆ T. Conversely if G is a group in T and N = GX , then N = NF Em (N ). But since T is a Fitting class, Em (G) = 1 = Em (N ). Then N ∈ F. Therefore G ∈ H. Hence H = T. 2 implies 1. We see that, under these hypotheses, the class T is a Fitting class. Let N be a normal subgroup of a T-group G. Clearly NX ≤ GX ∈ F, and then N ∈ T. Consider now a group G = N M such that N and M are normal T-subgroups of G. Then NX , MX ∈ F and the subgroup F = NX MX ∈ F. By [DH92, IX, 1.1], we have that GX /F ≤ Z(G/F ), and then GX ∈ X ∩ FA = F. Therefore G ∈ T. Thus, T is a Fitting class. Suppose that N is a normal T-subgroup of a group G, such that G/N ∈ A. Then NX ∈ F. Since GX /NX = GX /(N ∩ GX ) ∼ = N GX /N ∈ A, we have that GX ∈ X ∩ FA = F. Therefore G ∈ T. This implies that T = TS. Finally, observe that in this situation F = X ∩ T. Therefore GF = GT ∩ GX. Thus GT ≤ CG (GX /GF ) = C. Obviously (C ∩ GX )/GF is an abelian group and then CX = C ∩ GX ∈ F, since F = X ∩ FA. Therefore C ∈ T and C = GT . Corollary 7.2.9. Let T be a Fitting class such that T = TS. Then Fit b(T) ∩ T = Tb . Proof. Set m = b(T) and consider again the Fitting classes F = Tb and X = Fit m. By the above arguments, if a group G is in X ∩ T, then G = GF Em (G) ∈ T. Hence Em (G) ∈ T, and this implies that Em (G) = 1. Thus G ∈ F. Therefore X ∩ T = F. The following proposition is motivated by a result due to W. Gasch¨ utz (see [DH92, X, 3.14]). Proposition 7.2.10. Let F and G be two Fitting classes in the same Lockett section such that F ⊆ G. For each group G denote ψ : GG /GF −→ (GG G )/(GF G ) the natural epimorphism. If p is a prime divisor of |Ker(ψ)|, then GSp = G.

320


Proof. Observe that Ker(ψ) = (GG /GF )∩(G/GF ) . Let p be a prime divisor of |Ker(ψ)| and suppose that GSp = G. If P/GF is a Sylow p-subgroup of G/GF , then P ∈ FSp ⊆ GSp = G. Since F and G are in the same Lockett section and F ⊆ G, the groups P/PF and GG /GF are abelian, by [DH92, X, 1.21]. Thus P ≤ PF and P ∩ GG is a normal subgroup of GG . Hence P ∩ GG ∈ F and P ∩ GG is subnormal in GG . Therefore P ∩ GG ≤ (GG )F = GF . Then (P/GF ) ∩ (GG /GF ) = 1. By [DH92, X, 1.21] again, GG /GF ≤ Z(G/GF ) and then (P/GF ) ∩ (G/GF ) ∩ (GG /GF ) ≤ (P/GF ) ∩ (G/GF ) ∩ Z(G/GF ) ≤ (P/GF ) by [Hup67, IV, 2.2]. Thus, (P/GF ) ∩ (G/GF ) ∩ (GG /GF ) = 1 and this contradicts the choice of P . Lemma 7.2.11. Let T be a Fitting class such that TS = T. Then Tb ⊆ T∗ ⊆ T = T∗ . Proof. By [DH92, X, 1.8], we have that T = T∗ . If X ∈ b(T), then X is perfect. By Proposition 7.2.10, XT = XT∗ . Then Cosoc(X) ∈ T∗ and Tb ⊆ T∗ . Theorem 7.2.12 (see [ILPM04]). Let T be a Fitting class such that TS = T. The correspondence F −→ F · Fit b(T) , for every Fitting class F ∈ Sec(Tb , T), defines a bijection Sec(Tb , T) −→ Sec Fit b(T) , T · Fit b(T) whose is defined by G −→ G ∩ T, for every G ∈ Sec Fit b(T) , T · inverse Fit b(T) . Moreover, the restriction of this bijection to the Lockett section Locksec(T) gives a bijection Locksec(T) −→ Locksec T · Fit b(T) . Proof. Set m = b(T), M = Fit m, B = Tb and R = T · M. If F ∈ Sec(B, T), then F · M is a Fitting class by [DH92, XI, 4.7] and Lemma 7.2.6. Obviously F · M ∈ Sec(M, R) and F ⊆ F · M ∩ T. Let G be a group in F · M ∩ T. Then GM ∈ M ∩ T = B, by Corollary 7.2.9. Hence G = GF GM ∈ F. Thus, F = F · M ∩ T. On the other hand, if G ∈ Sec(M, R), then T ∩ G ∈ Sec(B, T) by Corollary 7.2.9 and (T ∩ G) · M ⊆ G. Let G be a group in G. Then GT = GT∩G and, since G ⊆ R, we have that G = GT GM = GT∩G GM ∈ (T ∩ G) · M and then G = (T ∩ G) · M.


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Hence it only remains to prove the properties of the second bijection. We have to prove that R is a Lockett class and R∗ = T∗ · M. If G and H are groups, then it is clear that Em (G × H) = Em (G) × Em (H). Since T is a Lockett class, by Theorem 7.2.11, we also have that (G × H)T = GT × HT . Hence (G × H)R = (G × H)T Em (G × H) = GT Em (G) × HT Em (H) = GR × HR , and R is a Lockett class. Let s(R) denote the largest Fitting subclass of R which has a generating system of perfect groups. Then M ⊆ s(R) ⊆ R∗ . Hence T∗ · M ⊆ R∗ . On the other hand, for an arbitrary group G, we have that [GR , G] = [GT GM , G] = [GT , G][GM , G] ≤ GT∗ GM , by [DH92, X, 1.3]. Hence T∗ · M ∈ Locksec(R) by [DH92, X, 1.21]. Therefore T∗ · M = R∗ and we conclude the proof. Lemma 7.2.13. Let T be a Fitting class such that T = TS. 1. Set M = Fit b(T) . If U is an M-subgroup of a group G containing GM , then U is a subgroup of GM G T . 2. The class T · Fit b(T) is a normal Fitting class. Proof. Denote m = b(T) and B = Tb . 1. We can assume that G ∈ / T and then bm (G) = {X1 , . . . , Xn } is a non-empty set and Em (G) = X1 · · · Xn ≤ GM ≤ U . Hence bm (U ) = {X1 , . . . , Xn , . . . , Xt }, for n ≤ t, and Em (U ) = Em (G)L, for L = Xn+1 · · · Xt . As in the proof of Theorem 7.2.8, GM = GB Em (G) and U = UB Em (U ). Since Xi ≤ UB for each index i, we have that [UB , Xi ] ≤ UB ∩ Xi ≤ (Xi )B. Thus [Em (G), UB ] = [X1 , UB ] · · · [Xn , UB ] ≤ (X1 )B · · · (Xn )B = Em (G)B , by Lemma 7.2.5 (2a). Analogously, by Lemma 7.2.5 (1a), [Xi , L] ≤ Xi ∩ L ≤ (Xi )B , for each i. Hence [Em (G), L] ≤ Em (G)B . Therefore [GM , UB L] = [GB Em (G), UB L] ≤ GB [Em (G), UB ][Em (G), L] ≤ GB . By Theorem 7.2.8, UB L ≤ GT and U = UB Em (G)L ≤ Em (G)GT = GM GT . 2. To see that the class R = T · M is a normal Fitting class consider a group G and suppose that U is an R-subgroup such that GR ≤ U ≤ G. By Statement 1, UM ≤ GM GT = GR . On the other hand, using the arguments of the proof of Statement 1, [Em (G), UT ] ≤ UT ∩ Em (G) ≤ Em (G)B . Then [GM , UT ] = [GB Em (G), UT ] ≤ GB [Em (G), UT ] ≤ GB Em (G)B ≤ GB . Hence UT ≤ CG (GM /GB ) = GT , by Theorem 7.2.8. Thus, U = UM UT ≤ GR and U = GR .

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Lemma 7.2.14. If T is a Fitting class such that T = TS, X is a group in b(T) and F ∈ Locksec(T), then XF is not F-maximal in X. Proof. If F ∈ Locksec(T), then, in particular, Tb ⊆ F ⊆ T by Lemma 7.2.11. Moreover b(T) ⊆ b(F) by [DH92, XI, 4.7]. Since X ∈ b(T), then Cosoc(X) = XF . Suppose that XF is F-maximal in X. Consider a soluble subgroup Y /XF of X/XF . Then Y ∈ TS = T, and by maximality of XF in X, we have that XF = YF . Since F ∈ Locksec(T), the quotient Y /XF is abelian, by [DH92, X, 1.21]. Then X/XF is soluble, and this is a contradiction. Theorem 7.2.15 Let T be a Fitting class such that T = (see [ILPM04]). TS. If H ∈ Sec T∗ , T · Fit b(T) , then 1. H is an injective Fitting class; 2. H is a normal Fitting class if and only if H ∈ Locksec T · Fit b(T) . Proof. 1. Write m = b(T), F = T ∩ H and G = F · Fit m. If H ∈ H, then H = HT Em (H), by Lemma 7.2.6, since H ⊆ T · Fit m. Thus, HT ∈ H ∩ T = F. Hence H = HF Em (H) ∈ F · Fit m = G. Hence H ⊆ G. To see that H is injective, let G be a group and let us prove that G possesses H-injectors. If bm (G) = ∅, then G ∈ T. Hence GF = GH . Since F ∈ Locksec(T) by Theorem 7.2.12, the quotient G/GH is abelian. Therefore GH is a normal H-injector of G. Assume that bm (G) = ∅. Since GH is a normal subgroup of G we can assume that bm (GH ) = {X1 , . . . , Xr } and bm (G) = {X1 , . . . , Xn }, for r ≤ n. (G ) = GF Em (G) = GG . By Theorem 7.2.12, If r = n, then GH = GF E m H G ∈ Locksec T·Fit b(T) . Since, by Lemma 7.2.13, T·Fit b(T) is a normal Fitting class , we deduce that so is G , by [DH92, X, 3.3]. Therefore GG is G-injector of G and GH is H-injector of G. Now assume that r < n. Fix an index i ∈ {r + 1, . . . , n}. Clearly, Xi is / H. In addition, Cosoc(Xi ) ∈ a perfect comonolithic group such that Xi ∈ H, by virtue of Lemma 7.2.11. In particular, Xi is an H-component of G, By Proposition 7.2.2, Xi possesses H-injectors. Consider H = Hr+1 · · · Hn , with Hi ∈ InjH (Xi ) (note that Hi normalises Hj , i, j ∈ {r + 1, . . . , n}, by Lemma 7.2.3). By induction on the order of G, if NG (H) is a proper subgroup of G, then NG (H) possesses H-injectors. Then G possesses H-injectors by Theorem 7.2.4. Therefore we can suppose that H is a normal subgroup of G. Then Hi is a normal subgroup of Xi and then Hi = Cosoc(Xi ) = (Xi )H . Thus (Xi )H is an F-maximal subgroup of Xi , which contradicts Lemma 7.2.14. 2. It is shown in Theorem 7.2.12 that T·Fit m is a Lockett class. Moreover, by Lemma 7.2.13, it is a normal Fitting class. If H ∈ Locksec(T · Fit m), then H is also a normal Fitting class by [DH92, X, 3.3]. For the converse, consider H∈ / Locksec(T·Fit m). Observe that (T·Fit m)∗ = T∗ ·Fit m, by Theorem 7.2.12 and then Fit m ⊆ H. Let X be a group in m \ H. Then X is a perfect and comonolithic group and Cosoc(X) ∈ H ∩ T = F. Hence XF = Cosoc(X). Since


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T∗ is contained in F, it follows that F ∈ Locksec(T). By Lemma 7.2.14, XF is not F-maximal in X. Therefore H is not a normal Fitting class. Corollary 7.2.16 (see [ILPM04]). If F is a Fitting class in Locksec(S), then F is injective. Proof. If F ∈ Sec S∗ , S · Fit b(S) = Sec S∗ , S∗ · Fit b(S) , then F is an injective Fitting class. In particular if F ∈ Locksec(S) = {F : S∗ ⊆ F ⊆ S = S∗ }, then F is injective. Remarks 7.2.17. The example of a non-injective Fitting class in Section 7.1 affords counterexamples to possible extensions of Theorem 7.2.15: b , Fit b(T) need not be injective; 1. Fitting classes H ∈ Sec T 2. if T = TS, then Fit b(T) need not be injective; 3. Fitting classes H ∈ Sec Tb , Fit b(T) need not be normal. There are normal Fitting classes which does not belong to Sec Tb , Fit b(T) . Proof. Let S and T be non-abelian simple groups such that D0 (S, T, 1) is a non-injective Fitting class. 1. Let R be a non-abelian simple group and consider the regular wreath product W = (S × T ) R. Then W is a perfect comonolithic group (see [DH92, A, 18.8]). Hence m = (W ) is a preboundary and T = h(m) is a Fitting class such that T = TS by Theorem 2.4.12 (3). Note that Tb = Fit Cosoc(W ) = D0 (S, T ) is not injective. 2. If m = (S, T, 1) and T = h(m), then T = TS and Fit b(T) = D0 (S, T, 1) is a non-injective Fitting class. 3. Let D denote the class of all direct products of non-abelian simple groups. Let E and F be any two non-abelian simple groups. The regular wreath product W = E F is a perfect comonolithic group. Set m = (W ), T = h(m) and H = S∗ D. Then Tb = D0 (E, 1) ⊆ H. Moreover, H is the smallest normal Fitting class, by [DH92, X, 3.27], and then H ⊆ T · Fit b(T) by Lemma 7.2.13. If R is a non-abelian simple group, R ∼ F , then the regular = ∼R wreath product G = E R ∈ T. The base subgroup is E = GH and G/G H= is non-abelian. Therefore T∗ ⊆ H, by [DH92, X, 1.2]. Clearly Fit b(T) = Fit(W ) ⊆ H. Note that Tb is not normal. Corollary 7.2.18. If F is a Fitting class such that FS = F, then F is injective. In particular, the class S of all soluble groups is injective. Corollary 7.2.19. A group G possesses a single conjugacy class of S-injectors if and only if G is soluble. Proof. Applying Theorem 2.4.26, only the necessity of the condition is in doubt. Assume that a group G possesses a single conjugacy class of S-injectors. Let p and q be two different primes dividing the order of ES(G)

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and let P and Q be a Sylow p-subgroup and a Sylow q-subgroup of ES (G) respectively. Applying Proposition 7.2.2 (2) and Theorem 7.2.4, there exist S-injectors V and W of G such that P ≤ V and Q ≤ W . Since V and W are conjugate in G and ES (G) is normal in G, it follows that V ∩ ES (G) contains a Sylow q-subgroup of ES (G) for each prime q dividing |ES (G)|. Therefore ES (G) is contained in V and so ES (G) = 1. This yields that G is soluble. Theorem 7.2.20. Let X be a class of quasisimple groups and consider the class K(X) = (G : every component of G is in X). Then K(X) is an injective Fitting class. Proof. Let X be a class of quasisimple groups and denote K = K(X). We first prove that K is a Fitting class. If G ∈ K and N is a normal subgroup of G, then every component of N is a component of G. Hence every component of N is in X and then N ∈ K. Suppose that a group G is product G = N M , where N and M are normal K-subgroups of G. Let E be a component of G. Assume that E is not contained in M and E is not contained in N . Applying Proposition 7.2.3, it follows that E centralises M N . Hence E is central in G. This is a contradiction. Therefore either E is contained in M or E is contained in N . Hence E belongs to X. It implies that G ∈ K. Let E be a component of a group G ∈ KS. Then E ∈ KS. Since E is perfect, it follows that E ∈ K. Hence K = KS and therefore K is injective by Corollary 7.2.18. Let K be a Fitting class as in Theorem 7.2.20. By Proposition 2.4.6 (5) and Proposition 2.4.6 (2), F K S = F K for each Fitting class F. Hence we have the following: Corollary 7.2.21. Let X be a class of quasisimple groups and consider the class K = K(X) as in Theorem 7.2.20. Then F K is an injective Fitting class for any Fitting class F. Note that [F¨ or87, 2.5(b)] is a consequence of the above corollary. In the following, we describe another injective Fitting class, the class of all F-constrained groups. Proposition 7.2.22. Let F be a Fitting class. In a group G, the following statements are equivalent: 1. CG (GF ) ≤ GF , 2. F∗ (G) ∈ F. Proof. 1 implies 2. Suppose that E is a component of G such that E ≤ GF . Then [GF , E] = 1, by Proposition 7.2.3. Therefore E ≤ CG (GF ) ≤ GF . This contradiction yields E(G) ≤ GF .


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Denote π = char Proposition 2.2.22 (2) we have that F∗ (G) = F. Applying Oπ F(G) E(G). On the other hand, F(G) E(G) = Oπ F(G) the normal F subgroup Oπ F(G) ∩ GF is a nilpotent π -group. Hence O π F(G) ∩ GF = 1 and then Oπ F(G) ≤ CG (GF ) ≤ GF . Therefore Oπ F(G) = 1 and F(G) = Oπ F(G) ∈ F. Then F∗ (G) ∈ F. 2 implies 1. Since F∗ (G) ∈ F, it follows that F∗ (G) ≤ GF . Thus, by Proposition 2.2.22 (4), CG (GF ) ≤ CG F∗ (G) ≤ F∗ (G) ≤ GF . Corollary 7.2.23. Let F be a Fitting class. Let G be a group such that CG (GF ) ≤ GF . Then for any subnormal subgroup S of G, we have that CS (SF ) ≤ SF . Corollary 7.2.24 ([IPM86]). Let F be a Fitting class and π = char F. For ¯ = G/ Oπ (G) and adopt the “bar convention:” if H ≤ G, any group G, write G ¯ then H = H Oπ (G)/ Oπ (G). The following statements are pairwise equivalent: ¯F) ≤ G ¯F, 1. CG¯ (G ¯ ∈ F, 2. E(G) ¯ ∈ F. 3. F∗ (G) Definition 7.2.25. For a Fitting class F, a group G is said to be F-constrained if G satisfies one condition of Corollary 7.2.24. Note that every group is Q-constrained by Proposition 2.2.22 (4) and a group G is N-constrained if CG (F (G)) ≤ F (G). Corollary 7.2.26. Let F be a Fitting class. The class of all F-constrained groups is an injective Fitting class. Proof. Let X be the class of all quasisimple F-groups and consider the Fitting class K = K(X). A group G is F-constrained if and only if E G/ Oπ (G) ∈ F. This is equivalent to say that every component of the group G/ Oπ (G) ∈ X. This happens if and only if G/ Oπ (G) ∈ K, or, in other words, if and only if G ∈ Eπ K. Therefore the class of all F-constrained groups is the Fitting class Eπ K. By Corollary 7.2.21, is an injective Fitting class. Recall that the first result of existence and conjugacy of N-injectors in a universe larger that the soluble groups is due to Mann working on N-constrained groups [Man71]. Theorem 7.2.1 proves that every group, i.e. every Q-constrained group, possesses a unique conjugacy class of Q-injectors. Thus it seems that for every Fitting class F, the property of being an F-constrained group is closely related to the conjugacy of F-injectors. In general the equivalence does not hold as we observed in Corollary 7.2.19 inasmuch as the class S of all soluble groups is properly contained in the class of all S-constrained groups (which is the same as the class of all N-constrained groups). For Fitting classes F such that N ⊆ F ⊆ Q, we have the following result.

