Fundamentals of Group Theory: An Advanced Approach

Steven Roman Fundamentals of Group Theory An Advanced Approach Steven Roman Irvine, CA USA ISBN 978-0-8176-8300-9 e...

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Steven Roman

Fundamentals of Group Theory An Advanced Approach

Steven Roman Irvine, CA USA

ISBN 978-0-8176-8300-9 e-ISBN 978-0-8176-8301-6 DOI 10.1007/978-0-8176-8301-6 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011941115 Mathematics Subject Classification (2010): 20-01 © Springer Science+Business Media, LLC 2012 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhäuser Boston, c/o Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper

Birkhäuser Boston is part of Springer Science+Business Media (www.birkhauser.com)

To Donna

Preface

This book is intended to be an advanced look at the basic theory of groups, suitable for a graduate class in group theory, part of a graduate class in abstract algebra or for independent study. It can also be read by advanced undergraduates. Indeed, I assume no specific background in group theory, but do assume some level of mathematical sophistication on the part of the reader. A look at the table of contents will reveal that the overall topic selection is more or less standard for a book on this subject. Let me at least mention a few of the perhaps less standard topics covered in the book: 1) An historical look at how Galois viewed groups. 2) The problem of whether the commutator subgroup of a group is the same as the set of commutators of the group, including an example of when this is not the case. 3) A discussion of xY-groups, in particular, a) groups in which all subgroups have a complement b) groups in which all normal subgroups have a complement c) groups in which all subgroups are direct summands d) groups in which all normal subgroups are direct summands. 4) The subnormal join property, that is, the property that the join of two subnormal subgroups is subnormal. 5) Cancellation in direct sums: A group K is cancellable in direct sums if E { K ¸ F { Lß

K¸L

Ê

E¸F

(The symbol { represents the external direct sum.) We include a proof that any finite group is cancellable in direct sums. 6) A complete proof of the theorem of Baer that a nonabelian group K has the property that all of its subgroups are normal if and only if KœUEF where U is a quaternion group, E is an elementary abelian group of exponent # and F is an abelian group all of whose elements have odd order.

vii

viii

Preface

7) A somewhat more in-depth discussion of the structure of :-groups, including the nature of conjugates in a :-group, a proof that a :-group with a unique subgroup of any order must be either cyclic (for :  #) or else cyclic or generalized quaternion (for : œ #) and the nature of groups of order :8 that have elements of order : 8" . 8) A discussion of the Sylow subgroups of the symmetric group (in terms of wreath products). 9) An introduction to the techniques used to characterize finite simple groups. 10) Birkhoff's theorem on equational classes and relative freeness. Here are a few other remarks concerning the nature of this book. 1) I have tried to emphasize universality when discussing the isomorphism theorems, quotient groups and free groups. 2) I have introduced certain concepts, such as subnormality and chain conditions perhaps a bit earlier than in some other texts at this level, in the hopes that the reader would acclimate to these concepts earlier. 3) I have also introduced group actions early in the text (Chapter 4), before giving a more thorough discussion in Chapter 7. 4) I have emphasized the role of applying certain operations, namely intersection, lifting, quotient and unquotient to a “group extension” L Ÿ K . A couple of random notes: Unless otherwise indicated, any theorem not proved in the text is an invitation to the reader to supply a proof. Also, sections marked with an asterisk are optional, meaning that they can be skipped without missing information that will be required later. Let me conclude by thanking my graduate students of the past five years, who not only put up with this material in manuscript form but also put up with the many last-minute changes that I made to the manuscript during those years. In any case, if the reader should find any errors, I would appreciate a heads-up. I can be contacted through my web site www.romanpress.com. Steven Roman

Contents

Preface, vii

1

Preliminaries, 1 Multisets, 1 Words, 1 Partially Ordered Sets, 2 Chain Conditions and Finiteness, 5 Lattices, 8 Equivalence Relations, 10 Cardinality, 12 Miscellanea, 15

2

Groups and Subgroups, 19 Operations on Sets, 19 Groups, 19 The Order of a Product, 26 Orders and Exponents, 28 Conjugation, 29 The Set Product, 31 Subgroups, 32 Finitely-Generated Groups, 35 The Lattice of Subgroups of a Group, 37 Cosets and Lagrange's Theorem, 41 Euler's Formula, 43 Cyclic Groups, 44 Homomorphisms of Groups, 46 More Groups, 46 *An Historical Perspective: Galois-Style Groups, 56 Exercises, 58

3

Cosets, Index and Normal Subgroups, 61 Cosets and Index, 61 Quotient Groups and Normal Subgroups, 65 Internal Direct Products, 72 Chain Conditions and Subnormality, 76 ix

x

Contents Subgroups of Index #, 78 Cauchy's Theorem, 79 The Center of a Group; Centralizers, 81 The Normalizer of a Subgroup, 82 Simple Groups, 84 Commutators of Elements, 84 Commutators of Subgroups, 93 *Multivariable Commutators, 95 Exercises, 98

4

Homomorphisms, Chain Conditions and Subnormality, 105 Homomorphisms, 105 Kernels and the Natural Projection, 108 Groups of Small Order, 108 A Universal Property and the Isomorphism Theorems, 110 The Correspondence Theorem, 113 Group Extensions, 115 Inner Automorphisms, 119 Characteristic Subgroups, 120 Elementary Abelian Groups, 121 Multiplication as a Permutation, 123 The Frattini Subgroup of a Group, 127 Subnormal Subgroups, 128 Chain Conditions, 136 Automorphisms of Cyclic Groups, 141 Exercises, 144

5

Direct and Semidirect Products, 149 Complements and Essentially Disjoint Products, 149 Product Decompositions, 151 Direct Sums and Direct Products, 152 Cancellation in Direct Sums, 157 The Classification of Finite Abelian Groups, 158 Properties of Direct Summands, 161 xY-Groups, 164 Behavior Under Direct Sum, 167 When All Subgroups Are Normal, 169 Semidirect Products, 171 The External Semidirect Product, 175 *The Wreath Product, 180 Exercises, 185

6

Permutation Groups, 191 The Definition and Cycle Representation, 191 A Fundamental Formula Involving Conjugation, 193 Parity, 194 Generating Sets for W8 and E8 , 195

Contents

xi

Subgroups of W8 and E8 , 197 The Alternating Group Is Simple, 197 Some Counting, 200 Exercises, 202

7

Group Actions; The Structure of : -Groups, 207 Group Actions, 207 Congruence Relations on a K-Set, 211 Translation by K, 212 Conjugation by K on the Conjugates of a Subgroup, 214 Conjugation by K on a Normal Subgroup, 214 The Structure of Finite :-Groups, 215 Exercises, 229

8

Sylow Theory, 235 Sylow Subgroups, 235 The Normalizer of a Sylow Subgroup, 235 The Sylow Theorems, 236 Sylow Subgroups of Subgroups, 238 Some Consequences of the Sylow Theorems, 239 When All Sylow Subgroups Are Normal, 240 When a Subgroup Acts Transitively; The Frattini Argument, 243 The Search for Simplicity, 244 Groups of Small Order, 250 On the Existence of Complements: The Schur–Zassenhaus Theorem, 252 *Sylow Subgroups of W8, 258 Exercises, 261

9

The Classification Problem for Groups, 263 The Classification Problem for Groups, 263 The Classification Problem for Finite Simple Groups, 263 Exercises, 271

10

Finiteness Conditions, 273 Groups with Operators, 273 H-Series and H-Subnormality, 276 Composition Series, 278 The Remak Decomposition, 282 Exercises, 288