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Proposition 7.2.27 ([IPM86]). Let F be a Fitting class such that N ⊆ F ⊆ Q. If G is an F-constrained group, then 1. G possesses a single conjugacy class of F-injectors, and 2. the F-injectors and the Q-injectors of G coincide. Conversely, if G is a group such that the Q-injectors are in F, then G is an F-constrained group. Proof. Let G be an F-constrained group. Then, since char F = P, we have that F∗ (G) = GF , by Corollary 7.2.24. Let V be an Q-injector of G. Then V is an Q-maximal subgroup containing F∗ (G) [BL79]. Observe that, since F∗ (G) ≤ VF , we have that CV (VF ) ≤ CV F∗ (G) ≤ F∗ (G) ≤ VF , and V is an F-constrained group. Thus V = F∗ (V ) = VF and V is an F-maximal subgroup of G. If S is a subnormal subgroup of G, then V ∩ S is an Q-injector of S. Since F is contained in Q, we have that V ∩ S is F-maximal in S. In order to obtain the conjugacy of all F-injectors of G, it is enough to prove that each F-injector of G is an Q-injector of G. Let H be an F-injector of G, then H is an F-maximal subgroup of G containing GF = F∗ (G). Hence H is an Q-subgroup of G containing F∗ (G) and there exists a Q-injector V of G such that H ≤ V . By the previous arguments, V = H. The converse is obvious. Lemma 7.2.28. Let H and F be Fitting classes and let G be a group such that CG (GHF /GH ) ≤ GHF . Let J be subgroup of G containing GHF . Then 1. J ∈ MaxHF (G) if and only if J/GH ∈ MaxF (G/GH ). 2. J ∈ InjHF (G) if and only if J/GH ∈ InjF (G/GH ). ¯F) ≤ G ¯F Proof. The condition CG (GHF /GH ) ≤ GHF is equivalent to CG¯ (G ¯ = G/GH . Let S be a subnormal subgroup of G. for the quotient group G By Corollary 7.2.23 we have that CS¯ (S¯F ) ≤ S¯F , for S¯ = SGH /GH . But, since SH = GH ∩S, we have that S¯ ∼ = S/SH . Therefore, for any subnormal subgroup S of G, CS (SHF /SH ) ≤ SHF . Let K be a subgroup of G such that GHF ≤ K. Observe that GH ≤ K implies that GH ≤ KH ∩ GHF . On the other hand KH ∩ GHF is a normal H-subgroup of K and then of GHF , i.e. KH ∩ GHF ≤ (GHF )H ≤ GH and therefore GH = KH ∩ GHF . Thus [KH , GHF ] ≤ GH . This implies that


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KH ≤ CG (GHF )/GH ≤ GHF and then GH = KH . Using this fact, the proof is a routine checking.

Corollary 7.2.29. Let F be a Fitting class containing the class of all nilpotent groups N. Assume that every F-constrained group possesses F-injectors. Then, for every Fitting class H, the class H F is injective. Proof. We have to prove that InjHF (G) = ∅ for every group G. Let G be a minimal counterexample. First we notice that a subgroup E is an H F-component of G such that N(E) ∈ H if and only if EGH /GH is a component / F. of G/GH such that EGH /GH ∈ Let E = {E1 , . . . , En } be the set of all H F-components of G such that N(Ei ) ∈ H and suppose that E = ∅. For Ji ∈ InjHF (Ei ), i = 1, . . . , n, construct the product J = J1 · · · Jn . If NG (J) is a proper subgroup of G, then InjHF NG (J) = ∅, by minimality of G. Since the set E is invariant by conjugation of the elements of G, we can apply Theorem 7.2.4 and then InjHF (G) = ∅. This contradicts our assumption. Therefore J is a normal subgroup of G and then each Ji is normal in Ei , for i = 1, . . . , n. This implies that Ji ≤ Cosoc(Ei ). Let P/(Ei )H be a Sylow subgroup of Ei /(Ei )H . Then P ∈ H F. Observe that, since Ji / N(Ei ) ≤ Z Ei / N(Ei ) , the subgroup P is normal in P Ji . Then P Ji ∈ H F. By maximality of Ji , we have that P ≤ Ji . Since this happens for any Sylow subgroup of Ei , we have that Ei ≤ Ji , which is a contradiction. Hence E = ∅ and every component of G/GH is in F. Therefore E(G/GH ) ∈ F. This implies that G/GH is F-constrained, i.e. CG (GHF )/GH ≤ GHF by Corollary 7.2.24 . By hypothesis, the group G/GH , possesses F-injectors. By Lemma 7.2.28, the group G possesses H F-injectors. This is the final contradiction. Corollary 7.2.30 (M. J. Iranzo and F. P´ erez-Monasor). Let F be a Fitting class such that N ⊆ F ⊆ Q. Then, for every Fitting class H, the class H F is injective. In particular, the class N of all nilpotent groups is injective (P. F¨ orster [F¨ or85a]). Observe that Eπ Nπ = Eπ N. This leads us to the following. Corollary 7.2.31. Let π be a set of prime numbers. The Fitting class Eπ Nπ is injective. In particular, for any prime p, the Fitting class Ep Sp of all p-nilpotent groups is injective. Remark 7.2.32. Let p be a prime. We say that a group G is p-constrained if G is Sp -constrained group. M. J. Iranzo and M. Torres proved in [IT89] that

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a group G possesses a unique conjugacy class of p-nilpotent injectors if and only if G is p-constrained. Moreover, in this case, InjEp Sp (G) = {Op ,p (G)P : P ∈ Sylp (G)}, and the p-nilpotent injectors of G are the p-nilpotent maximal subgroups of G containing Op ,p (G). Theorem 7.2.33 ([IPM88]). Every extensible saturated Fitting formation is injective. Proof. Assume the result is false and let G be counterexample of least order. Clearly π = char F = π(F) and Nπ ⊆ F ⊆ Eπ since F is saturated. Assume the result is false and let G be counterexample of least order. Since G possesses F-injectors if and only if G/ Oπ (G) possesses F-injectors, it follows that Oπ (G) = 1. Also, since F is an extensible homomorph, G has F-injectors if and only if G/GF possesses F-injectors. Therefore GF = 1. Consider, as in Theorem 7.2.4, the set E = {E1 , . . . , En } of all F-components of G and suppose that E = ∅. Observe that, since GF = 1, the F-components of G are just the components. Let i = 1, . . . , n. Then every F-maximal subgroup Ji of Ei containing the F-radical of Ei is an F-injector of Ei by Proposition 7.2.2 (2). Consider the subgroup J = J1 , . . . , Jn . By Theorem 7.2.4, we have that J is normal in G. Moreover, J is an F-group. Hence J is contained in GF and then Ji = 1. This implies that Ei ∈ Eπ and, since Ei is subnormal E(G) = 1 in G, we obtain that Ei = 1. Then andF∗ (G) = F(G) = Oπ F(G) × Oπ F(G) . But Oπ F(G) ≤ GF = 1 and Oπ F(G) ≤ Oπ (G) = 1. Hence F∗ (G) = 1. This contradiction proves the theorem. It is not difficult to prove that every extensible saturated Fitting formation F is of the form F = G : all composition factors of G belong to F ∩ J . The most popular extensible saturated Fitting formations are the class Eπ , π a set of primes, and the class S of all soluble groups. Applying the above result, every finite group possesses Eπ -injectors. In general, if V is an Eπ -injector of a group G, then V is a maximal π-subgroup of G containing Oπ (G); but |G : V | need not to be a π -number. If G possesses Hall π-subgroups, in particular if G is soluble, then the Eπ -injectors of G are the Hall π-subgroups of G. Concluding Remarks 7.2.34. There are many other injective Fitting classes closely related to the ones presented in the section. For instance, for each prime p, let us consider the class Ep∗ p , the p∗ p-groups, defined by H. Bender (see [HB82b]). This is the class composed by all groups G factorising as G = N C∗G (P ) for any normal subgroup N and any P ∈ Sylp (N ), where C∗G (P )

7.3 Supersoluble Fitting classes

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is the largest normal subgroup of NG (P ) acting nilpotently on P . A group G ∈ Ep∗ p such that Op (G) = G is said to be a p∗ -group and the class of all p∗ -groups is denoted by Ep∗ . The class Ep∗ p is an injective Fitting class and, in fact, any Fitting class F such that Ep∗ p ⊆ F ⊆ Ep∗ Sp is injective (see [IT89]). all Other examples of injective Fitting classes are the class Ep Q of p-quasinilpotent groups and the class Op = G : G/ CG Op (G) ∈ Sp (see [MP92]). These classes satisfy the following chain Ep Q ⊂ Ep∗ p ⊂ Ep∗ Sp ⊂ Op where all containments are strict. Finally let us mention the contribution of M. J. Iranzo, J. Medina, and F. Pérez-Monasor in [IMPM01] that, using that the class Eπ is injective, proves that the class of all p-decomposable groups is an injective Fitting class. Bearing in mind Salomon’s example in Section 7.1 and the results of the present section, the following question arises: Open question 7.2.35. Is it possible to characterise the injective Fitting classes?

7.3 Supersoluble Fitting classes It is well-known that the product of two supersoluble normal subgroups of a group need not to be supersoluble. In other words, the class U of all supersoluble groups is not a Fitting class, although U is closed for subnormal subgroups. This failure is the starting point of two fruitful lines of research. 1. Obviously the direct product of supersoluble subgroups is always supersoluble; hence the study of different types of products, with extra conditions, such that those special products of supersoluble subgroups give a new supersoluble subgroup makes sense; following these ideas a considerable amount of papers has been published in the last years dealing with totally permutable products, mutually permutable products, . . . (see, for instance, [AS89], [BBPR96a]) 2. On the other hand we can analyse the properties of supersoluble Fitting classes, i.e. those Fitting classes contained in the class U of all supersoluble groups. This investigation was encouraged by the excellent results obtained in metanilpotent Fitting classes due to T. O. Hawkes, T. R. Berger, R. A. Bryce, and J. Cossey (see [DH92, XI, Section 2]). The question of the existence of Fitting classes composed of supersoluble groups was settled by M. Menth in [Men95b]. In this paper he presented a family of supersoluble non-nilpotent Fitting classes. These Fitting classes are constructed via Dark’s method (see [DH92, IX, Section 5]). Terminology and notation are mainly taken from [DH92, IX, Sections 5 and 6] and the papers of Menth [Men94, Men95b, Men95a, Men96].

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Following Dark’s strategy, we start with a identification of the universe of groups to consider. Let p be a prime such that p ≡ 1 (mod 3), and n a primitive 3rd root of unity in the field GF(p). The universe to consider will be the class Sp S3 . Now the ingredients are: 1. The key section κ(G) of a group G ∈ Sp S3 is κ(G) = Op (G). 2. The associated class X. Consider the groups T = a, b : ap = bp = [a, b, a, a] = [a, b, a, b] = [a, b, b, b] = 1 and V = T, s : s3 = 1, as = an , bs = bn . These groups have the following properties: a) |T | = p5 , T = Z2 (T ) and the factors of the central series are T /T ∼ = Cp × Cp , T / Z(T ) ∼ = Cp , and Z(T ) ∼ = Cp × Cp ; b) Z(V ) = Z(T ) and the conjugation by s induces on T /T the power automorphism x −→ xn , on T / Z(T ) the power automorphism x −→ 2 xn , and centralises Z(T ); c) every extension of T by an elementary abelian 3-group is supersoluble; in particular V is supersoluble. Let V0 be the class of all finite groups G which can be factorised as G = XY where a) X = Op (G) is a central product of copies Ti of T (the empty product, i.e. the case Op (G) = 1, is admitted); ∼ b) Y ∈ Syl3 (G) and for every index i, we have that Y / CY (Ti ) = C3 and ∼ [Ti ] Y / CY (Ti ) = V . 3. The class V = Dp (V0 ) = G ∈ Sp S3 : κ(G) ∈ V0 . The following result is due to Menth. We quote it here without proof. Theorem 7.3.1 ([Men95b, 4.2]). The class V = Fit(V ) is the Fitting class p generated by V . If G ∈ V and write P = Op (G), V0 = O (G), and C = O3 Z∞ (V0 ) , then 1. G is supersoluble; 2. F(G) = P C and G/ F(G) is an elementary abelian 3-group; 3. G = CP (Y )V0 for every Sylow 3-subgroup Y of G; 4. Soc(G) ≤ Z(G). Moreover, V is a Lockett class ([Men94, 2.2]). This supersoluble Fitting class is contained in Sp S3 . The above construction can be generalised to include examples of supersoluble Fitting classes in Sp Sq for other odd primes q. In [Tra98], G. Traustason gives an example of a supersoluble Fitting class in Sp S2 . This class is also constructed following Dark’s strategy.


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In contrast with metanilpotent Fitting classes, supersoluble Fitting classes are extremely restricted in additional closure properties. This is also proved by M. Menth in [Men95a]. In this section we will present the most relevant results of this paper. Lemma 7.3.2. Let G be a supersoluble group. Then, Fit(G) is supersoluble if and only if Fit(G) ⊆ lform(G). Proof. Denote G = lform(G). Since G is supersoluble, G ⊆ U. Hence Fit(G) is a supersoluble Fitting class. For the converse, observe that since G is supersoluble, the quotient group G/ Op ,p (G) is an abelian group of exponent e(p) dividing p−1 for each prime p by [DH92, IV, 3.4 (f)]. Applying Theorem 3.1.11, the saturated formation G is locally defined by the formation function f , where f (p) = form G/ Op ,p (G) , if p divides |G|, and f (p) = ∅ if p does not divide |G|. It is rather easy to see that f (p) = A e(p) , where A(m) denotes the class of all abelian groups of exponent dividing m. Since f (p) is subgroup-closed for all primes p, the formation G = LF(f ) is subgroup-closed by [DH92, IV, 3.14]. Hence the class Fit(G) ∩ G is Sn -closed. Let X be a group which is the product of two normal subgroups N1 , N2 of X such that N1 , N2 ∈ Fit(G) ∩ G. For each prime p, we have that X/ Op ,p (X) is the normal product of N1 Op ,p (X)/ Op ,p (X) and N2 Op ,p (X)/ Op ,p (X). Since X ∈ Fit(G), then X is supersoluble and so X/ Op ,p (X) is abelian by [DH92, IV, 3.4 (f)]. Moreover, for i = 1, 2, we have that Ni Op ,p (X)/ Op ,p (X) ∼ = Ni / Op ,p (Ni ) ∈ A e(p) , since Ni ∈ LF(f ). Hence X/ Op ,p (X) ∈ A e(p) . Hence X ∈ G. This is to say that the class Fit(G) ∩ G is N0 -closed. Therefore Fit(G) ∩ G is a Fitting class containing G. Thus, Fit(G) ⊆ lform(G). Lemma 7.3.3. Let X be a group such that the regular wreath product W = XC is a supersoluble group for some non-trivial group C. Then X is nilpotent. Proof. Suppose that the result is false and let X be a counterexample of minimal order. Then X is a non-nilpotent group and the regular wreath product W = X C is a supersoluble group for some non-trivial group C. Denote by X the base of group of W . If Y is a subgroup of X, denote by Y the corresponding subgroup of X . Let N be a minimal normal subgroup of X. Then (X/N ) C ∼ = W/N by [DH92, A, 18.2(d)]. Moreover (X/N ) C is supersoluble. By minimality of X, we have that X/N is nilpotent. Since X is non-nilpotent, it follows that X ∈ b(N) and so X is a primitive group. Since X is a supersoluble non-nilpotent primitive group, then X possesses a unique minimal normal subgroup Y which is a cyclic group of prime order, q say, and Z(X) = 1. Then Y is a minimal normal subgroup of W by [DH92, A, 18.5(a)]), and W is primitive by [DH92, A, 18.5(b)]. In particular, the order

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of the minimal normal subgroup of W is a prime. Note that the order of Y is q |C| . This contradiction proves the lemma. Theorem 7.3.4. Let F be a supersoluble Fitting class. Assume that X is a group and p is a prime such that the regular wreath product X Cp ∈ F. Then X is a p-group. Proof. Set G = X Cp ∈ F. We can assume, without loss of generality that F = Fit(G). By Lemma 7.3.2, F ⊆ lform(G). We can apply now some results due to P. Hauck (see [DH92, X, 2.9 and 2.10]) to deduce that X P ∈ F ⊆ lform(G), for every p-group P . Suppose further that X is not a p-group. Then there exists a prime divisor q = p of |X|. Since X Cp is supersoluble, it follows that X is nilpotent by Lemma 7.3.3. Therefore X = Oq ,q (X). Applying Theorem 3.1.11, lform(G) = LF(f ) is locally defined by the formation function f , where f (r) = form G/ Or ,r (G) , if r divides |G|, and f (r) = ∅ if r does not divide |G|. Then P ∈ f (q) for all p-groups P . Hence Sp ⊆ f (q) = form G/ Oq ,q (G) . (e) Observe that for every natural number e, the class Sp = G ∈ Sp : exp(G) ≤ pe is a subformation of Sp . Hence form G/ Oq ,q (G) has infinitely many subformations, and this contradicts the theorem of R. M. Bryant, R. A. Bryce, and B. Hartley ([DH92, VII, 1.6]). Fitting classes with the property of Theorem 7.3.4 are called abstoßend by P. Hauck. This term is translated into English as repellent (see [DH92, X, 2, Exercise 4]). Proposition 7.3.5. Let F be a Fitting class of soluble groups. Suppose that the group G is a semidirect non-direct product G = [N ]A of the normal subgroup N by a q-subgroup A, q a prime. Suppose that A induces the automorphism group A∗ on N and consider the semidirect product G∗ = [N ]A∗ . Then G ∈ F if and only if G∗ ∈ F. Proof. First observe that A∗ ∼ = A/ CA (N ) and C = CA (N ) is a normal subgroup of G. Thus, the group G∗ ∼ = G/C is an epimorphic image of G. Moreover, since the semidirect product is non-direct, C = A. Suppose that G ∈ F. Then Sq ⊆ F, by [DH92, IX, 1.9], and G/N ∼ = A ∈ F. Moreover N ∩ C = 1 and G/N C ∼ = A∗ is nilpotent. By Lemma 2.4.2, the G∗ ∼ = G/C ∈ F. The same arguments show that G is in F if G∗ ∼ = G/C ∈ F. Proposition 7.3.6. Let F be a Fitting class and suppose that G is an F-group such that G is the semidirect product G = [N ]s where N = N1 × · · · × Nn , Ni normal in G, 1 ≤ i ≤ n. Let σi be the automorphism of Ni induced by conjugation of s. For each i = 1, . . . , n, consider a copy Ni ∼ = Ni and construct the semidirect product Hi = [Ni × Ni ]s, where s induces on Ni the automorphism σi−1 . Then Hi ∈ F.


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Proof. Without loss of generality, we can argue with the normal subgroup N1 . Consider the direct product N ∗ = N1 ×N1 ×· · ·×Nn and a cyclic group t such that s ∼ = t. Construct the semidirect product G∗ = [N ∗ ](s × t), where N1 and all factors Ni are normal in G∗ and the operation of s and t on the Ni is as follows: s centralises N1 and acts on Ni in the same way as σi ; t centralises N1 , operates on Ni in the same way as σi for 2 ≤ i ≤ n and on N1 as σ1 . Since N1 ∈ F, we have that N1 ∈ F. Therefore N ∗ , s ∼ = N ∗ , t ∼ = N1 × G ∈ F. ∗ ∗ −1 Then G ∈ F. The normal subgroup N , st of G∗ is an F-group. Finally, observe that H1 ∼ = N1 × N1 , st−1 and this is normal in N ∗ , st−1 . Hence H1 ∈ F. Remarks and notation 7.3.7. Let p and q be different primes, p odd, such that q divides p − 1. Let e and r be natural numbers. 1. Recall that Aut(Cpe ) ∼ = Cpe−1 (p−1) (see [DH92, A, 21.1]). Each natural number m, with gcd(m, p) = 1 and 1 ≤ m ≤ pe can be uniquely written in the form m = tp + k, for 0 ≤ t ≤ pe−1 − 1 and 1 ≤ k ≤ p − 1. The pair (t, k) uniquely determines the automorphism σ(t, k) of the cyclic group Cpe = x of order pe , defined by xσ(t,k) = xtp+k = xm . 2. Therefore there exists an automorphism α = σ(t, k) of Cpe of order q. This means that n = tp + k = 1 is an integer such that nq ≡ 1 (mod pe ). Moreover any automorphism of Cpe of order q is of the form αt for t t 1 ≤ t ≤ q − 1. If x is a generator of the cyclic group Cpe , then xα = xn . 3. Let Xr be the direct product of r copies of the cyclic group of order pe . Construct the semidirect product Gr = [Xr ]C of Xr and a cyclic group C = s ∼ = Cq where s raises all elements of Xr to the same n-th power. If {x1 , . . . , xr } is a set of r generators (a basis) of Xr , observe that all subgroups of the form xi , s, for i = 1, . . . , r, are isomorphic to E(q|pe ) (see [DH92, B, 12.5]). Lemma 7.3.8. Consider the Fitting class, Fit(Gr ), generated by the group Gr . For any natural number k, let Hk = [Xk ]C denote a group which is a semidirect product of the homocyclic abelian group Xk of exponent pe and rank k ≥ 1 by a cyclic group C = α such that α is an automorphism of Xk of order q and det(α) = 1. Then Hk ∈ Fit(Gr ). Proof. The prime q is a divisor of p − 1 and then gcd(q, p) = 1. By [DH92, A, 11.6], Xk has a direct decomposition Xk = Xk(1) × · · · × Xk(s) into αadmissible subgroups Xk(i) with the following properties for each i = 1, . . . , s: 1. Xk(i) is indecomposable as a α-module; 2. Yk(i) = Xk(i) /Φ(Xk(i) ) is an irreducible GF(p)α-module. The finite field GF(p) contains a primitive q-th root of unity n. This implies that every irreducible representation of the cyclic group Cq over the field GF(p) is linear ([DH92, B, 8.9 (d)]). Therefore Yk(i) ∼ = Cp for each i = 1, . . . , s. Therefore Xk(i) ∼ = Cpe for each i = 1, . . . , s. This is to say that there exists a

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basis of Xk such that the action of α on Xk , according to this basis, can be written as a diagonal matrix diag(nλ1 , . . . , nλk−1 , nλ ), where λ = −(λ1 + · · · + λk−1 ). Consider the homocyclic group Xk+r−1 of exponent pe and rank k + r − 1 and fix a basis {x1 , . . . , xk , y1 , . . . , yr−1 } of Xk+r−1 . For each j = 1, . . . , k − 1, λj α consider the extension Lj = [Xk+r−1 ]αj of Xk+r−1 such that xj j = xnj , α

λj

α

xl j = xl , if l ∈ {1, . . . , k} \ {j}, and ys j = ysn , for s = 1, . . . , r − 1. Consider αk nλ k also the extension Lk = [Xk+r−1 ]αk of Xk+r−1 such that xα k = xk , xl = λ xl , if l ∈ {1, . . . , k − 1}, and ysαk = ysn , for s = 1, . . . , r − 1. In other words, the action of the automorphism αj on Xk+r−1 , in the fixed basis, can be written as a diagonal matrix αj = diag(1, . . . , 1, nλj , 1, . . . , 1, nλj , . . . , nλj ), $ "# ! "# $ ! ! "# $ j−1

if 1 ≤ j ≤ k − 1,

r−1

k−j

and αk = diag(1, . . . , 1, nλ , . . . , nλ ). ! "# $ ! "# $ k−1

r

∼ Gr × Xk−1 and therefore Hence, for all j = 1, . . . , k, we have that Lj = Lj ∈ Fit(Gr ). Set L = [Xk+r−1 ]α1 , . . . , αk . Clearly L is a normal product of L1 , . . . , Lk . Hence L ∈ Fit(Gr ). Consider the product α=

k j=1

αj = diag(nλ1 , nλ2 , . . . , nλk−1 , nλ , 1, . . . , 1) ! "# $ r−1

and the normal subgroup L0 = [Xk+r−1 ]α of L. Identify Xk = x1 , . . . , xk and observe that the subgroup Xk , α is isomorphic to Hk and L0 ∼ = Hk × Xr−1 . Therefore Hk is isomorphic to a subnormal subgroup of L. Hence Hk ∈ Fit(Gr ). Lemma 7.3.9. Let α be any nontrivial automorphism of Xr of order a power of q and write G = [Xr ]α. Then Gq ∈ Fit(G). m−1

is q. Proof. If the order of α is q m and m > 1, then the order of αq m−1 m−1 is normal in G, then Xr , αq ∈ Fit(G). Therefore Since Xr , αq we can assume that the order of α is q. As in Lemma 7.3.8, there exists a basis {x1 , . . . , xr } of Xr such that the matrix of α with respect to this basis is diagonal and α = diag(nλ1 , . . . , nλr ). Since α = id, not all λi are equal to 0. Without loss of generality we can assume that λ1 = 1. As a consequence of Proposition 7.3.6, the class Fit(G) contains the group E1 = [Xq ]β1 which is an extension of Xq by the automorphism β1 such that in a fixed basis of Xq has a diagonal matrix expression as follows: β1 = diag(n, n−1 , 1, . . . , 1). Clearly, this group is isomorphic to E2 = [Xq ]β2 ,


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where the automorphism β2 in the fixed basis of Xq has a diagonal matrix expression β2 = diag(1, n2 , n−2 , 1, . . . , 1). Hence E2 belongs to Fit(G). Therefore the class Fit(G) contains the extensions of Xq by the automorphisms βj , for j = 1, . . . , q − 1 such that in the fixed basis have diagonal matrix expressions as follows: β1 = diag(n, n−1 , 1, . . . , 1) β2 = diag(1, n2 , n−2 , 1, . . . , 1) ... βq−1 = diag(1, . . . , 1, nq−1 , n)

Thus Fit(G) contains the extension of Xq by the automorphism β=

q−1

βi = diag(n, . . . , n)

i=1

and then Gq = [Xq ]β ∈ Fit(G).

Lemma 7.3.10. Let X be a homocyclic group of exponent pe and let G = [X]Q be a semidirect non-direct product of X and a q-group Q. 1. If q ≥ 3, then Cpe Cq ∈ Fit(G). 2. If q = 2, then Fit(G) contains the extension of X4 by α, β, where α and β are automorphisms of X4 , i.e. members of the group GL(4, Z/pe Z), such that in a fixed basis {x1 , x2 , x3 , x4 } of X4 have matrix expressions ⎛ ⎞ ⎛ ⎞ 0010 −1 ⎜0 0 0 1⎟ ⎜ ⎟ −1 ⎟ ⎟ α=⎜ β=⎜ ⎝1 0 0 0⎠ , ⎝ ⎠ 1 0100 1 3. In both cases 1 and 2 the Fitting class Fit(G) is not supersoluble. Proof. By Proposition 7.3.5, we can assume that Q is a group of automorphisms of X. Since the semidirect product is non-direct, there exists an element s ∈ Q which is a non-trivial automorphism of X of order a power of q. It is clear that [X]s is subnormal in G and then H = [X]s ∈ Fit(G). By Lemma 7.3.9, we have that Fit(Gq ) ⊆ Fit([X]s) ⊆ Fit(G). By Lemma 7.3.8, the class Fit(G) contains all extensions of a homocyclic group X of exponent pe by α ∈ Aut(X) of order q such that det α = 1. 1. Suppose that q is odd. Observe that the regular wreath product Cpe Cq is isomorphic to a extension of the homocyclic group Xq of exponent pe and rank q by an automorphism α of order q whose action on Xq has matrix

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⎛

0100 ⎜0 0 1 0 ⎜ ⎜0 0 0 1 ⎜ ⎜0 0 0 0 ⎜ ⎜ .. .. .. .. ⎜. . . . ⎜ ⎝0 0 0 0 1000

⎞ 00 0 0⎟ ⎟ 0 0⎟ ⎟ 0 0⎟ ⎟ .. .. ⎟ . .⎟ ⎟ . . . 0 1⎠ ... 0 0

... ... ... ...

whose determinant is (−1)q−1 = 1. Hence Cpe Cq ∈ Fit(G). 2. Since α and β have both order 2 and determinant 1, the extensions X4 , α and X4 , β are in Fit(G). The group α, β is isomorphic to a dihedral group of order 8. Therefore the extension H = [X4 ]α, β is a subnormal product of X4 , α and X4 , β and then H ∈ Fit(G). 3. In Case 1, the Fitting class Fit(G) is not supersoluble by Theorem 7.3.4. In Case 2, suppose that the group H is supersoluble and consider the Frattini quotient Y4 = X4 /Φ(X4 ). The group H ∗ = [Y4 ]α, β is an epimorphic image of H and then H ∗ is supersoluble. Denote Y4 = y1 , y2 , y3 , y4 , where yi = xi Φ(X4 ), for i = 1, 2, 3, 4. Now the respective actions of α and β on the 4-dimensional GF(p)-vector space Y4 have the same matrix representation, but now considered in GL(4, p). Let N be a minimal normal subgroup of H ∗ contained in Y4 . Since H ∗ is supersoluble, the group N is cyclic, N = y say. This is to say that y is an eigenvector for α and for β. Since y is an eigenvector for β, then either y = xn1 1 xn2 2 or y = xn3 3 xn4 4 . But then y is not an eigenvector for α. Hence H is not supersoluble and Fit(G) is not a supersoluble Fitting class. Theorem 7.3.11. If F is a supersoluble Fitting class, then every metabelian F-group is nilpotent. Proof. Assume that the result is not true and let G be a metabelian nonnilpotent F-group of minimal order. Note that N = G is abelian. For every element x ∈ / N , N x is a metanilpotent normal subgroup of G. If N x were a proper subgroup of G for each element x ∈ G, then G would be nilpotent. This would contradict the choice of G. Therefore G = N x, for some element x∈ / N . By the same argument, we can assume that x is a q-element for some prime q. Clearly N is not a q-group and G = Oq (N )Q for some Q ∈ Sylq (G) such that x ∈ Q. The subgroup G0 = Oq (N )x is subnormal in G. Hence G0 ∈ F. If G0 were nilpotent, then G = N G0 would be a product of two subnormal nilpotent subgroups and therefore G would be nilpotent, contrary to supposition. The minimal choice of G implies that G = G0 , i. e., we can assume that N is a q -group. We also may suppose that x is of order q. For a prime p with p = q, the subgroup Op (N ) is normal in G. If Op (N )x is nilpotent, then x centralises Op (N ). In this case G = N ∗ x × Op (N ), where N ∗ is the Hall p -subgroup of N . By minimality of G, N ∗ x is nilpotent. Thus G is nilpotent, and this contradicts our choice of G. Hence, we can assume


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that N = N1 × · · · × Nn , where Ni ∈ Sylpi (N ), for all primes pi dividing |N |, and x induces on each Ni a non-trivial automorphism σi . Since x does not centralise N1 , it follows that x does not centralise some chief factor of G below N1 . This implies that q divides p1 − 1 since G is supersoluble. Consider the semidirect product H = [P ]C, where P = N0 × N1 , with N0 ∼ = N1 , and C = x. Suppose that x induces on N1 the automorphism σ1 and on N0 the automorphism σ1−1 . By Proposition 7.3.6, we have that H ∈ F and H is non-nilpotent. By [DH92, A, 11.6], we have that N0 has a direct decomposition N0 = A1(0) × · · · × Ak(0) with the following properties for each i = 1, . . . , k: 1. Ai(0) is indecomposable as a C-module; 2. Ai(0) /Φ(Ai(0) ) is an irreducible GF(p1 )C-module; 3. Ai(0) is homocyclic. Note that Ai(0) /Φ(Ai(0) ) is a faithful C-module and so its dimension is 1 because q divides p1 − 1 ([DH92, B, 8.9 (d)]). Therefore Ai(0) ∼ = Cpe1 for each i = 1, . . . , k. Moreover x induces on each Ai(0) an automorphism σ1−1 . Analogously N1 = A1(1) × · · · × Ak(1) , Ai(1) ∼ = Cpe1 for each i = 1, . . . , k and x induces on each Ai(1) the automorphism σ1 . By Lemma 7.3.6, we have that [A1(0) × A1(1) ]C ∈ F. Hence Lemma 7.3.10 implies that F is not supersoluble. This contradiction proves the theorem. Theorem 7.3.12. Let F be a supersoluble non-nilpotent Fitting class. Then F is not closed with respect to any of the operators Q, S, and EΦ . Proof. Assume that F is a Q-closed non-nilpotent supersoluble Fitting class. Let H be a supersoluble non-nilpotent F-group of minimal order. Then H/N is nilpotent F-group for every minimal normal subgroup N of H. Consequently H ∈ b(N) and so H is a primitive group. Then, by Theorem 1, N = Soc(H) is a minimal normal subgroup of H and N = CH (N ) and N is cyclic of prime order. In particular, H is metabelian. This contradicts Theorem 7.3.11. Therefore the class F is not Q-closed. Suppose that F is an EΦ -closed supersoluble non-nilpotent Fitting class. Since F is composed of metanilpotent groups we can apply the theorem [DH92, XI, 2.16] to conclude that F is S-closed. Applying Theorem 2.5.2, F is a saturated formation. In particular, F is Q-closed. This contradiction proves that F is not EΦ -closed. Note that F cannot be subgroup-closed either. Recall that a Fischer class is a Fitting class F satisfying the following property: if G is a group in F and H/K is a normal nilpotent subgroup of G/K for some normal subgroup K of G, it follows that H ∈ F. These classes were originally introduced by Fischer in the soluble universe. If F is a Fischer class of soluble groups, then the F-injectors of a soluble group are exactly the Fischer F-subgroups, which are the natural duals of Gasch¨ utz’s covering subgroups (see [DH92, IX, Section 3]).

338


Corollary 7.3.13. Let F be a supersoluble non-nilpotent Fitting class. Then F is not a Fischer class. Proof. Assume that F is a Fischer class. We shall prove that F is subgroupclosed. Suppose that this is not true and let G be a group of minimal order such that G ∈ F but M ∈ / F for some subgroup M of G. Among the subgroups of G which are not in F, we choose M of maximal order. Clearly M is a maximal subgroup of G. If G is contained in M , then M is normal and so M ∈ F, contrary to supposition. Consequently, G = M G . Since, by [DH92, VII, 2.2], M has prime index, it follows that M/M ∩ G is a cyclic group of prime order. Note that G is nilpotent and M ∩ G has prime index in G . This implies that M ∩ G is normal in G . Therefore M ∩ G is normal in G. Since F is a Fischer class, we have that M ∈ F, contrary to the choice of M . Then F is subgroup-closed. This contradicts Theorem 7.3.12. Consequently, F is not a Fischer class. Since metanilpotent R0 -closed Fitting classes need not be Q-closed, the exclusion of the R0 -closure cannot be argued in the same way. What Menth shows is that the supersoluble Fitting class V introduced at the beginning of the section is not R0 -closed. Theorem 7.3.14. The class V is not

R0 -closed.

Proof. We will use the notation introduced at the beginning of the section. Let us consider the direct product W = V × V ϕ of two copies of V . The diagonal subgroup D = {(x, xϕ ) : x ∈ V } of W is isomorphic to V . The subgroups A = {(x, 1) : x ∈ T } and B = {(1, xϕ ) : x ∈ T } are normal in W and A ∩ B = (1, 1). Observe that the subgroup G = A, D is a semidirect product G = [A]D = [B]D and G/A ∼ = G/B ∼ = V ∈ V. Next we see that G∈ / V. The element (s, sϕ ) is a 3-element and then (s, sϕ ) ∈ Op (G). Hence the commutator [(a, aϕ ), (s, sϕ )] = (a, aϕ )n−1 ∈ Op (G) and also (b, bϕ )n−1 ∈ Op (G) for the generators a, b of T . Therefore D is contained in Op (G). There 2 exists an element t ∈ T \ Z(T ) such that ts = tn . Hence [(t, 1), (s, sϕ )] = 2 ([t, s], 1) = (tn −1 , 1) ∈ Op (G). Since n is a primitive cube root of unity in GF(p), we have that p divides n3 − 1 but gcd(p, n2 − 1) = 1. Therefore (t, 1) ∈ Op (G). Then [(t, 1), (a, aϕ )] = ([t, a], 1) and [(t, 1), (b, bϕ )] = ([t, b], 1) are in Op (G). Then A ≤ Op (G). Therefore G = Op (G) and the group G is p-perfect. Observe that the subgroup Z(T ) × Z(T )ϕ is a subgroup of Z Op (G) of order p4 . If we suppose that G ∈ V, then G ∈ V0 and then Op (G) is a central product of copies of T . Since |Op (G)| = p8 , we need exactly two copies of T, T1 , T2 say, such that |T1 ∩ T2 | = p2 . Therefore Z(T1 ) = Z(T2 ) = Z Op (G) / V. We has order p2 . This contradicts the previous observation. Hence G ∈ conclude then that V is not R0 -closed.