11

Solvable and Nilpotent Groups, 291 Classes of Groups, 291 Operations on Series, 293 Closure Properties of Groups Defined by Series, 295 Nilpotent Groups, 297 Solvability, 305 Exercises, 313

xii

12

Contents

Free Groups and Presentations, 319 Free Groups, 319 Relatively Free Groups, 326 Presentations of a Group, 336 Exercises, 350

13

Abelian Groups, 353 An Abelian Group as a ™-Module, 355 The Classification of Finitely-Generated Abelian Groups, 355 Projectivity and the Right-Inverse Property, 357 Injectivity and the Left-Inverse Property, 359 Exercises, 363

References, 367 List of Symbols, 371 Index, 373

Chapter 1

Preliminaries

In this chapter, we gather together some basic facts that will be useful in the text. Much of this material may already be familiar to the reader, so a light skim to set the notation may be all that is required. The chapter then can be used as a reference.

Multisets The following simple concept is much more useful than its infrequent appearance would indicate. Definition Let W be a nonempty set. A multiset Q with underlying set W is a set of ordered pairs Q œ ÖÐ=3 ß 83 Ñ ± =3 − Wß 83 − ™ ß =3 Á =4 for 3 Á 4× where ™ œ Ö"ß #ß á ×. The positive integer 83 is referred to as the multiplicity of the element =3 in Q . A multiset is finite if the underlying set is finite. The size of a finite multiset Q is the sum of the multiplicities of its elements. For example, Q œ ÖÐ+ß #Ñß Ð,ß $Ñß Ð-ß "Ñ× is a multiset with underlying set W œ Ö+ß ,ß -×. The element + has multiplicity #. One often writes out the elements of a multiset according to their multiplicities, as in Q œ Ö+ß +ß ,ß ,ß ,ß -× Two multisets are equal if their underlying sets are equal and if the multiplicities of each element in the multisets are equal.

Words We will have considerable use for the following concept. Definition Let \ be a nonempty set. A finite sequence = œ ÐB" ß á ß B8 Ñ of elements of \ is called a word or string over \ and is usually written in the form S. Roman, Fundamentals of Group Theory: An Advanced Approach, DOI 10.1007/978-0-8176-8301-6_1, © Springer Science+Business Media, LLC 2012

1

2

Fundamentals of Group Theory

= œ B" âB8 The number of elements in A is the length of A, denoted by lenÐAÑ. There is a unique word of length !, called the empty word and denoted by %. The set of all words over \ is denoted by \ ‡ and \ is called the alphabet for \ ‡ . A subword or substring of a word = is a subsequence of = consisting of consecutive elements of =. The empty word is considered a subword of all words. The set \ ‡ of words over \ has an algebraic structure. In particular, the operation of juxtaposition (also called concatenation) is associative and has identity %. Any nonempty set with an associative operation that has an identity is called a monoid. Thus, \ ‡ is a monoid under juxtaposition. It is customary to allow the use of exponents other than " when writing words, where B8 œ î BâB

8 factors

for 8  !. Note, however, that this is merely a shorthand notation. Also, it does not affect the length of a word; for example, the length of B# C$ D is '.

Partially Ordered Sets We will need some basic facts about partially ordered sets. Definition A partially ordered set is a pair ÐT ß Ÿ Ñ where T is a nonempty set and Ÿ is a binary relation called a partial order, read “less than or equal to,” with the following properties: 1) (Reflexivity) For all + − T , +Ÿ+ 2) (Antisymmetry) For all +ß , − T , + Ÿ ,ß

,Ÿ+

Ê

+œ,

Ê

+Ÿ-

3) (Transitivity) For all +ß ,ß - − T , + Ÿ ,ß

,Ÿ-

Partially ordered sets are also called posets. Sometimes partially ordered sets are more easily defined using strict order relations. Definition A strict order  on a nonempty set T is a binary relation that satisfies the following properties:

Preliminaries

3

1) (Asymmetry) For all +ß , − T , +,

Ê

, y +

,-

Ê

2) (Transitivity) For all +ß ,ß - − T , +  ,ß

+-

Theorem 1.1 If ÐT ß Ÿ Ñ is a partially ordered set, then the relation +,

if + Ÿ ,ß + Á ,

is a strict order on T . Conversely, if  is a strict order on T , then the relation +Ÿ,

if +  , or + œ ,

is a partial order on T . It is customary to use a phrase such as “Let T be a partially ordered set” when the partial order is understood. Also, it is very convenient to extend the notation a bit and define W Ÿ + for any subset W of T to mean that = Ÿ + for all = − W . Similarly, + Ÿ W means that + Ÿ = for all = − W and W Ÿ X means that = Ÿ > for all = − W and > − X . Note that in a partially ordered set, it is possible that not all elements are comparable. In other words, it is possible to have Bß C − T with the property that B Ÿ ± C and C Ÿ ± B. Here are some special kinds of partially ordered sets. Definition Let ÐT ß Ÿ Ñ be a partially ordered set. 1) The order Ÿ is called a total order or linear order if every two elements of T are comparable. In this case, ÐT ß Ÿ Ñ is called a totally ordered set or linearly ordered set. 2) A nonempty subset of T that is totally ordered is called a chain in T . The family of chains of T is ordered by set inclusion. 3) A nonempty subset of T for which no two elements are comparable is called an antichain in T . 4) A nonempty subset H of a partially ordered set T is directed if every two elements of H have an upper bound in H. Definition Let ÐT ß Ÿ Ñ be a poset and let +ß , − T . 1) The closed interval Ò+ß ,Ó is defined by Ò+ß ,Ó œ Ö: − T ± + Ÿ : Ÿ ,× 2) The open interval Ð+ß ,Ñ is defined by Ð+ß ,Ñ œ Ö: − T ± +  :  ,×

4


3) The half open intervals are defined by Ò+ß ,Ñ œ Ö: − T ± + Ÿ :  ,×

and

Ð+ß ,Ó œ Ö: − T ± +  : Ÿ ,×

Here are some key terms related to partially ordered sets. Definition (Covering) Let ÐT ß Ÿ Ñ be a partially ordered set. If +ß , − T , then , covers +, written + ¡ , , if + Ÿ , and if there are no elements of T between + and , , that is, if +ŸBŸ,

B œ + or B œ ,

Ê

Definition (Maximum and minimum elements) Let ÐT ß Ÿ Ñ be a partially ordered set. 1) A maximal element is an element 7 − T with the property that there is no larger element in T , that is : − Tß7 Ÿ :

Ê

7œ:

A maximum (largest or top) element 7 − T is an element for which T Ÿ7 2) A minimal element is an element 8 − T with the property that there is no smaller element in T , that is : − Tß: Ÿ 8

Ê

:œ8

A minimum (smallest or bottom) element 8 in T is an element for which 8ŸT

Definition (Upper and lower bounds) Let ÐT ß Ÿ Ñ be a partially ordered set. Let W be a subset of T . 1) An element ? − T is an upper bound for W if WŸ? The smallest upper bound ? for W , if it exists, is called the least upper bound or join of W and is denoted by lubÐWÑ or 1W . Thus, ? has the property that W Ÿ ? and if W Ÿ B then ? Ÿ B. The join of a finite set W œ Ö+" ß á ß =8 × is also denoted by lubÖ+" ß á ß +8 × or +" ” â ” +8 . 2) An element j − T is a lower bound for W if jŸW The largest lower bound j for W , if it exists, is called the greatest lower bound or meet of W and is denoted by glbÐWÑ or 3W . Thus, j has the property that j Ÿ W and if B Ÿ W then B Ÿ j. The meet of a finite set W œ Ö+" ß á ß +8 × is also denoted by glbÖ+" ß á ß +8 × or +" • â • +8 .