7.4 Fitting sets, Fitting sets pairs, and outer Fitting sets pairs

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Let F be a Fitting class of soluble groups. If π is a set of primes, F is said to be Hall-π-closed provided that whenever H is a Hall π-subgroup of G and G ∈ F, then H ∈ F. The class F is said to be Hall-closed if it is Hall-π-closed for all sets of primes π. Theorem 7.3.15. Every metanilpotent Lockett class is Hall-closed. Proof. Assume that the result is false and let F be a metanilpotent Fitting class that is not Hall-closed. There exists a set π of primes and a group G ∈ F such that G has a Hall π-subgroup H ∈ / F. Set F = F(G), and let p1 , . . . , pn be the prime divisors of |F |. Then F is the direct product of its Sylow pi -subgroups Pi , 1 ≤ i ≤ n, and G/F is nilpotent. Having numbered the primes suitably, there is an integer k (1 ≤ k ≤ n) such that p1 , . . . , pk are elements of π. Note that k < n because otherwise H would be subnormal in G. Then P = H ∩ F = P1 · · · Pk . The quotient H/P is isomorphic to a subgroup of G/F and therefore nilpotent. Hence H/P is generated by cyclic subgroups xi P . At least one of the subgroups P, xi is not an F-group. Let us choose H ∗ = P, x such that |H ∗ | is of minimal order. Then HF∗ is a normal maximal subgroup of H ∗ . Now we replace G by G∗ = F, x, because G∗ ∈ F and H ∗ is a Hall π-subgroup of G∗ . Set Q = Pk+1 · · · Pn . We define a direct product D = P, x1 × Q, x2 , where P, x1 is a copy of H ∗ and Q, x2 is a copy of Qx. Then K = P Qx1 x2 is a normal subgroup of D isomorphic to G∗ . Hence K is contained in DF = P, x1 F × Q, x2 F . Since |P, x1 : P, x1 F | = P and |Q, x2 : Q, x2 F | = p, it follows that |D : DF | = p2 . However |D : K| = p. This contradiction proves the theorem. Not every supersoluble Fitting class is a Lockett class ([Men96, Example 1]). In the following we shall prove that every supersoluble Fitting class is contained in a supersoluble Lockett class. Theorem 7.3.16. Every supersoluble Fitting class is contained in a supersoluble Lockett class. Proof. Assume that F is a supersoluble Fitting class. If G ∈ F∗ , then D = {(g, g −1 ) : g ∈ G} is a subgroup of (G × G)F by [DH92, X, 1.5, 1.9]. Therefore D is supersoluble. Since G is an epimorphic image of D, it follows that G is supersoluble. Therefore F∗ is a supersoluble Lockett class.

7.4 Fitting sets, Fitting sets pairs, and outer Fitting sets pairs This section has two main themes. The first is connected with Fitting sets and injectors. The second subject under investigation is the localised theory of Fitting pairs and outer Fitting pairs developed in [AJBBPR00].

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As mentioned in Section 2.4, the theory of Fitting classes has been enriched by the introduction of Fitting sets by W. Anderson in [And75]. Recall that a subgroup H of a group G is an injector of G if H is an F-injector of G for some Fitting set F of G. One the most important motivating questions in the theory of Fitting sets is to determine which subgroups are injectors. Some results in this direction are presented in [DH92, VIII, Section 3]. There Doerk and Hawkes proposed the problem of describing injectors of soluble groups without explicit use of the concept of a Fitting set. This problem is complicated by the general nature of injectors: there are likely to be many Fitting sets for a given group, often leading to different sets of injectors. For example, the set of injectors of a soluble group includes all its normal subgroups, all its Hall subgroups, and all its maximal subgroups [DH92, VIII, 3.5]. An injector A of a finite soluble group B must have rather strong properties that can be described without direct reference to Fitting sets: A ∩ K must be a CAP subgroup of K and pronormal (see [DH92, Section I, 6]) in B for each normal subgroup K of B [DH92, VIII, 2.14]. However, these properties are inadequate to characterise injectors [DH92, Exercise 2, p. 553]. We present here the best attempt to accomplish that task. This characterisation, unpublished at the moment of writing this, was communicated privately by its authors, R. Dark and A. Feldman ([DF]), to us. If G is a group, denote by Inj(G) the set of all injectors of G. The following result is a very useful characterisation of this set. Recall that if H is a subgroup of G then Sn

H G = {S ≤ G : S is a subnormal subgroup of H g , for some g ∈ G}.

Lemma 7.4.1 ([DH92, VIII, 3.3]). Let G be a soluble group and H a subgroup of G. Then any two of the following statements are equivalent 1. H ∈ Inj(G) 2. Sn H G is a Fitting set of G. 3. Sn H G is the smallest Fitting set of G which contains H. Lemma 7.4.2. Suppose S and T are pronormal subgroups of a soluble group G and x, y ∈ G. If S and T are subnormal in S, T and S x and T y are subnormal in S x , T y , then there exists z ∈ G with S x = S z and T y = T z . Proof. Let Σ be a Hall system of G which reduces into S, T . Applying [DH92, I, 6.3], S and T are normal in S, T = ST . By [DH92, I, 4.21], Σ reduces into both S and T . Analogously, S x and T y are normal in S x , T y = S x T y . Then by [DH92, I, 6.11], S x T y = (ST )z for some z ∈ G. This implies that Σ z , which reduces into (ST )z , reduces into the subnormal subgroups S x and S z and T y and T z of that group. But the pronormality of S and T then implies, by [DH92, I, 6.6], that S x = S z and T y = T z , as claimed. Now we prove a result that will supply the inductive step in our eventual characterisation of injectors:


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Theorem 7.4.3. Let G be a soluble group and suppose H is a subgroup of G and M is a normal subgroup of G. Assume that the following condition holds: Whenever S is a subnormal subgroup of H, g ∈ G, S g ≤ HM and S1 = H ∩ S g M is subnormal in H, then S1 and S g are conjugate in (7.1) J = S1 , S g . Then 1. if S is a subnormal subgroup of H, then S is pronormal in NG (SM ) and 2. if HM ∈ Inj(G), then H ∈ Inj(G). Proof. 1. Let g be an element of NG (SM ), so that S g M = (SM )g = SM. Note that if S is subnormal in H, then SM is subnormal in HM , and therefore S1 = H ∩ S g M = H ∩ SM = S(H ∩ M ) is subnormal in H. Applying (7.1) with g = 1 yields S and S1 = S(H ∩ M ) are conjugate. Now, by order considerations, S = S1 . By (7.1) then, S and S g are conjugate in S, S g ; i.e. S is pronormal in NG (SM ). 2. Suppose that S and T are subnormal subgroups of H and a, b ∈ G with S a and T b normal in S a T b . By Lemma 7.4.1, it suffices to find an element w such that S a T b is subnormal in H w . Now SM and T M are subnormal subgroups of HM and S a M and T b M are normal in Y = S a T b M = S a M T b M , and because HM ∈ Inj(G), there exists c ∈ G such that Y is subnormal in (HM )c = H c M . Let H0 = H c and S0 = S c . Note that condition (7.1) still holds when H is replaced by the conjugate H0 . Replacing S and g by S0 and c−1 a we have S0 is subnormal in H0 , S0g = S a ≤ H0 M , and S0g M = S a M is normal in Y which is subnormal in H0 M . Hence S0g M is subnormal in H0 M , and S1 = H0 ∩ S0g M is subnormal in H0 . Then by (7.1), S1 and S a are conjugate in S1 , S a ≤ S a M ≤ Y . Similarly, T1 = H0 ∩ T b M is subnormal in H0 , and T b is conjugate in Y to T1 ; hence there are elements x, y ∈ Y such that S1x = S a and T1y = T b . Now S a M = H0 M ∩ S0g M = (H0 ∩ S0g M )M = S1 M and then Y ≤ NG (S a M ) = NG (S1 M ), and if follows from Assertion 1 that S1 is pronormal in Y . Similarly, T1 is pronormal in Y . We also have that S1 and T1 are subnormal in S1 , T1 and S1x , T1y normal in S1x T1y . By Lemma 7.4.2, there exists z ∈ Y with S1x = S1z and T1y = T1z . Hence S a T b = S1x T1y = (S1 T1 )z is subnormal in H0z = H cz , so setting w = cz yields our result. Now we are ready to prove that two properties that do not involve Fitting sets are equivalent to that of being an injector. Not surprisingly, conjugation, which is crucial to the definition of Fitting set and normality (and therefore indirectly, subnormality) play an important role in these properties. In particular, for convenience we introduce the following definition: Definition 7.4.4. If H and X are subgroups of a soluble group G and g ∈ G, we say H is (X, g)-pronormal if H ∩ X and H g ∩ X are conjugate in J = H ∩ X, H g ∩ X.

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Note that H is a pronormal subgroup of G if and only if H is (G, g)pronormal for all g ∈ G. We now can prove: Theorem 7.4.5 (R. Dark and A. Feldman). Let G be a soluble group, and suppose that H is a subgroup of G. Then any two of the following conditions are equivalent: 1. H is an injector of G; −1 2. whenever H ≤ K ≤ G, g ∈ G, and X and X g are subnormal subgroups of K, then H is (X, g)-pronormal; 3. whenever M/N is a chief factor of G which is not covered by H, S is a subnormal subgroup of H such that H ∩ N ≤ S, g ∈ G, and S g ≤ HM with S1 = H ∩ S g M subnormal in H, then S1 and S g are conjugate in J = S1 , S g . Proof. 1 implies 2. Suppose that H is an F-injector of G for some Fitting set F of G. Then, with K and X as in 2 and J as in the definition of (X, g)pronormal, H is an FK -injector of K by [DH92, VIII, 2.13], and then H ∩X is an FX -injector of X by [DH92, VIII, 2.6], and hence H ∩ X is an FJ -injector of J by [DH92, VIII, 2.13] again. Similarly, H g is an FK g -injector of K g , and X is subnormal in K g by hypothesis, and then H g ∩ X is an FX -injector of X, and H g ∩ X is an FJ -injector of J. Thus by Theorem 2.4.26, H ∩ X and H g ∩ X are conjugate in J, establishing 2. 2 implies 3. First observe that, in these hypotheses, we certainly have that H avoids M/N . With X = M , we see that H ∩ M and H g ∩ M are conjugate in J = H ∩ M, H g ∩ M , and then (H ∩ M )N and (H g ∩ M )N are conjugate in JN . But JN/N ≤ M/N , which is abelian, and it follows that (H ∩ M )N = (H g ∩ M )N . This holds for all g ∈ G because X = M is normal in G, and then (H ∩ M )N is normal in G. Since H does not cover the chief factor M/N of G, we have that (H ∩ M )N < M . Then (H ∩ M )N = N , establishing the result. Assume the hypotheses of 3 and take X = S g M . Then X is subnormal −1 in H g M and X g is subnormal in HM . Also, X ≤ HM , and X = HM ∩ S g M = (H ∩ S g M )M = S1 M is subnormal in HM . Moreover, H ∩ X = S1 by definition, and H g ∩ X = H g ∩ S g M = S g (H g ∩ M ), which equals S g (H g ∩ N ) inasmuch as H g avoids M/N . But H ∩ N ≤ S by hypothesis, and then H g ∩ N ≤ S g , and H g ∩ X = S g . Thus 2 yields that S1 and S g are conjugate in S1 , S g , as claimed. To see that 3 implies 1, we pass through an intermediate Step 4. 4. Whenever M/N is a chief factor of G which is not covered by H, and such that CoreG (H) ≤ N < M ≤ H G , and S is a subnormal subgroup of H such that H ∩ N ≤ S, g ∈ G, and S g ≤ HM with S1 = H ∩ S g M subnormal in H, then S1 and S g are conjugate in J = S1 , S g .


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It is clear that 3 implies 4. Hence we have to prove that 4 implies 1. Note first that if C = CoreG (H), it is easy to see that if 4 holds for H ≤ G, then 4 also holds for H/C ≤ G/C. Moreover, if H/C ∈ Inj(G/C), then H ∈ Inj(G) by [DH92, VIII, 2.17]. Thus it suffices to prove that if 4 holds for H/C in G/C, then H/C ∈ Inj(G/C), and we may assume that C = 1, i.e. H is core-free in G. We proceed by induction on the index |H G : H|. If |H G : H| = 1, then H = 1 inasmuch as H is core-free. In this case H is obviously an injector of G. Hence we may assume that |H G : H| > 1. Let M1 be a minimal normal subgroup of G such that M1 ≤ H G . Since H is core-free, H does not cover M1 . We see next that because 4 holds for H, it also holds for HM1 . Suppose that M/N is a chief factor of G which 1 < M1 ≤ CoreG (HM1 ) ≤ N < M ≤ (HM1 )G = H G and M/N is not covered by HM1 . Now suppose that g ∈ G and S¯ is a subnormal subgroup of HM1 such ¯ g ∈ G, and S¯g ≤ (HM1 )M with S¯1 = HM1 ∩ S¯g M that HM1 ∩ N ≤ S, subnormal in HM1 , ¯ ¯ Then S is subnormal in H. Since M1 ≤ HM1 ∩N ≤ S, Consider S = H ∩ S. g g ¯ ¯ ¯ ¯ then S = HM1 ∩ S = (H ∩ S)M1 = SM1 , and then S M = S M . Observe also that H ∩ N = H ∩ HM1 ∩ N ≤ H ∩ S¯ = S and S g ≤ S¯g ≤ HM . Finally, it is clear that S1 = H ∩ S g M = H ∩ (HM1 ∩ S¯g M ) = H ∩ S¯1 is subnormal in H. Thus the hypotheses of 4 hold, implying S1 and S g are conjugate in J = S1 , S g . Moreover, S¯ = SM1 , and S¯g = S g M1 , and S¯1 = HM1 ∩ S¯g M = (H ∩ S g M )M1 = S1 M1 , and J¯ = S¯1 , S¯g = JM1 . Hence S¯1 and S¯g are ¯ conjugate in J. Observe that |H G | = |(HM1 )G : HM1 | < |H G : H|. Thus the induction hypothesis implies that HM1 ∈ Inj(G). To complete the proof, we apply Theorem 7.4.3 (2) with M = M1 . With N = 1, and by 4 applied to the chief factor M1 /N , Condition (7.1) of Theorem 7.4.3 holds. Thus, Theorem 7.4.3 (2) shows that H ∈ Inj(G). Corollary 7.4.6. Let G be a soluble group. Suppose that H is an injector of G and M a normal subgroup of G. Then H ∩ M is pronormal in G. Applying [DH92, VIII, 3.5], a maximal subgroup of a group is always an injector. Hence, in particular, in a soluble group the intersection of a maximal subgroup and a normal subgroup is pronormal in the group. By [DH92, VIII, 3.8] every normally embedded subgroup of a soluble group is an injector. In the following we give a proof of this fact using Theorem 7.4.5. Corollary 7.4.7. Suppose H is a normally embedded subgroup of a soluble group G. Then H ∈ Inj(G). −1

Proof. Assume that H is normally embedded in G, H ≤ K ≤ G, and X, X g are subnormal in K for some g ∈ G. We shall show that H is (X, g)-pronormal.

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First we show that H ∩ X and H g ∩ X are locally conjugate in X. For an arbitrary prime, p, let P ∈ Sylp (H); let P1 ∈ Sylp (K) such that P1 ∩ H = P . Because X is subnormal in K, P1 ∩ X ∈ Sylp (X), by [DH92, I, 4.21]. Also, H ∩ X is subnormal in H, and P ∩ X = P ∩ (H ∩ X) ∈ Sylp (H ∩ X). Now H normally embedded in G implies P ∈ Sylp (P G ), and P ≤ P G ∩P1 ≤ P G , and then P = P G ∩ P1 . Because P G ∩ X is normal in X, (P1 ∩ X) ∩ (P G ∩ X) ∈ Sylp (P G ∩ X). But (P G ∩ X) ∩ (P1 ∩ X) = (P G ∩ P1 ) ∩ X = P ∩ X ∈ Sylp (H ∩ X). Hence any Sylow p-subgroup of H ∩ X is a Sylow p-subgroup of P G ∩ X. By similar arguments, P g ∈ Sylp (H g ) implies P g ∩ X ∈ Sylp (P g )G ∩ X = Sylp (P G ∩ X) and P g ∩ X ∈ Sylp (H g ∩ X). Thus we have Sylow p-subgroups of H ∩ X and H g ∩ X that are Sylow p- subgroups of the same subgroup of X, and they are conjugate in X, as desired. Now note that P G ∩X is normal in X, and since this works for all primes p, H ∩ X and H g ∩ X are normally embedded in X. Thus H ∩ X and H g ∩ X are locally pronormal [DH92, I, 7.13] and therefore pronormal [DH92, I, 6.14] in X. Thus H ∩ X and H g ∩ X are locally conjugate and locally pronormal subgroups in X, and they are conjugate in X [DH92, I, 6.16]. Finally, the pronormality of H ∩ X in X implies that H ∩ X and H g ∩ X are conjugate in their join; i.e. H is (X, g)-pronormal, establishing the result. Let F be a Fitting class. Blessenohl and Gasch¨ utz [BG70] introduced the notion of F-Fitting pair which turns out to be useful for the construction of normal Fitting classes in the Lockett section of F. We need to deal with arbitrary (possibly infinite) groups. Hence if we denote a group by G, we are assuming that the group G is finite. Otherwise, we put G. Definition 7.4.8. If N and M are groups, an embedding is a group monomorphism ν : N −→ M . If N ν is a normal subgroup of M , then ν is said to be a normal embedding. Definition 7.4.9 ([BG70]). Let F be a Fitting class. An F-Fitting pair is a pair (d, A) which consists of a group A and a family dU ∈ Hom(U, A) : U ∈ F such that for each normal embedding ν : U −→ V ∈ F, the assertion dU = νdV holds. It can be proved that in this case {(g)dG : g ∈ G, G ∈ F} is an abelian subgroup of A ([DH92, IX, 2.12 (b)]). Hence, without loss of generality, we may assume that A is abelian. In the same paper, Blessenohl and Gasch¨ utz gave examples of Fitting pairs and proved the following result, which remains valid in the general finite universe. Proposition 7.4.10 (see [DH92, IX, 2.11]). Let F be a Fitting class and let (d, A) be an F-Fitting pair. Then the class R = Ker(d, A) of all groups G ∈ F such that GdG = 1 is a normal Fitting class such that F∗ ⊆ R ⊆ F.


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Lausch [Lau73] showed that every non-trivial normal Fitting class in the soluble universe can be described as the kernel of a Fitting pair. He also described a universal F-Fitting pair, leading to the so-called Lausch group. He carried out the construction for the case F = S, but as Bryce and Cossey pointed out in [BC75], Lausch’s method applies to an arbitrary Fitting class (see [DH92, X, Section 4] for details). J. Pense, in his Dissertation [Pen87], generalised the concept of an F-Fitting pair to that of outer F-Fitting pair. Definition 7.4.11 (see [Pen88]). Let F be a Fitting class. An outer F- Fitting pair is a pair (d, A) which consists of a group A and a family dU ∈ Hom(U, A) : U ∈ F such that for each normal embedding ν : U −→ V ∈ F, there exists an inner automorphism α of A such that dU α = νdV . Obviously, if A is an abelian group, then an outer F-Fitting pair is just an F-Fitting pair. Pense extended the definition of a Fitting set to an infinite group by requiring it to mean a set of finite subgroups closed under conjugation and under the usual operations of taking normal subgroups and forming finite normal products. He also introduced the concept of F-Fitting sets pair (d, A), where A is an abelian group, to develop a local version of the Lausch group in certain type of groups ([Pen87]). Definition 7.4.12. If N and M are finite subgroups of G, a G-embedding is a group monomorphism ν : N −→ M which is the restriction to N of an inner automorphism of G. If N ν is a normal subgroup of M , then ν is said to be a normal G-embedding. Definition 7.4.13. Let F be a Fitting set of a group G. An F-Fitting sets (d, A) which consists of a group A and a family pair relative to G is a pair dU ∈ Hom(U, A) : U ∈ F such that for each normal G-embedding ν : U −→ V ∈ F, the assertion dU = νdV holds. Note that, in our definition of F-Fitting sets pair, we do not require that A is an abelian group. An outer F-Fitting sets pair is defined as follows: Definition 7.4.14 ([AJBBPR00]). Let F be a Fitting set of a group G. An outer F-Fitting sets pair relative to G is a pair (d, A) which consists of a group A and a family dU ∈ Hom(U, A) : U ∈ F such that for each normal G-embedding ν : U −→ V ∈ F, there exists an inner automorphism α of A such that dU α = νdV . If F is a Fitting class, then TrF (G) is a Fitting set of the group G, and if (d, A) is an (outer) F-Fitting pair, then the pair (d, A), for dU ∈ Hom(U, A) : U ∈ TrF (G) , is an (outer) TrF (G)-Fitting sets pair relative to G.

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Definition 7.4.15. Two outer F-Fitting sets pairs (di , Ai ), i = 1, 2, are equivalent if there exists an isomorphism σ : A1 −→ A2 , such that for each U ∈ F, there exists αU ∈ Inn(A2 ) such that d2U = d1U σαU . In [AJBBPR00], P. Arroyo-Jord´ a, A. Ballester-Bolinches, and M. D. PérezRamos made a complete study of outer Fitting sets pairs. In the sequel, we will present the main results of this paper. To begin with, we point out that there are some differences between Fitting pairs and Fitting sets pairs. We shall show two of them. Remarks 7.4.16. 1. In Definition 7.4.9 of Fitting pair, the group A can be assumed abelian without loss of generality. This is not true for Fitting sets pairs in general. Let G be the alternating group of degree 5, G = Alt(5), and F the trace in G of the Fitting class F = S3 S5 S2 . In other words, the Fitting set F is composed of all subgroups of G of prime-power order, and the normalisers of the Sylow 5- and 3-subgroups. Consider the symmetric group S = Sym(3) of degree 3. If X is a subgroup of prime-power order of G, then put dX : X −→ S to be the trivial homomorphism: xdX = 1 for all x ∈ X. If P ∈ Syl3 (G) and N3 = NG (P ), then put dN3 : N3 −→ S to be a homomorphism such that P = Ker(dN3 ) and Im(dN3 ) = (12). If Q ∈ Syl5 (G) and N5 = NG (Q), then put dN5 : N5 −→ S to be a homomorphism such that Q = Ker(dN5 ) and Im(dN5 ) = (23). The pair ({dH : H ∈ F}, S) is an F-Fitting sets pair relative to G. Observe that S is not abelian and S = hdH : H ∈ F, h ∈ H. 2. Pense [Pen87, Kollollar 3.30] shows that if (d, A) is a outer Fitting pair with A finite, then it is equivalent to a Fitting pair. This is not true for outer Fitting sets pairs. Let Q = x, y : x4 = 1, x2 = y 2 , xy = x−1 be a quaternion group of order 8 and fix a subgroup C = x of order 4 of Q. The set of all subgroups of C is a Fitting set F of Q. The inclusion ι : C −→ Q induces a family of monomorphisms between the members of F and Q. The pair (ι, Q) is an outer F-Fitting sets pair relative to Q. The inner automorphism αy of Q induced by y gives a normal Q-embedding of ν : C −→ C such that xν = x−1 and ιαy = νι. If (ι, Q) were equivalent to a F-Fitting sets pair (d, A), there would exist an isomorphism ψ : Q −→ A such that for each subgroup T of C there would exist αT ∈ Inn(A) such that dT = ιT ψαT . Since dC = νdC , we have that x2 ∈ Ker(dC ). But ιC ψαC is a monomorphism and therefore dC = ιC ψαC . Thus (ι, Q) cannot be equivalent to an F-Fitting sets pair (d, A). The following result is the “Fitting sets” version of [Pen87, Satz 3.2]. Theorem 7.4.17. Let (d, A) be an outer F-Fitting sets pair relative to G and let H be a Fitting set of A.


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1. The collection Hd−1 = {U ∈ F : U dU ∈ H} of finite subgroups of G is a Fitting set of G. d−1 2. If U ∈ F, then UHd−1 = (U dU )H U . Proof. 1. If N is a normal subgroup of U ∈ Hd−1 , then N dN is conjugate in A to the normal subgroup N dU of U dU ∈ H. Thus N ∈ Hd−1 . Assume that N1 and N2 are subgroups of G which are normal in T = N1 N2 and Ni ∈ Hd−1 , for i = 1, 2. Then T dT = N1dT N2dT and NidT is normal in T dT , dN for i = 1, 2. Moreover, NidT is conjugate in A to Ni i , for i = 1, 2. Therefore T ∈ Hd−1 . d−1 2. Let C = (U dU )H U . By Statement 1, C is a normal Hd−1 -subgroup of U . If M is a normal subgroup of U , with M ∈ Hd−1 , then M dM ∈ H and it is conjugate in A to M dU . Hence M dU ≤ (U dU )H and then M ≤ C. Definition 7.4.18. For an outer F-Fitting sets pair relative to G, (d, A), and a homomorphism ϕ : A −→ B, we define the induced outer F-Fitting pair relative to G, (dϕ, B), by (dϕ)T = dT ϕ, for every T ∈ F. The next theorem provides a criterion for the Fitting sets constructed by means of outer Fitting sets pairs to be injective. Theorem 7.4.19 ([AJBBPR00]). Let G be a group and denote by EG the Fitting set composed of all finite subgroups of G. Let (d, A) be an outer EG Fitting sets pair relative to G. Suppose that F is a Fitting set of A and the pair (d, A) satisfies the following condition: For each G-embedding ν : V −→ U , for U , V ∈ EG such that UF d−1 ≤ (7.2) V ν , there exists η ∈ Inn(A) such that νdU = dV η. Let X ∈ EG . If the group X dX possesses a single conjugacy class of F-injectors, then X also possesses a single conjugacy class of Fd−1-injectors. Proof. Let X be a subgroup of G and assume that T is an F-injector of −1 X dX . Denote by U = T dX . We shall see that U is an Fd−1 -injector of X. Since T is an F-injector of X dX , it follows that (X dX )F is a subgroup of −1 T . Hence XF d−1 = ((X dX )F )dX by Theorem 7.4.17 (2) and it is contained in U . By property (7.2) there exists a ∈ A such that (U dU )a = U dX . Since T = U dX ∈ F it follows that U ∈ Fd−1 . Let N be a subnormal subgroup of X and suppose that U ∩ N ≤ W ≤ N , where W ∈ Fd−1 . Since N is a subnormal subgroup of X, it holds that NF d−1 = N ∩ XF d−1 ≤ N ∩ U ≤ W . By (7.2), the subgroup W dN is conjugate in A to W dW which is in F. On the other hand, since (d, A) is an outer EG -Fitting sets pair relative to G, there exists θ ∈ Inn(A) such that dN is dX θ restricted to N . Hence W dN is conjugate in A to W dX . Consequently W dX ∈ F. Now Ker(dX ) ≤ XFd−1 ≤ U . Hence (U ∩ N )dX = T ∩ N dX which is contained in W dX ≤ N dX . Since T is an F-injector of X dX and W dX ∈ F,

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it follows that T ∩ N dX = W dX . Therefore W ≤ U and U ∩ N = W . This means that U is an Fd−1 -injector of X. Suppose now that X dX has a single conjugacy class of F-injectors. Let U ˜ be two Fd−1 -injectors of X. A straightforward proof using analogous and U ˜ dX are F-injectors of X dX . By hypothesis, arguments provides that U dX and U dX ˜ x )dX . Since Ker(dX ) ≤ U ∩ U ˜ , it follows there exists x ∈ X such that U = (U x ˜ that U = U . The rest of the section is devoted to construct injective Fitting sets using outer Fitting sets pairs. We shall give some examples of outer Fitting sets pairs which are local versions of the outer Fitting pairs constructed in [Pen88, Sections 4 and 5]. These local constructions provide further information and show that Fitting sets pairs are worth investigating. Our first example leads to a p-supersoluble Fitting set, p a prime, in every group. This Fitting set is dominant in the set of all p-constrained groups (see Definition 2.4.29). Example 7.4.20. Let G be a group and let J be a simple group. Suppose that nG is the largest natural number such that |J|nG divides |G|. Denote of J. If nG = 0, we agree by DJ (nG ) the direct product of nG copies that DJ (nG ) = 1. Let AJ (nG ) = Aut DJ (nG ) and OJ (nG ) = Out DJ (nG ) . It is known that 1. if J is non-abelian, then AJ (nG ) is isomorphic to the natural wreath product AJ (nG ) ∼ = Aut(J) nat Sym(nG )

and

OJ (nG ) ∼ = Out(J) nat Sym(nG ).

2. if J ∼ = Cp , for a prime p, then AJ (nG ) ∼ = GL(p, nG ). Also let DJ be the restricted direct product of countably infinitely many copies of J and let AJ = Aut0 (DJ ) be the group of all automorphisms of DJ with finite support Denote OJ the group of outer automorphisms of DJ with finite support. Let F and G be two Fitting classes such that G ⊆ F. 1. ([Pen88, Theorem II]) For any group G and any chief series Γ of G through GF and GG , let DJ (Γ, F/G) be the direct product of all the J-chief factors of Γ between GF and GG , taken in the order of occurrence in Γ . We consider this group as the subgroup of DJ consisting of the first direct components of DJ . The group G operates on every such DJ (Γ, GF /GG ) and by identical continuation also on DJ . This action defines a homomorphism J,F/G

dG

: G −→ AJ .

Then the pair (dJ,F/G , AJ ) is an outer E-Fitting pair. This is called the chief factor product Fitting pair . The construction is dependent on the inherent choices only within equivalence of outer Fitting pairs.


349

2. ([AJBBPR00, Ex. IV]) Let G be a group. Let EG denote the Fitting set composed of all subgroups of G. For each T ∈ EG , i.e. for each subgroup T of G, we consider a chief series ΓT of T through TF and TG . Let DJ (ΓT ) be the direct product of all the J-chief factors of T taken in the order of occurrence in ΓT . We consider this group as the subgroup of DJ (nG ) consisting of the first direct components of DJ (nG ). T acts by conjugacy on DJ (ΓT ) and in trivial way on the rest of components of DJ (nG ). This action defines a homomorphism J,F/G

: T −→ AJ (nG ). Then the pair dJ,F/G , AJ (nG ) is an outer EG -Fitting sets pair relative to G. This is called the chief factor product Fitting sets pair relative to G. The construction is dependent on the inherent choices only within equivalence of outer Fitting sets pairs.

dT

Remark 7.4.21. With the above notation, if F is a Fitting set of AJ , then F = Fd−1 is a Fitting class defined by the chief factor product Fitting pair by [Pen87, Satz 3.2]. Then TrF (G) is the Fitting set of G defined by the chief factor product Fitting sets pair relative to G (see Theorem 7.4.17). There exist Fitting sets associated with chief factor product Fitting sets pairs which cannot be obtained in this way. Let G be a group and p a prime dividing |G|. Following the notation the above example, we take J = Cp , the cyclic group of order p, F = E the class of all finite groups, and G = (1), the trivial class. Let nG be the natural number such that pnG is the order of a Sylow p-subgroup of G. Then DJ (nG ) is an nG and AJ (nG ) = GL(nG , p). Denote elementary abelian p-group of order p by d, GL(nG , p) the chief factor product Fitting sets pair relative to G of Cp ,E/(1) Example 7.4.20 .

(2), that is d = d Let F = U ≤ GL(nG , p) : U ≤ Z GL(nG , p) . Since Z GL(nG , p) is a normal subgroup of GL(nG , p), it is clear that F is a Fitting set of GL(nG , p). By Theorem 7.4.17 we have that FZ = Fd−1 is a Fitting set of G. It is proved in [AJBBPR00, Ex. VI]) that there exist groups G for which FZ is not the trace in G of any Fitting class. In particular, FZ is not the trace in G of the Fitting class obtained by the inverse image of a Fitting set of ACp through the chief factor product Fitting pair. We study the Fitting set FZ in a group G. We assume that nG = 0. For any subgroup B ≤ G, write pnB the order of a Sylow p-subgroup of B. If x ∈ B, then M (x) 0 , xdB = 0 InG −nB where M (x) ∈ GL(nB , p) is the matrix of the action of x on the p-chief factors of a fixed chief series of B. If B ∈ FZ , then xdB = λInG , for some non-zero scalar λ of GF(p). Hence, the p-chief factors of B are simple and all of them are B-isomorphic. In particular, Ker(dB ) = Op ,p (B). Moreover, B/ Ker(dB ) is a subgroup of Z GL(nG , p)

350


and then it is isomorphic to a cyclic group of order dividing p − 1. If B does not contain any Sylow p-subgroup of G, then B is p-nilpotent; that is, B = Ker(dB ). Note that all p-nilpotent subgroups of G are in FZ . If H is a subgroup of G, the order of a Sylow p-subgroup of H is denoted by |H|p . Lemma 7.4.22. Let H be a subgroup of G. Assume that H is a p-soluble group of p-length at most 1. Then: 1. HFZ is the unique FZ -maximal subgroup of H containing Op ,p (H); in particular HFZ is the unique FZ -injector of H. 2. If |H|p < pnG , then HFZ = Op ,p (H). 3. If |H|p = pnG , then HFZ is the set of all m ∈ H such that m has scalar action on the direct product of the p-chief factors of H in a chief series of H. Proof. 1. Let M be an FZ -subgroup of H containing O p ,p(H).We claim that M is normal in H, so that the conclusion is clear. Since the p-length of H is smaller than or equal to 1, then M/ Op ,p (H) is a p -group. Consequently the p-chief factors of H are completely reducible GF(p)M -modules. Hence the direct product of the p-chief factors of H in a chief series of H , viewed as a GF(p)M -module in the natural way, is GF(p) M-isomorphic to the direct product of the p-chief factors of M in a chief series of M . Since M ∈ FZ , then M has scalar action on the above mentioned direct product of the p-chief factors of H. Therefore [M, H] ≤ Op ,p (H) ≤ M . In particular M is normal in H. 2. If |H|p < pnG , it is clear that HFZ is p-nilpotent and then HFZ = Op ,p (H). 3. Assume now that |H|p = pnG . Denote by S the set of all m ∈ H such that m has scalar action on the direct product of the p-chief factors of H in a chief series of H. It is clear that S is a normal subgroup of H containing Op ,p (H). Note that the p-chief factors of H are completely reducible as GF(p)HFZ -modules and also as GF(p)S-modules because HFZ and S are normal subgroups of H. Moreover, since |H|p = pnG we can easily deduce that S ∈ FZ and also that S = HFZ . Recall that the class Ep Sp of all p-nilpotent groups is injective, and a group G possesses a unique conjugacy class of Ep Sp -injectors if and only if G is p-constrained (see Corollary 7.2.31 and Remark 7.2.32). Moreover, in this case, InjEp Sp (G) = {Op ,p (G)P : P ∈ Sylp (G)}, and the p-nilpotent injectors of G are the p-nilpotent maximal subgroups of G containing Op ,p (G). Lemma 7.4.23. Let H be a p-constrained subgroup of G such that |H|p = pnG . Suppose that M is an FZ -maximal subgroup of H containing Op ,p (H).


351

1. There exists a p-nilpotent injector I of H such that I = Op ,p (M ). 2. Moreover, M is the FZ -radical of NH (I) and is the set of all elements m ∈ NH (I) such that m has scalar action on the direct product of the p-chief factors of NH (I) in a chief series of NH (I). Proof. Suppose that |M |p < pnG . In this case since M ∈ FZ , we have that M is a p-nilpotent group and then M is contained in a p-nilpotent injector, X say, of H, because Op ,p (H) ≤ M . But clearly X ∈ FZ , which implies X = M . In particular M = Op (H)Hp for some Hp ∈ Sylp (H), which is a contradiction. Consequently there exists a Sylow p-subgroup Hp of H such that Op ,p (H)Hp ≤ Op ,p (M ). But I = Op ,p (H)Hp is a p-nilpotent injector of H, which implies that I = Op ,p (M ). Observe that I ≤ M ≤ NH (I) and I = Op ,p NH (I) . Since NH (I) is a psoluble group of p-length at most 1, the conclusion follows from Lemma 7.4.22. Theorem 7.4.24. Let H be a p-constrained subgroup of G. Then H has a unique conjugacy class of FZ -injectors. Moreover, the FZ -injectors of H are exactly the FZ -maximal subgroups of H containing Op ,p (H), or equivalently, the FZ -radical of H. Moreover, we have: 1. If |H|p < pnG , then the FZ -injectors of H are exactly the p-nilpotent injectors of H. 2. If |H|p = pnG , then the set of FZ -injectors of H is exactly

InjFZ (G) = NH (I) F : I ∈ InjEp Sp (H) Z

In particular, the FZ -injectors of H are the subgroups composed of all elements m ∈ NH (I) such that m has scalar action on the direct product of the p-chief factors of NH (I) in a chief series of NH (I), where I is a p-nilpotent injector of H. Proof. Note that if |H|p < pnG , then the FZ -subgroups of H are exactly the p-nilpotent subgroups. On the other hand, if |H|p = pnG , it is clear by of H containing Op ,p (H) Lemma 7.4.23 that

the set of FZ -maximal subgroups is exactly the set NH (I) FZ : I ∈ InjEp Sp (H) which is a conjugacy class of subgroups of H. Since Op ,p (H) ≤ HFZ , we deduce that this set also coincides with the set of all FZ -maximal subgroups of H containing HFZ . Therefore the Fitting set FZ is dominant in the set X = {H ≤ G : H is p-constrained}. J. Pense ([Pen87, 4.14]) presented a type of Fitting classes, constructed by means of Fitting pairs, with respect to which every finite group has a unique conjugacy class of injectors. An improved version of this result is presented in [Pen90c]. We shall show in the sequel that Pense’s result is actually a particular case of a more general one.