Preliminaries

5

Note that the join of the empty set g is, by definition, the least upper bound of the elements of g. But every element of T is an upper bound for the elements of g and so the least upper bound is the minimum element of T , if it exists. Otherwise g has no join. Similarly, the meet of the empty set is the greatest lower bound of g and since all elements of T are lower bounds for g, the meet of g is the maximum element of T , if it exists. Now we can state Zorn's lemma, which gives a condition under which a partially ordered set has a maximal element. Theorem 1.2 (Zorn's lemma) If T is a partially ordered set in which every chain has an upper bound, then T has a maximal element. Zorn's lemma is equivalent to the axiom of choice. As such, it is not subject to proof from the axioms of ZF set theory. Also, Zorn's lemma is equivalent to the well-ordering principle. A well ordering on a nonempty set \ is a total order on \ with the property that every nonempty subset of \ has a least element. Theorem 1.3 (Well-ordering principle) Every nonempty set has a well ordering.

Order-Preserving and Order-Reversing Maps A function 0 À T Ä U between partially ordered sets is order preserving (also called monotone or isotone) if BŸC

Ê

0B Ÿ 0C

BŸC

Í

0B Ÿ 0C

and an order embedding if

Note that an order embedding is injective, since 0 B œ 0 C implies both 0 B Ÿ 0 C and 0 C Ÿ 0 B, which implies that B Ÿ C and C Ÿ B, that is, B œ C . A surjective order-embedding is called an order isomorphism. Similarly, a function 0 À T Ä U is order reversing (also called antitone) if BŸC

Ê

0B 0C

Í

0B 0C

and an order anti-embedding if BŸC

An order anti-embedding is injective and if it is surjective, then it is called an order anti-isomorphism.

Chain Conditions and Finiteness The chain conditions are a form of finiteness condition on a poset.

6


Definition Let T be a poset. 1) T has the ascending chain condition (ACC) if it has no infinite strictly ascending sequences, that is, for any ascending sequence :" Ÿ :# Ÿ :$ Ÿ â there is an index 8 such that :85 œ :8 for all 5 !. 2) T has the descending chain condition (DCC) if it has no infinite strictly descending sequences, that is, for any descending sequence :" :# :$ â there is an index 8 such that :85 œ :8 for all 5 !. 3) T has both chain conditions (BCC) if T has the ACC and the DCC. The following characterizations of ACC and DCC are very useful. Definition Let T be a poset. 1) T has the maximal condition if every nonempty subset of T has a maximal element. 2) T has the minimal condition if every nonempty subset of T has a minimal element. Theorem 1.4 Let T be a poset. 1) T has the ACC if and only if it has the maximal condition. 2) T has the DCC if and only if it has the minimal condition. Proof. Suppose T has the ACC and let W © T be nonempty. Let =" − W . If =" is maximal we are done. If not, then we can pick =# − W such that =#  =" . Continuing in this way, we either arrive at a maximal element in W or we get a strictly increasing ascending chain that does not become constant, which contradicts the ACC. Hence, T has the maximal condition. Conversely, if T has the maximal condition then any ascending sequence in T has a maximal element, at which point the sequence becomes constant. The proof of part 2) is similar. A poset can express “infinitness” by spreading vertically, via an infinite chain or by spreading horizontally, via an infinite antichain. The next theorem shows that these are the only two ways that a poset can express infiniteness. It also says that if a poset has an infinite chain, then it has either an infinite ascending chain or an infinite descending chain. This theorem will prove very useful to us as we explore chain conditions on subgroups of a group. Theorem 1.5 Let T be a poset. 1) The following are equivalent: a) T has no infinite chains. b) T has both chain conditions.