352


Definition 7.4.25. Let G be a group and let S be a perfect comonolithic group whose head is isomorphic to a simple group J. Let L be the subgroup generated by all subnormal subgroups of G isomorphic to S L = T : T is subnormal in G and T ∼ = S and let

M = Cosoc(T ) : T is subnormal in G and T ∼ = S

(which is a normal subgroup of L by Theorem 2.2.19). The factor group L/M is called the S-head-section of G. By Theorem 2.2.19, L = T1 · · · Tm , where all Ti are normal subgroups of L and Ti ∼ = S. Note that if S a perfect comonolithic subnormal subgroup of a group which is the join of two subnormal subgroups S1 and S2 , then either S is contained in S1 or S is contained in S2 ([Wie39]). This implies that Ti ∩ M = Cosoc(Ti ) and then Ti M/M ∼ = J. Hence L/M is a group in the Fitting class Fit(J) generated by J, i.e. L/M is isomorphic to a direct product of copies of J, by Example 2.2.3 (1). Example 7.4.26. Let G be a group and let S be a perfect comonolithic group whose head is isomorphic to a simple group J. Let DJ (nG ), AJ (nG ), F, and G be as in Example 7.4.20. 1. ([Pen88, Theorem III]) For any group G fix an embedding of the S- head-section of GF /GG as the first components of DJ . Then G operates on DJ via this embedding, and therefore we have a homomorphism S,F/G

HG

: G −→ AJ .

The pair (H S,F/G , AJ ) is an outer E-Fitting pair. 2. ([AJBBPR00, Ex. V] ) For each subgroup T of the group G, we fix an embedding of the S-headsection of TF /TG as the first components of DJ (nG ). Then T operates on DJ (nG ) via this embedding, and therefore we have a homomorphism S,F/G

hT

: T −→ AJ (nG ).

Denote by EG the Fitting set of all subgroups of G. Thus the pair S,F/G , AJ (nG ) h is an outer EG -Fitting sets pair relative to G. Let S be a perfect comonolithic group whose head is isomorphic to a nonabelian simple group J. Consider the Fitting classes F = E, the class of all finite groups, and G = (1), the trivial class. Write H S,F/G = H S . Then it appears the outer E-Fitting pair, (H S , AJ ) say. Consider the projection from


353

˜ S , OJ ) be the induced outer Fitting pair from the pair AJ to OJ and let (H S (H , AJ ). Analogously, the projection from AJ (nG ) onto OJ (nG ) = if we consider S ˜ , OJ (nG ) be the induced outer Fitting sets pair Out DJ (nG ) and let h relative to G from the pair hS , AJ (nG ) . Theorem 7.4.27. With the notation introduced above, let F be a Fitting set of OJ (nG ) all whose elements are subgroups of the base groupof OJ (nG ) and ˜ S , OJ (nG ) . ˜ S )−1 be the Fitting set corresponding to the pair h let T = F(h If Out(J) is soluble, then each subgroup of G has exactly a conjugacy class of T -injectors. Proof. Note that for every subgroup B of OJ (nG ), the F-injectors of B ∩ Out (J) , where Out(J) is the base group of OJ (nG ), are exactly the F-injectors of B. Therefore each subgroup of OJ (nG) possesses a single conjugacy class of F-injectors by Theorem 2.4.26. Then it is enough to show that the ˜ S , OJ (nG ) satisfies the property (7.2) of Theorem 7.4.19. pair h ˜ S . Let ν : V −→ U be a G-embedding between subgroups Write f = h U and V of G such that UT ≤ V ν . We consider LU /MU and LV ν /MV ν the S-head-section of U and V ν respectively. It is clear that LU /MU is the S-headfL section of LU and so LU U = 1 ∈ F. Then LU ∈ Ff −1 = T and LU ≤ UT and so also LU ≤ V ν . This implies that LU ≤ LV ν . Now suppose that there exists a subnormal subgroup X of V ν such that X ∼ = S and X is not subnormal in U . Then, for any subnormal subgroup T of U such that T ∼ = S, we have that X and T are normal in XT , by Theorem 2.2.19, and then [X, ). T ] ≤ Cosoc(T Hence [X, LU ] ≤ MU . Therefore X ≤ CV ν LU /MU ≤ CU LU /MU . Since CU (LU /MU ) ≤ Ker(fU ) ≤ UT ≤ V ν , it follows that X is subnormal in CU (LU /MU ) and also is in U , contrary to supposition. Therefore the S-head-section of V ν coincides with the S-head-section of U and then it is conjugate to the S-section of V . By construction of the Fitting sets pair, it follows that there exists η ∈ Inn OJ (nG ) , such that νfU = fV η. Now we deduce the aforesaid result of J. Pense. Theorem 7.4.28 ([Pen90c]). Let S be a perfect comonolithic group with ˜ S , OJ ). Let F be a Fitting set in head J. Consider the outer Fitting pair (H S −1 ˜ the base group of OJ and let F = F(H ) be the corresponding Fitting class. If the outer automorphism group of J is soluble, then every finite group has exactly a conjugacy class of F-injectors. (n copies)

" $! # Proof. First of all, note that AJ = limn→∞ Aut(J × · · · × J) and so AJ is the (restricted, natural) wreath product limn→∞ Aut(J) nat Sn with base group Aut(J) . Then OJ is AJ / Inn(J) with base group Out(J) .

354


For each group G we consider OJ (nG ) as a subgroup of OJ. With respect S ˜ , OJ (nG ) and for each to the outer EG -Fitting sets pair relative to G, h subgroup T of G, we have ˜S

˜S

for every t ∈ T . (t)hT = (t)HT ∈ OJ (nG ) ≤ OJ S −1 ˜ ) . Applying TheTherefore it follows that TrG (F) = TrOJ (nG ) (F) (H orem 7.4.27, G has a conjugacy class of F-injectors. Recall finally Schreier’s conjecture, whose validity has been proved using the classification of finite simple groups, which states that the group Out(J), of all outer automorphisms of a non-abelian simple group J, is always soluble (see [KS04, page 151]).

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¨ H. Wielandt. Uber die Existenz von Normalteilern in endlichen Gruppen. Math. Nachr., 18:274–280, 1958. H. Wielandt. Kriterien f¨ ur Subnormalit¨ at in endlichen Gruppen. Math. Z., 138:199–203, 1974. H. Wielandt. Mathematische Werke-Mathematical Works, volume 1: Group Theory. Walter de Gruyter, 1994. Edited by B. Huppert and H. Schneider. H. Wielandt. Subnormale Untergruppen endlicher Gruppen. In Mathematical Works [Wie94a], pages 413–479. Vorlesung Univ. T¨ ubingen, 1971. T. Yen. On F-normalizers. Proc. Amer. Math. Soc., 26:49–56, 1970.

List of symbols

(G1 , . . . , Gn ) 87 (S) 87 (d, A) 344 A/B C/D, C/D A/B 53 A \ B set/class difference of A and B A × B direct product of the (sub)groups A and B Aϕ , Aφ, φ(A) image of A by φ A(n) nth term of the derived series of A C ∗ /RX 64 Cm cyclic group of order m Eλ 12 E E K 210 E T H 230 Eg 11 F canonical definition of the local formation F F (p) 131 F1 ∼ =G F2 42 G derived subgroup of the group G G∼ = H G is isomorphic to H GF 95 GF 111 GΦ,p 5 GX (p) 154 GF 114 Gωd 163 H F-sn G 236 H ≤ G H is a subgroup of G H ≤ G H is not a subgroup of G H G H is a normal subgroup of G H S 352

367

H x x−1 Hx, conjugate of H by x HG 2 KG 120 RX 64 Sn 8 U K-F-sn G 236 V ⊕ W direct sum of the modules V and W V ⊗ W a tensor space over K of the K-vector spaces V and W V ⊗A W a tensor product of the right A-module V and the left A-module W VA the G-module V restricted to A Vj1 ⊗ · · · ⊗ Vjn 120 W ⊗ K 120 W G induced module of the A-module W from A up to G W K 120 X =U Y 239 X =¨G Y 239 X σ Y 239 X σ ∞ Y 239 X H 8 X ϕ H 8 X 8 Xi : i ∈ I f 207 [A/B C/D] 54 [H, K] commutator subgroup of H and K [H/K] ∗ G 45 [N ]H semidirect product of the H-group N with H

List of symbols

[N ]α H semidirect product of the H-group N with H via the action α [a, b] commutator of a and b, a−1 b−1 ab AGL(n, p) 5 AJ (nG ) 348 Ap (G) 5 A 87 Alt(n) 3 Aut(G) group of automorphisms of the group G AutG (H/K) 41 BF 281 BLF(f ) 97 B(F) 288 CX p (G) 128 char X 88 CoreG (H) 2 Cosoc(G) 98 CovH d (G) 226 CovH (G) 101 CritS (Z) 267 DJ (ΓT ) 349 DJ (nG ) 348 ∆K (G) 123 E 87 E(π) 231 E(n) 91 E(p) 135 E(G) 98 E(q|pe ) 333 EF (G) 113 F ◦ G 95 F × G 95 F (G) 78 F(G) Fitting subgroup of the group G F∗ (G) 97 Fit Z 110 GF(q) finite field of q elements GL(n, q) general linear group of dimension n over GF(q) Hallπ (G) 99 Hom(U, A) set of homomorphisms between U and A HomKG (V, W ) set of KGhomomorphisms from V to W I(G) 258 IK (E) 223

InjF (G) 114 InjF (G) 114 Inn(G) group of inner automorphisms of the group G Irr(M ) 293 J 87 J(KG) 5 KG (X) 239 Kn (G) 93 K(X) 324 Ker(d, A) 344 Ker(f ) kernel of the homomorphism f Ker(x on U ) kernel of the action of the element x ∈ G on the KG-module U LF(f ) 97 LFX (f ) 126 L(X) 142 L(G) 197 LH (G) 197 Locksec(X) 112 M(G) 120 Mp 295 Max(G) 52 Max(G)n N 197 Max∗ (G) 53 Max∗ (G)aH 192 Max∗ (G)aN 197 MaxH d (G) 225 MaxH (G) 101 ModF (U ) 293 N, set of all natural numbers N 87 Nc 91 NorH (G) 171 NG (Σ) 183 NG X(Σ)) 85

)

368

Op (G) smallest normal subgroup of G of p -index p O (G) smallest normal subgroup of G of p-index OJ (nG ) 348 OY (G) 126 Oπ (G) largest normal π-subgroup of G Op ,p (G) largest p-nilpotent normal subgroup of the group G Op (G) largest normal p-subgroup of G

List of symbols

U 87 W(X, F) 288 W(G, X) 192 X 126 XY 88 X · Y 111 X Y 111 X ⊆ Y 87 X∗ 111 Xn 88 Xb 112 X∗ 112, 148 Y 126 YG 158 Yp 126 Z set of all integer numbers Zm set of all integer numbers modulo m Zf 260 f¯ 135 bform(Y) 128 i∈I Vi direct sum of the modules Vi Xi∈I Ai direct product of the groups Ai , i ∈ I Xi∈I Fi 96 ¯ b(X) 112 b(H) (for a class of groups) 102 b(X) (for a Fitting class) 112 b3 (F) 295 bn (F) 306 bp (X) 112 bX (F) 134 bm (G) 113 C∗G (H/K) 41 CG (H) centraliser of H in G Z(G) centre of the group G ZF (G) 179 ZF G mod Φ(G)) the subgroup S of G with S/Φ(G) = ZF G/Φ(G)) χf 263 A B, B A 90 C 89 C = Aλ : λ ∈ Λ 90 2 C 89 C1 ≤ C2 89 D0 (F, S) 92 D0 X 89 EΦ X 89 K X 88

)

)

PGL(n, q) projective general linear group of dimension n over GF(q) P 87 P1 87 P1 , P2 , P3 87 P2 87 P3 87 PSL(n, q) projective special linear group of dimension n over GF(q) Para(G) 200 ParaX (G) 199 Φ(G mod N ) the subgroup S of G with S/N = Φ(G/N ) Φ(G) Frattini subgroup of G ΦX (G) 52 Pref XL (G) 193 Pref X (G) 192 P 87 Pro(G) 200 ProX (G) 199 ProjH d (G) 225 ProjH (G) 101 Q 97 R 140 RX 134 R(G) 197 RH (G) 197 Rad T radical of the module T SL(n, q) special linear group of dimension n over GF(q) Sec(X, Y) 111 ΣN/N 184 Soc(G) socle of G Soc G mod Φ(G)) the subgroup S of G with S/Φ(G) = Soc G/Φ(G)) S 87 S(d) 91 Sπ 100 Supp(f ) 296 Sylp (G) set of all Sylow p-subgroups of the group G T(G) 281 TG (H; F) 245 T(1, M) 120, 292, 297 T(1, r, P, X1 , X2 ) 297 T M(K, P, X )) 123 TK (G) 120, 292 Tπ 233 TrF (G) 114

369

)

)

)

370 N0 X

List of symbols

89 102 Q X 89 R0 X 89 S(X) 205 S X 89 Sn (G) 206 Sn X 89 Sp (G) 206 SX (G) 142 d(G) 205 det(x on U ) determinant of the element x ∈ G acting on the KG-module U diag(n1 , . . . , nr ) diagonal matrix with elements n1 , . . . , nr in its diagonal ∅ 87 η(G, M ) 191 e ≤ f 206 r (G) 207 r (G) 207 r(G) 207 formX (Y) 128 f 128 t (G) 207 t (G) 207 t(G) 207 h(Y) (for a Fitting class) 112 h(Y) (for a class of groups) 102 hπ (G) 232 id, idK identity automorphism (of K) Im(f ) image of the homomorphism f κ(G) 330 X G normal closure of X in G A, B 90 A 90 a1 , a2 , . . . subgroup generated by the elements a1 , a2 , . . . lform(Y) 128 |G : H| index of the subgroup H in the group G |X| cardinal/order of X Xπ 192 A 344 AJ 348 DJ 348 DJ (Γ, F/G) 348 G 344 S(Σ) 83 PQX

X 40 X/N 52 Xϕ 52 Xg 74 XM 80 Xp 192 XaH 191 Xn H 191 Y(F, N, T ) 75 −1 Yϕ 74 Ei 63 FH 114 Hd−1 347 P 87 P(H) 108 R(Z, F) 303 W(F) 303 M(K, P, X ) 121 C(M) 123 M(K, P, X ) 121 max S maximum of the set S min S minimum of the set S N(E) 113 NG (H) normaliser of H in G π(G) 88 π(X) 88 πS 9 j∈S Xj product of the subgroups Xj with j ∈ S ψ B 16 ψ G 26 SnF (G) 236 SnK-F (G) 236 SG (X) 239 SG (X; F) 239 SG (X; K-F) 239 a∗ 26 aϕ , aφ, φ(a) image of a by φ J,F/G 348 dJ f (H) 106 f |K restriction of f to K f ∗ 130 f1 (H) 106 f1 ≤ f2 128 sV direct sum of s copies of the module V v ⊗ w a generator of a tensor product V ⊗A W

Index of authors

Cameron, P. J. 24 Carocca, A. 204 Carter, R. W. 40, 99, 100, 114, 169, 171, 188, 268 Chambers, G. A. 170 Clifford, A. H. 286 Cossey, J. VIII, 118–121, 123, 286, 300, 329, 345

Alejandre, M. J 281, 282 Anderson, W. 114, 116, 340 Arroyo-Jord´ a, M. 264, 265 Arroyo-Jord´ a, P. 346, 349, 352 Aschbacher, M. 22, 24 Baer, R. VIII, 1, 3, 4, 89, 96, 97, 119, 125, 127, 133, 144, 148, 155, 161, 250, 296 Ballester-Bolinches, A. IX, 93, 119, 132, 133, 143, 146–148, 154, 157, 160, 161, 163–166, 170, 171, 189–192, 198, 244, 245, 247, 248, 250, 258, 260, 264, 265, 268, 269, 272–277, 279, 281, 282, 284, 286, 301, 306, 307, 346, 349, 352 Barnes, D. W. 40, 92, 93 Bartels, D. 239 Bechtell, H. VIII, 197 Beidleman, J. C. 171 Bender, H. 97, 328 Berger, T. R. 123, 329 Berkovich, Y. 281 Birkhoff, G. 53 Blackburn, N. 97, 98, 297, 328 Blessenohl, D. 118, 315, 344 Bolado-Caballero, A. 118 Brandis, A. 49, 170, 193 Brewster, B. 171 Bryant, R. M. 93 Bryce, R. A. 93, 118, 119, 329, 345

Dade, E. C. 315 Dark, R. 329, 330, 340, 342 Dark, R. S. 110 De Cervantes Saavedra, M. VII Deskins, W. E. 191 Doerk, K. VII, VIII, 1, 40, 78, 84, 87, 90, 91, 95, 97, 100, 101, 103, 104, 106, 111, 114–116, 118–120, 123, 128–130, 133, 140, 141, 146, 153, 157, 161, 166, 171, 173–177, 179, 181, 182, 188, 208, 235, 244, 247, 248, 250, 254, 258, 264, 266–268, 272, 284, 292, 315, 319, 329, 332, 333, 340, 343, 345 Erickson, R. P. 100, 170 Esteban-Romero, R. VIII, 132, 133, 146–148, 154, 161, 164–166, 268 Ezquerro, L. M. IX, 119, 181, 190–192, 198, 204, 284, 286, 307 Feit, W. 219 Feldman, A. VIII, 340, 342 Fischer, B. 40, 109, 110, 114, 116, 268, 309, 315, 337, 338

Calvo, C. 132, 133, 146–148, 154, 161, 164–166

371

372

Index of authors

Fitting, H. VIII, 78, 90, 97, 99, 109–114, 116–121, 123, 126, 134, 135, 153, 181, 213, 248, 250, 252, 254, 257, 259, 260, 264, 265, 282, 284–288, 292, 293, 295–297, 300, 301, 303, 306, 307, 309, 314–333, 335–341, 344–349, 351–353 F¨ orster, P. 32, 42, 46, 68, 69, 100, 101, 103, 104, 109, 125, 126, 134, 138, 140, 141, 144, 147, 153, 170, 206, 208, 315, 324, 327 Frattini, G. 5, 40, 41, 44, 52, 54, 56–60, 62, 66, 70, 92, 96, 119, 125, 139, 144, 147, 148, 153, 169, 174, 175, 183, 194, 196–199, 273 Frobenius, G. 217, 221, 277, 281 Gajendragadkar, D. 120 ´ 5 Galois, E. Gasch¨ utz, W. 344 Gasch¨ utz, W. VIII, 62, 69, 73, 95–97, 100, 101, 104, 110, 111, 114, 116, 118, 125, 144, 148, 152, 169, 170, 192, 309, 319, 337, 344 Gillam, J. D. 181, 182 Griess, R. L. 175, 273 Gross, F. 1, 10, 13, 18, 21, 309 Haberl, K. L. 123, 300 Hall, P. 73, 83, 84, 89, 98–100, 118, 153, 163, 169–172, 182–187, 195, 218, 231, 235, 250, 257, 268, 328, 339, 340 Harris, M. E. 115 Hartley, B. 93, 110, 114, 116, 309 Hauck, P. 332 Hawkes, T. O. VII, VIII, 1, 40, 50, 78, 84, 87, 90, 91, 95, 97, 100, 103, 104, 106, 111, 114–116, 118–120, 123, 128–130, 133, 140, 141, 146, 153, 157, 161, 166, 169–171, 173–177, 179, 181, 182, 188, 202, 208, 235, 247, 250, 254, 266–268, 272, 284, 292, 300, 315, 319, 329, 332, 333, 340, 343, 345 Hawthorn, I. 114, 116 Heineken, H. 123, 300 Higman, G. 188

H¨ older, O. L. VIII, 40, 41, 52, 53, 61, 73 Huppert, B. 2, 5, 7, 93, 96–98, 166, 217, 219, 221, 231, 268, 277, 297, 309, 328 Iranzo, M. J. VIII, 315–318, 327, 329 Isaacs, I. M. 120 Itˆ o, N. 277 Jacobson, N. 5 Jiménez-Seral, P. VIII, 25 Jordan, C. VIII, 40, 41, 52, 53, 61, 73 Kamornikov, S. F. 248, 253, 257, 260, 262–264, 269, 272, 286, 296, 301, 306, 307 Kanes, C. 119–121, 123, 300 Kattwinkel, U. 106 Kegel, O. H. 92, 93, 236, 248, 283, 284, 301 Khukhro, E. I. 157, 158, 248, 265, 268, 284 Klimowicz, A. A. 170 Kov´ acs, L. G. 1, 10, 13, 18, 21, 25, 29, 31, 309 Kurzweil, H. 1, 27, 97, 98, 170, 193–195, 354 Lacasa-Esteban, C. VIII Lafuente, J. VIII, 6, 34, 40–42, 53, 61, 62, 103, 106, 315, 317, 318 Laue, H. 315 Lausch, H. 345 Lennox, J. C. 235 Liebeck, M. W. 25 Lizasoain, I. VIII Lockett, P. 111, 112, 116, 257, 319, 320, 330, 339, 344 Lubeseder, U. VIII, 96, 97, 125, 144, 148, 152 Maier, R. 204 Makan, A. 170 Mann, A. 84, 169, 171, 176, 315, 325 Mart´ınez-Pastor, A. 260, 284, 329 Mart´ınez-Verduch, J. R. 118 Mazurov, V. D. 157, 158, 248, 265, 268, 284 Medina, J. 329 Menth, M. 329–331, 339

Index of authors O’Nan, M. 1, 24, 25, 28, 34, 39 Ormerod, E. O. 123 Pedraza, T. VIII Pedraza-Aguilera, M. C. VIII, 268, 284, 301, 306, 307 Pense, J. 345, 346, 348, 351–353 Pérez-Monasor, F. VIII, 315–318, 327, 329 Pérez-Ramos, M. D. 93, 157, 160, 244, 245, 247, 248, 250, 258, 260, 264, 265, 268, 269, 272–277, 279, 301, 306, 307, 346, 349, 352 Plotkin, B. I. 89 Praeger, C. 25 Robinson, D. J. S.

268

Salomon, E. VIII, 133, 134, 138, 140, 153–155, 166, 205, 206, 208, 220, 309, 329 Saxl, J. 25 Schaller, K.-U. 106 Schmid, P. VIII, 96, 97, 100, 125, 133, 144, 148, 152, 170, 175, 180, 189, 273 Schmidt, O. J. 268, 269, 272–274, 281, 282 Schreier, O. 27, 354 Schubert, H. 100 Schunck, H. VIII, 99–110, 112, 118, 123, 169–171, 177, 192, 193, 197, 202, 205, 206, 224, 226–231 Schur, I. 156, 231 Scott, L. 1, 22, 24, 25, 28, 34, 39

373

Sel’kin, M. V. 264 Semenchuk, V. N. 248, 253, 257, 265, 268 Shemetkov, L. A. 96, 97, 133, 152, 153, 157, 158, 160–165, 171, 189, 248, 268, 269, 286, 296, 315 Skiba, A. N. 157, 158, 161, 163, 164, 269 Soler-Escriv` a, X. 204 Stellmacher, B. 1, 27, 97, 98, 354 Stonehewer, S. E. 235 Suzuki, M. 156, 309 Sylow, P. L. M. 3, 98–100, 166, 170, 176, 187, 189, 206, 219, 237, 247, 273, 301, 330, 346, 349, 350 Thompson, J. G. 219 Tomkinson, M. J. 170, 203 Torres, M. 315, 316, 327, 329 Traustason, G. 330 Vasil’ev, A. F. 248, 253, 257, 260, 262–264, 268, 283, 284 Vedernikov, V. A. 157 Vorob’ev, N. T. 157 Wieladt, H. 170 Wielandt, H. VII, VIII, 98, 170, 215, 217, 221, 235, 243, 244, 247, 268, 285–288, 295–297, 300, 301, 306 Yen, T.

183

Zassenhaus, H.

231

Index

ACAP 194, 195 action 2, 8, 12, 14, 16, 17, 23, 25–28, 34, 39, 41, 92, 93, 96, 99, 224, 286, 348, 349, see also representation induced 17, 26 regular 31 scalar 350, 351 transitive 310, 311 automorphism 16, 22, 332–335, 348 group of see group, automorphism inner 7, 25, 41, 97, 223, 224, 230, 309, 345–347 outer 26, 309, 310, 345, 353, 354 power 330 Baer function 97, 119, 127, 296 Baer-local formation 96, 97, 125, 127, 133, 153, 155, 161, 269, 272, see also solubly saturated formation defined by f 97 block 1, 23, 25, 27, 28, 34, 39, 96 non-trivial 2 trivial 1 U -invariant 28 boundary 101, 102, 103, 104, 106, 107, 112, 113, 155, 174, 180, 226–231, 266, 275, 277, 286, 288, 293, 295, 303, 304, 306 X-wide 134, 135, 136, 138–140 XG-free 155, 156 centre 7, 17, 22, 98, 113, 140, 146, 153, 156, 174, 309, 330, 349

375

chain of classes of groups 329 of critical subgroups 171, 172, 180 of crucial critical subgroups 176 of subgroups 177, 207, 235, 236 character π-factorable 120 π-special 120 characteristic 88, 125–132, 134–138, 141, 142, 144, 146–149, 152–157, 159, 160, 164–166, 169, 235, 244, 250, 257–260, 264, 265, 267, 282, 284, 307, 308 of a field 120, 121, 123, 291 chief factor 40, 41, 42, 44, 45, 49, 50, 52–55, 61, 62, 66, 70, 74, 78–80, 96, 97, 126, 128, 134, 135, 139, 140, 147, 200, 342, 348–351 abelian 45, 74, 75, 120, 191 avoided by a subgroup 79, 99, 169, 170, 173, 181, 182, 186, 194, 196 central 99 complemented 44, 45, 49, 62, 74, 75, 92, 123, 169, 191, 200 composition type 152 covered by a subgroup 79, 99, 169, 170, 173–176, 178, 181, 182, 186, 194, 196, 208, 213, 214, 342 Frattini 40, 41, 44, 92, 139, 169, 199 G-connected 42, 42, 46, 48, 57, 62–66, 70, 71 G-isomorphic 40, 41, 42, 57, 62, 63, 65, 66, 74, 75, 126, 140, 291 H-central 106

376

Index

H-central 106, 108, 173, 174, 176, 178, 181, 182, 186, 293 H-eccentric 106, 108, 173–175, 181, 182, 186, 189, 293 supplemented 44, 45, 48, 52, 53, 63–65, 74, 78, 174, 176, 179 X-complemented 52, 60, 65, 194–196, 198–201 X-Frattini 52, 54, 56–60, 62, 66, 70, 194, 196 X-related 57 X-supplemented 52, 54–62, 64–66, 70, 71, 73, 75, 191, 193, 200, 201 chief series 40, 53, 54, 58, 64–66, 69–71, 73, 126, 178, 181, 348–351 class nilpotency 91 class of groups VII, VIII, 1, 87, 88–91, 96–98, 100–102, 106–112, 125, 126, 128, 129, 132–135, 138, 139, 142, 144, 147, 148, 152, 153, 156–158, 161, 163–166, 174, 191, 200, 205, 225, 231, 233, 252, 260, 264, 265, 267–269, 272, 281, 285, 286, 288, 294, 295, 300, 303, 306, 307, 314, 315, 318, 324, 338 closed for a closure operation 89, 90 d-projective 226, 227 D0 -closed 103 extreme 268 Fitting see Fitting class injective 115 Lockett see Lockett class power 88 product see product, class; product, Fitting; product, formation projective 100, 101, 103, 110, 170 Q-closed 91, 126, 140, 337, 338 R0 -closed 91, 103, 111, 338 residually closed 90 saturated 90, 91, 95, 101, 337 Schunck see Schunck class Sn -closed see class of groups, subnormal subgroup-closed subgroup-closed 90, 95, 96, 111, 118, 119, 142, 144, 263, 268, 337 subnormal subgroup-closed 90, 110 subnormally independent 112 trivial 349, 352

closure F-subnormal 239 N-subnormal 244 normal 239, 243 subnormal 239, 243 closure operation 89, 90, 91, 103, 109, 110, 112, 141, 142 generated by a set of operations 90 idempotent 90 complement 5, 38, 44, 45–50, 52, 60, 62, 70, 71, 73–75, 169, 170, 187–189, 210, 220–222, 224, 225, 230, 231, 309, 310 component 23, 34, 96, 98, 113, 315, 324 for a Fitting class 113, 315, 316 composition factor 40, 65, 88, 97, 120, 123, 125, 126, 163, 164, 211, 214, 219, 269, 291–293, 328 composition formation 97, 152, 153, see also Baer-local formation composition series 40, 62, 65, 235 conjecture Schreier 27, 354 conjugacy class 14, 18, 25, 98–100, 110, 116–118, 169, 170, 175, 179, 180, 182, 183, 187–189, 191, 194, 196, 228, 230–232, 241, 315, 323, 325, 326, 328, 347, 350, 351, 353 characteristic 169, 315 of complements 45, 46 of maximal subgroups 3, 5, 18, 37 conjugation 16, 23, 25, 26, 32, 41, 93, 99, 316, 330, 341 core 2, 17, 25, 26, 44–64, 70, 71, 73, 74, 80, 103, 108, 176, 179, 182, 191, 197, 198, 225, 236, 303 core-relation 73, 84 cosocle 98, 112 cover-avoidance property 170, 173, 175, 179, 181, 182, 191, 194, 202 covering subgroup see subgroup, covering (Cp )-local formation 133, 157, see also (Cp )-saturated formation (Cp )-saturated formation 150, 152, 162, see (Cp )-local formation crown VIII, 62, 63, 65–67, 73, 75, 103, 169, 170 CY -satellite 152

Index d-Schunck class 226, 227–229, 231 Duality Principle 53 element conjugate 220, 250 embedding 344, 352 normal 344, 345 Eπ -projector 231 E(π)-projector 233 E(π)-t-projector 231 epimorphism 12, 74, 75, 89, 93, 99, 128, 310, 312, 319 extension 11, 330, 335 field 120, 292, 295 Frattini see Frattini, extension induced 1, 10, 13, 15–18, 21, 26, 309–312 non-split 21, 96, 225, 309, 312 pull-back 11 split 18, 188, 189, 200, 311 F-Fitting class 265 F-hypercentre 178, 179, 182, 189, 197 F-injector 114, see also injector f-join 207 F-subgroup 114 factorisation solubly saturated 154 Fischer class 337, 338 Fischer F-subgroup see subgroup, Fischer Fitting class VIII, 90, 97, 99, 109, 110, 111–114, 118, 126, 248, 259, 260, 264, 265, 300, 309, 315–327, 329, 330, 332–335, 337–340, 344–346, 348, 349, 351–353 dominant 118, 257 formation see Fitting formation generated by a class of groups 110 Hall-closed 339 Hall-π-closed 339 injective 309, 315, 317, 322–325, 327–329, 350 metanilpotent 329, 331 non-injective VIII, 110, 309, 314, 323 normal 118, 252, 254, 321–323, 344, 345 repellent 332 soluble 332

377

subgroup-closed 111 supersoluble 329–332, 335–339 Fitting family of modules 119, 120, 123, 292, 293, 295, 297 Fitting formation 91, 110, 118, 119, 121, 123, 134, 135, 252, 284–288, 292, 295, 297, 300, 314 defined by a Fitting family of modules 119, 120, 121, 123, 292, 293, 295, 297 extensible 328 non-saturated 119 saturated 118, 119, 250, 254, 282, 301, 306, 307, 328, see also saturated formation soluble 119, 296 solubly saturated 119, 293, 295, see also solubly saturated formation subgroup-closed 118, 119, 296, 301, 303, 306, 307 Fitting pair 344, 345, 346, 351 chief factor product 348 kernel 344 outer 339, 345, 346, 348, 352, 353 equivalent 348 induced 347, 353 Fitting pairs 346 outer 348 Fitting set 114, 117, 339–341, 345–347, 349, 352, 353 dominant 348 injective 347, 348 p-supersoluble 348 Fitting sets pair 345, 346–348 chief factor product 349 outer 345, 346–349, 352 equivalent 346 induced 353 formation 90, 91, 92–97, 100, 101, 105–111, 118, 123, 125, 127, 128, 132, 134, 137, 139, 140, 142, 148, 149, 152–154, 156–158, 160–166, 171, 177, 202, 235, 237, 239, 247, 248, 253, 260, 262, 263, 265, 267–269,272, 277, 281, 283, 284, 286, 301, 302 Baer-local see Baer-local formation

378

Index

closed under taking triple factorisations see formation, with the Kegel property composition see composition formation (Cp )-local see (Cp )-local formation (Cp )-saturated see (Cp )-saturated formation Fitting see Fitting formation function see formation function generated by a class of groups 91 K-lattice 248, 250, 251, 262, 305 largest contained in a Schunck class 106, 107 lattice 247–254, 257–265, 268, 301–303, 305, 306 local see local formation maximal contained in a class of groups 138, 140 ω-local see ω-local formation ω-saturated see ω-saturated formation p-local see p-local formation p-saturated see p-saturated formation saturated 272, 273, see saturated formation Sn -closed see formation, subnormal subgroup-closed soluble 119, 260, 268, 272, 301, 306 solubly saturated see solubly saturated formation subgroup-closed 96, 141, 237, 244, 248, 252, 254, 260, 263–266, 268, 269, 272, 274, 277, 279, 281, 284, 286, 301, 303, 304, 306–308 subnormal subgroup-closed 95, 97, 254, 257, 260, 283, 284 totally nonsaturated 198 with the generalised Wielandt property for residuals 301, 303, 305–307 with the Kegel property 283, 284 with the Kegel-Wielandt property for residuals 301, 305 with the Shemetkov property 268, 269, 272–275, 277, 279, 281, 283, 284

with the Wielandt property for residuals 285, 286, 296, 297, 300 X-local see X-local formation X-saturated see X-saturated formation Xω -saturated see Xω -saturated formation formation function 97, 127, 140, 166, 190, 265–267, 274, 275, 277, 296, 307 Frattini argument 5, 183 extension 96, 148, 153, 163 maximal 5, 119, 174, 273 module 175 subgroup 96, 144–148, 153, 161, 163, 197 Frattini-like subgroup see also X-Frattini subgroup functor see subgroup functor G-embedding 345, 347 normal 345, 346 G-isomorphism 41, 74 G-set 1, 25, see also representation transitive 1, 2, see also representation, transitive Gasch¨ utz class 101, 104 group algebra 5, 146 group theoretical class see class of groups group theoretical property 87 group(s) π-soluble 171 p-soluble 5 abelian 93, 96, 119, 120, 125, 132, 134, 147, 150, 153, 157, 164, 165 class of 87, 91, 107, 111, 133, 260, 263 homocyclic 333, 335 affine general linear 5 almost simple 7, 25, 28, 29, 31, 34, 275 alternating 3, 21, 34, 39, 129, 133, 141, 142, 144, 146, 153, 161, 166, 172, 174–176, 179, 192, 211, 220, 225, 230, 236, 237, 244, 250, 253, 273, 274, 286, 309, 310, 312, 314, 346