Preliminaries

7

If these conditions hold, then for any +  , in T , there is a maximal finite chain from + to , . 2) The following are equivalent: a) T has no infinite chains and no infinite antichains. b) T is finite. Proof. It is clear that 1a) implies 1b). For the converse, suppose that T has BCC and let V be an infinite chain. The ACC implies that V has a maximal element B" , which must be maximum in V since V is totally ordered. Then V Ï ÖB" × is an infinite chain and we may select its maximum element B#  B" . Continuing in this way gives an infinite strictly descending chain, a contradiction to the DCC. Hence, 1a) and 1b) are equivalent. If 1a) and 1b) hold, then since Ð+ß ,Ó is nonempty, it has a minimal member +" , whence + ¡ +" is a maximal chain from + to +" . If +"  , , then Ð+" ß ,Ó has a minimal member +# and so + ¡ +" ¡ +# is a maximal chain from + to +# . This cannot continue forever and so must produce a maximal finite chain from + to , . For part 2), assume that T has no infinite chains or infinite antichains but that T is infinite. Using the ACC, we will create an infinite descending chain, in contradiction to the DCC. Since T has the maximal condition, it has a maximal element. Let ` œ Ö73 ± 3 − M× be the set of all maximal elements of T . Denote by Æ B the set of all elements of T that are less than or equal to B. (This is read: down B.) Since ` is a nonempty antichain, it must be finite. Moreover, the ACC implies that T œ .Ð Æ 73 Ñ

and so one of the sets, say Æ 73" , must be infinite. The infinite poset T" œ Ð Æ 73" Ñ Ï Ö73" × also has no infinite antichains and no infinite chains. Thus, we may repeat the above process and select an element 73# − T" such that T# œ Ð Æ 73# Ñ Ï Ö73# × is infinite. Note that 73"  73# . Continuing in this way, we get an infinite strictly descending chain. The presence of a chain condition on a poset T has consequences for meets and joins. Theorem 1.6 1) Let T be a poset in which every nonempty finite subset has a meet. If T has the DCC, then every nonempty subset of T has a meet.

8


2) Let T be a poset in which every nonempty finite subset has a join. If T has the ACC, then every nonempty subset of T has a join. Proof. For part 1), let W © T be nonempty. The family of all meets of finite subsets of W has a minimal member 7 in T and the minimality of 7 implies that 7 • + œ 7 for all + − T , that is, 7 Ÿ + for all + − T . Hence, 4

+−W

+œ7−T

We leave proof of part 2) to the reader.

Lattices Many of the partially ordered sets that we will encounter have a bit more structure. Definition 1) A partially ordered set ÐT ß Ÿ Ñ is a lattice if every two elements of T have a meet and a join. 2) A partially ordered set ÐT ß Ÿ Ñ is a complete lattice if every subset of T has a meet and a join. Thus, a complete lattice has a maximum element (the join of T ) and a minimum element (the meet of T ). Note that if 0 À T Ä U is an order isomorphism of the lattices T and U, then 0 preserves meets and joins, that is, 0 Š4:3 ‹ œ 40 :3

and

0 Š2:3 ‹ œ 20 :3

However, an order embedding need not preserve these operations. We will often encounter partially ordered sets T for which every subset of elements has a meet. In this case, joins also exist and T is a complete lattice. Theorem 1.7 Suppose that ÐT ß Ÿ Ñ is a partially ordered set for which every subset of T has a meet. Then ÐT ß Ÿ Ñ is a complete lattice, where the join of a subset W of T is the meet of all upper bounds for W . Proof. First, note that the meet of the empty set is the maximum element of T and the meet of T is the minimum element of T and so T is bounded. In particular, the join of g exists. Let W be a nonempty subset of T . The family Y of upper bounds for W is nonempty, since it contains the maximum element. We need only show that the meet 7 œ 3Y is the join of W . Since = Ÿ Y for any = − W , that is, any = − W is a lower bound for Y , it follows that = Ÿ 7, that is, 7 W . Moreover, if 8 W then 8 − Y and so 7 Ÿ 8. Hence, 7 is the least upper bound of W .