Index automorphism 5, 7–9, 16, 22, 23, 25, 26, 29, 32, 41, 79, 96 induced by conjugation 41, 79, 97, 174 base 8, 33, 140, 210, 211, 224, 230, 353 class see class of groups comonolithic 98, 112, 113, 215, 217, 221, 315, 352, 353 critical for a class of groups 252, 254, 267–269, 272, 274, 281 cyclic 5, 31, 39, 70, 126, 140, 200, 219, 224, 252, 254, 268, 269, 272, 274, 295, 333, 350 class of 88 d-primitive 225, 226, 227 dihedral 31 elementary abelian 147, 150, 161 F-constrained 325, 326, 327 class of 324, 325 factorised 53, 75, 77, 218, 220–222, 265, 284, 307, 330 finite 96, 98, 119, 133, 152, 344, 346 class of 87, 349, 352 Frobenius 217, 219, 221 Frobenius-Wielandt 217, 221 general linear 5, 309, 335, 349 induced 17 infinite 344, 345 Lausch 345 meta-X 88 metabelian 336 metanilpotent 284 monolithic 91, 126, 128, 147, 252, see also group(s), primitive, monolithic N-constrained 315, 325 class of 325 nilpotent 91, 93, 95, 97, 99, 100, 111, 250, 260, 268, 272, 283, 307, 331, 336, 337 class of 87, 100, 110, 118, 169, 172, 235, 236, 239, 263–265, 281, 327 non-abelian 91, 92, 98, 134, 140, 146, 147, 156, 164 non-soluble 96, 101, 170, 171, 190, 194 ω-separable 165, 166 p -perfect

379

class of 88 p-constrained 327, 328, 348, 350, 351 p-decomposable class of 329 p-nilpotent 277, 350 class of 247, 265, 268, 277, 281, 307, 327, 350 p-quasinilpotent class of 329 p-soluble 119, 192, 193, 215, 218, 225, 231, 350 p-length 350 perfect 98, 113, 215, 221, 315, 317, 352, 353 permutation transitive 29 π-perfect class of 118 π-soluble 170 primitive VIII, 1, 2, 3–5, 8, 24, 29, 31, 44, 45, 62, 91, 100, 101, 103, 107, 108, 194, 208, 220, 284 associated with a chief factor 45 class of 1, 87 monolithic 4, 45, 215, 222, 225, 226, 254, 266 of type 1 4, 5, 6, 44, 180, 215, 220, 221, 224, 225 of type 2 4, 5–7, 10, 24, 25, 28–34, 37–39, 44, 60, 66, 104, 107, 119, 143, 144, 147, 200, 220, 224, 225, 254, 273 of type 3 4, 5–7, 42, 44, 46, 61, 63, 224, 228 soluble 5 with small maximal subgroups 32 projective general linear 309, 310 projective special linear 309, 310, 314 quasinilpotent 97, 198, 209, 213, 224, 226, 228 class of 97, 297, 315, 325–327, 329 quasisimple 98, 113 class of 324 quaternion 219, 346 r -primitive 225 r-soluble 120, 123, 292 S-constrained class of 325

380

Index

S-perfect class of 175 Schmidt 268, 269, 272–274, 281, 282 semisimple 208, 222 simple 7, 21, 34, 41, 96, 97, 125, 129, 132, 133, 138, 144, 147, 148, 152, 153, 156, 157, 161, 163–166, 170, 208, 213, 214, 223, 269, 272, 295, 310, 311, 314, 317, 348, 349, 352, 354 class of 87, 125, 126, 142, 144, 152 non-abelian 5, 7–9, 22–25, 28, 29, 31, 32, 41, 62, 91, 92, 98, 103, 106, 108, 110, 146, 147, 156, 164, 196, 198, 200, 208, 210–214, 218, 219, 224, 225, 230, 233, 253, 266, 275, 295 soluble VII, VIII, 5, 27, 40, 45, 62, 73, 74, 78, 83–85, 87, 91, 96–101, 109, 110, 112, 114–117, 119–121, 123, 133, 153, 166, 169–177, 179–188, 190–196, 198, 200–202, 204, 205, 208, 212, 214, 215, 220, 221, 224, 225, 230, 231, 233, 235, 236, 244, 245, 248, 250, 253, 257–260, 262–269, 272, 274, 281–284, 295, 297, 301, 303, 307, 309, 315, 323, 325, 328, 332, 337, 339–343, 345,353, 354 class of 87, 100, 110, 139, 268, 323, 325, 328, 345 soluble p-nilpotent class of 295 special linear 140, 146, 153, 166 strictly semisimple 208, 209, 211, 215, 222 supersoluble 329–331 class of 87, 91, 111, 268, 329 symmetric 9, 12, 14, 22, 25, 39, 58, 60, 68, 69, 172, 211, 220, 230, 237, 247, 250, 286, 301, 309–311, 346 t-primitive 225, 230, 231 triply factorised 283, 284 X-dense 134 Y-perfect class of 102, 106 GWP-formation see formation, with the generalised Wielandt property for residuals

H-covering subgroup 101 H-d-covering subgroup 226, 227–232 H-d-projector 225, 226–233 H-projector 101, see also projector H-t-projector 230 Hall subgroup see subgroup, Hall Hall system 73, 83, 84, 99, 169–172, 182–187, 195, 235 reducing into a subgroup 73, 84, 169, 172, 184, 186, 235 head 98, 352, 353 homomorph 90, 101–103, 111, 112, 226, 227, 260 homomorphism 10–12, 16, 18, 26, 32, 41, 52, 285, 310–312, 346–349, 352 trivial 346 inductivity 103, see also subgroup functor, inductive injector 99, 109, 110, 114, 115–118, 257–259, 265, 309, 314–317, 323, 325–328, 337, 339–343, 347, 350, 351, 353 normal 118 p-nilpotent 328, 350, 351 inneriser 41, 46, 97 involution 220, 230, 309–311 kernel 10, 12, 14, 17, 25, 26, 41, 74, 93, 119, 140, 146, 175, 254, 297, 310–312, 319, 344, 345, 349, 350 Frobenius-Wielandt 217 KW-formation see formation, with the Kegel-Wielandt property for residuals lattice 207 modular 53, 61, 62 layer 98, 113, 325 length derived 91 nilpotent 100, 250, 265, 268, 301 local definition see formation function canonical 130 maximal 133, 134, 190 local formation 96, 97, 125, 127, 129, 132–134, 139–141, 152, 157, 160, 161, 163, 331 local function see formation function Lockett class 112, 330, 339

Index metanilpotent 339 supersoluble 339 M-set 61, 62 module 62, 65, 120, 140, 175, 290, 293 absolutely irreducible 297 completely reducible 62, 65, 290 faithful 143, 144, 150, 166, 233, 247, 273 homogeneous 62, 65, 297 induced 17, 291 irreducible 120, 121, 123, 140, 143, 144, 146, 150, 166, 233, 247, 273, 286, 291–293, 297 P-factorable 120, 121 π-factorable 121 π-special 120, 121 socle 175 trivial 120 monomorphism 10–12, 310, 311, 344–346 normaliser associated with a saturated formation 169–171, 174–183, 187–190, 204 associated with a Schunck class VIII, 78, 169, 170, 171, 172–179, 203 associated with a system of maximal subgroups 172, 173, 175, 203, 235 crucial 176 ω-local formation 161, 163–167 satellite 163, 164 canonical 164 minimal 164 ω-saturated formation 161, 162–164,166 operation 89, 90 closure see closure operation idempotent 89 product 89 operator Wielandt see Wielandt operator orbit 1, 217 p∗ -group 329 class of 329 p∗ p-group class of 328, 329

381

p-chief factor 119, 120, 132, 150 p-group 237, 281, 284, 289–291, 332 class of 158, 162 p-local formation 163, see also p-saturated formation p-rank 119 p-saturated formation 161–163, 165, see also p-local formation p-subgroup strongly closed 114, 115, 116 partially saturated formation VIII, 125 partition 2, 22, 23, 25, 28, 39, 120, 121, 254, 257, 262, 264, 297 trivial 22 U -invariant 23, 25, 27, 28, 34 π-complement 308 π-group 244, 245, 250, 279 class of 118, 328 soluble 257–259 class of 253 preboundary 112, 113, 317 precrown 45, 45, 48, 52, 63 primitive group see group, primtive primitive pair 3, 29, 31, 32 of diagonal type 39 of type 2 32 with product action 31, 34, 35 with simple diagonal action 29, 31, 34 with twisted wreath product action 32, 35 product central 98, 330 class 88, 95, 111 direct 5, 7–10, 16, 21–23, 28, 31, 34, 39, 41, 62, 65, 90, 106, 108, 111, 112, 128, 140, 156, 175, 208, 212–214, 225, 254, 314, 317, 329, 349–352 restricted 348 Fitting 111, 324, 326, 327 formation 95, 111, 153, 158, 160, 162, 163 Gasch¨ utz see product, formation mutually permutable 329

382

Index

semidirect 6, 8, 17, 21, 44–46, 70, 93, 140, 143, 146, 166, 210, 221, 233, 247, 332, 333, 335 subdirect 90, 112, 214 totally permutable 329 wreath 8, 16, 22, 26, 29, 31, 140, 210 natural 348 regular 39, 140, 200, 210, 211, 224, 230, 331, 332 twisted 17, 18, 32, 35, 224 projector VIII, 99, 100, 101, 103–105, 109, 110, 118, 171, 176, 177, 179, 180, 182–187, 205, 206, 224, 229–231, 233, 235 property of groups see class of groups pull-back 11, 12, 309 r -subgroup see subgroup functor, r r -subgroup see subgroup functor, r r-subgroup see subgroup functor, r radical 111, 114, 118, 248, 250, 295, 351 E X-radical 147 E Y-radical 126 Jacobson 5 soluble 96, 125, 147, 148 representation faithful 1–3, 8 modular 96 permutation 1, 8, 140, 210 primitive 2, 3 transitive 1–3, 12, 25, 27, 99, 210 residual 94, 95, 171, 177, 179–188, 244, 245, 250, 269, 281, 285–288, 295–297, 300, 301, 306 nilpotent 281, 285, 287 soluble 201 right coset 1, 12, 14, 31, 41, 140, 210 right transversal 2, 11, 12, 17, 26, 310, 311 root of unity primitive 330 ˇ S-formation see formation, with the Shemetkov property satellite 96 saturated formation VIII, 95–97, 99–101, 106, 107, 109, 118, 119, 123, 137, 144, 148, 153, 166, 169–171, 174–180, 182, 183, 188, 190, 192,

197, 198, 203, 204, 235, 236, 239, 244, 248–254, 257, 259–261, 263–269, 272–275, 277, 279,281–284, 286, 296, 301, 303, 305–308, see also Fitting formation, saturated maximal contained in a class of groups 166 subgroup-closed 244, 277, 281 Schunck class VIII, 99, 100, 101, 102–110, 112, 118, 123, 169–171, 177, 192, 193, 197, 202, 205, 206, 224, 228–230 section 5, 34, 41, 53, 62, 79, 111, 211, 220 complemented 62 key 330 Lockett 111, 112, 319, 320, 322, 323, 344 S-head 352 simple 208, 213, 214 series central 330 set injective 115 JH-solid 52, 52, 53, 55, 57, 58, 60–66, 68–70, 73, 74, 194 solid VIII, 191, 192, 194, 195, 197–202 w-solid 190, 191, 192, 194–196, 202–204 weakly solid see set, w-solid socle 3, 5–8, 23–29, 31–35, 37–39, 45, 46, 61, 64, 65, 78, 98, 119, 126, 128, 134, 143, 144, 146, 147, 155, 156, 159, 163, 175, 191, 200, 215, 220–222, 224, 225, 231, 236, 252, 254, 273, 275, 277, 284, 288–290, 292, 303, 306 solubly saturated formation VIII, 96, 97, 119, 125, 153, 154, 165, 166, 264, 272, 292, 293, 295–297, see also Baer-local formation; Fitting formation, solubly saturated stabiliser 1, 26, 29, 85, 99 subformation 158, 162 subgroup CAP 194, 196, 340 Carter 99, 100, 114 chain see chain of subgroups

Index characteristic 95, 145, 148 conjugate 1, 5, 15, 18, 27, 39, 84, 98–100, 110, 170, 183, 188, 189, 194–196, 220, 230–232, 239, 284, 341, 342 covering VIII, 100, 101, 103, 104, 110, 169, 180, 205, 206, 230, 337 d-maximal 225 core-free 225 derived 307 diagonal 22, 31, 211, 214 F-Dnormal 264 F-hypercentral 178, 179 F-maximal 114, 118 F-normal 236, 264 F-subnormal VIII, 235, 236, 237, 239, 244, 247–250, 254, 258–260, 262, 264–266, 272, 284, 300, 301, 303, 306, 307 Fischer 110, 337 Fitting 78, 98, 105, 139, 153, 176, 181, 182, 250, 287, 315, 330 generalised 97, 98, 104, 213, 224, 239, 315, 324, 325 Frattini see Frattini subgroup Frattini-like 125, 144, 147, 148, see X-Frattini subgroup full diagonal 22, 23, 28, 31, 32, 34, 39 functor see subgroup functor H-d-maximal 225, 228 H-maximal 101, 104, 105, 118, 176, 179, 257, 265, 315, 322, 328, 350, 351 H-prefrattini 170 H-prefrattini 192, 193 Hall 99, 100, 101, 110, 118, 153, 163, 218, 231, 233, 250, 257, 258, 268, 328, 339, 340 K-F-subnormal 236, 239, 248, 250, 262, 301, 305, 307 L-prefrattini 193, 195 maximal 3, 18, 24, 25, 27–29, 31–35, 39–41, 44, 45, 50, 52–55, 57, 58, 60–63, 66, 70, 73, 75, 78, 79, 84, 101, 103, 108, 169, 174, 175, 179, 181, 191, 196–198, 200, 204, 236, 264, 268, 284, 328, 340, 343 conjugate subsystems 74, 84, 85

383

core-free 2–7, 25, 27–32, 34, 37, 39, 44, 61, 175, 176, 194, 224, 288 critical 78, 79–81, 83, 107, 108, 171, 204 crucial 176 frequent 37 H-abnormal 108, 169, 175, 179, 186, 197 H-critical 108, 169, 171–174, 176, 180 H-normal 108, 191, 195, 203, 235, 236, 247 JH-solid set see set, JH-solid monolithic 4, 52, 55, 57, 60–66, 68–70, 73, 74, 78–80, 108, 175, 176, 191, 192, 197, 198, 200 of diagonal type 39 of type 1 44, 74 of type 2 50, 74 of type 3 44, 50, 62 small 32, 34, 60 solid set see set, solid subsystem 73, 74, 75, 77, 80, 84, 191–195 system VIII, 73, 74, 77, 78, 81, 83–85, 171–173, 175, 190–193, 195, 197, 198, 202–204 w-solid set see set, w-solid weakly solid set see subgroup, maximal, w-solid set maximal normal 98, 200, 210, 217, 289 minimal normal 3, 5–7, 9, 21, 25, 40–42, 44, 46, 61–64, 66, 91, 108, 126, 139, 147, 156, 174–176, 180, 205, 210, 211, 215, 220–222, 224, 225, 227, 254, 266, 288, 303 abelian 3, 4 complemented 3, 5, 6, 24, 31, 33, 38, 40, 199, 215 non-abelian 3, 4, 7 self-centralising 3, 5 N-subnormal 236 normal parafrattini 198 profrattini 198 X-parafrattini 198, 199, 202 X-profrattini 198, 199 normally embedded 115, 343

384

Index

NTL-functor see subgroup NTL-functor of prefrattini type 73, 169, 191, 192, 194, 204 of soluble type 205, 206, 208, 214, 215, 224 p-prefrattini 193 prefrattini VIII, 73, 169, 170, 190, 191, 192, 194, 196, 197, 202, 204 pronormal 84, 340–343 strongly conjugate 239 subnormal 98, 112, 113, 118, 206–209, 214, 215, 219, 223, 224, 235–237, 239, 243, 244, 247, 285–288, 295, 303, 315, 325, 329, 340–342, 352 supplemented 3, 6 Sylow 3, 5, 98, 100, 110, 166, 176, 187, 189, 206, 219, 237, 247, 273, 301, 328, 330, 349, 350 U -invariant 23, 25, 27, 28, 210, 211 X-Frattini 125, 147, 148 F¨ orster 125, 144, 145–148 X-Frattini 52, 197, 198 X-parafrattini 199 X-prefrattini 192, 195, 197, 202 X-profrattini 199 (X, g)-pronormal 341, 342 subgroup functor 206, 207, 208, 210–212, 215, 221, 224, 226, 229, 230, 236, 263, 264 inductive 211, 212, 229 inherited 206, 207, 210, 230, 236 NTL see subgroup NTL-functor r 206, 208, 211, 212, 214, 215, 222, 224–226, 230 r 206, 208, 211–213, 222, 224–226, 231 r 208, 211, 212, 214, 215 S 205, 206, 208, 230, 231, 233, 266, 281 Sn 206, 207, 224, 236, 237 Sp 206 t 206, 208, 210, 211, 214, 215, 217–221, 224–226, 230–233 t 206, 208, 210–213, 215, 222–224, 230, 231 t 208, 210–212

w-inherited 206, 207, 210–212, 219, 237, 263 weakly inductive 211, 212, 224 subgroup NTL-functor 263, 264 submodule 121, 291 supplement 3, 5, 6, 18, 21, 22, 25, 27, 37, 38, 40, 41, 44, 45, 49, 50, 52, 53, 55, 63, 64, 70, 71, 73, 74, 78, 79, 108, 171, 179, 181, 220 monolithic 45, 49, 191 Sylow subgroup see subgroup, Sylow system normaliser 99, 100, 169, 183, 185 absolute 183, 187 t -subgroup see subgroup functor, t t’-Schunck class 230 t -subgroup see subgroup functor, t t-Schunck class 230, 231 t-subgroup see subgroup functor, t theorem Baer VIII, 3, 4, 7, 125, 144, 148 Clifford 286 Frobenius 277 Gasch¨ utz-Lubeseder-Schmid VIII, 96, 125, 144, 148 Jordan-H¨ older 40, 41, 52, 53, 60, 61, 73 O’Nan-Scott 24, 25, 34 Odd Order 219 Orbit-Stabiliser 1 Schur-Zassenhaus 231 trace 114, 345, 346, 349 transversal see right transversal Wielandt operator

285

X-by-Y-groups 88 X-chief factor 126 X-crossing 54 X-formation function 126, 127–136, 138–140, 154, 269, 274, 293, 297, see also X-local definition X-group 87 X-local definition 127, 129, 130, 133, 154, 295 canonical 132, 140, 141, 153, 166 full 130, 131, 134 integrated 130, 131, 132, 135, 154

Index maximal 133, 134, 136, 138–140 maximal integrated 131 minimal see X-local definition, smallest smallest 128, 129 X-local formation 125, 126, 127–138, 140–143, 148, 149, 151, 153–160, 163, 165, 166, 269, 272, see also X-saturated formation generated by a class of groups 154, 156, 159, 160

128,

385

smallest containing a class of groups 154 X-precrown 52, see also precrown X-saturated formation 125, 148, 149, 151–153, 161, 164–166, see also X-local formation (F) 148, 149 (N) 148 Xω -saturated formation 161 Xp -chief factor 126, 128, 154 Y-chief factor 126, 152 Yp -chief factor 126

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