Preliminaries

9

The previous theorem is very useful in many algebraic contexts. In particular, suppose that \ is a nonempty set and that Y is a family of subsets of \ that contains both g and \ and is closed under intersection. (Examples are the subspaces of a vector space, the subgroups of a group, the ideals in a ring, the subfields of a field, the sublattices of a latttice and so on.) Then Y is a complete lattice where the join of any subfamily Z of Y is the intersection of all members of Y containing the members of Z . Note that this join need not be the union of Z , since the union may not be a member of Y . However, if the union of Z is a member of Y , then it will be the join of Z . Example 1.8 1) The set ‘ of real numbers, with the usual binary relation Ÿ , is a partially ordered set. It is also a totally ordered set. It has no maximal elements. 2) The set  œ Ö!ß "ß á × of natural numbers, together with the binary relation of divides, is a partially ordered set. It is customary to write 8 ± 7 to indicate that 8 divides 7. The subset W of  consisting of all powers of # is a totally ordered subset of , that is, it is a chain in . The set T œ Ö#ß %ß )ß $ß *ß #(× is a partially ordered set under ± . It has two maximal elements, namely ) and #(. The subset U œ Ö#ß $ß &ß (ß ""× is a partially ordered set in which every element is both maximal and minimal. The partially ordered set  is a complete lattice but the set of all positive integers under division is a lattice that is not complete. 3) Let W be any set and let c ÐWÑ be the power set of W , that is, the set of all subsets of W . Then c ÐWÑ, together with the subset relation © , is a complete lattice.

Sublattices The subject of sublattices requires a bit of care, since a nonempty subset W of a lattice P inherits the order of P but not necessarily the meets and joins of P. That is, the meet of a subset X of W may be different when X is viewed as a subset of W than when X is viewed as a subset of P. For example, let P œ Ö"ß #ß $ß 'ß "#× under division and let W œ P Ï Ö'×. Then P and W are both lattices under the same partial order. However, in P we have # ” $ œ ' and in W we have # ” $ œ "#. Let us use the term P-meet to refer to the meet in P, and similarly for join. Definition Let P be a lattice and let Q © P be a nonempty subset of P. 1) Q is a sublattice of P if the Q -meet of any finite nonempty subset W © Q exists and is the same as the P-meet of W , and similarly for join, that is, if 4 Wœ4 W Q P

and

2 Wœ2 W Q P

2) If P is a complete lattice, then Q is a complete sublattice of P if theQ meet of any subset W © Q exists and is the same as the P-meet of W , and

10

Fundamentals of Group Theory similarly for join, that is, if

4 Wœ4 W Q P

and

2 Wœ2 W Q P

Theorem 1.9 1) A nonempty subset T of a lattice P is a sublattice of P if and only if the Pmeet and the P-join of any finite nonempty subset E © T are in T . 2) A nonempty subset T of a complete lattice P is a complete sublattice of P if and only if the P-meet and the P-join of any subset E © T are in T .

Equivalence Relations The concept of an equivalence relation plays a major role in mathematics. Definition Let W be a nonempty set. A binary relation µ on W is called an equivalence relation on W if it satisfies the following conditions: 1) (Reflexivity) For all + − W , +µ+ 2) (Symmetry) For all +ß , − W , +µ,

Ê

,µ+

3) (Transitivity) For all +ß ,ß - − W , + µ ,ß , µ -

Ê

+µ-

Definition Let µ be an equivalence relation on W . For + − W , the set of all elements equivalent to + is denoted by Ò+Ó œ Ö, − W ± , µ +× and is called the equivalence class of +. Theorem 1.10 Let µ be an equivalence relation on W . Then 1) , − Ò+Ó Í + − Ò,Ó Í Ò+Ó œ Ò,Ó 2) For any +ß , − W , we have either Ò+Ó œ Ò,Ó or Ò+Ó ∩ Ò,Ó œ g. Definition A partition of a nonempty set W is a collection c œ ÖE3 ± 3 − M× of nonempty subsets of W , called the blocks of the partition, for which 1) E3 ∩ E4 œ g for all 3 Á 4 2) W œ -3−M E3 A system of distinct representatives, abbreviated SDR, for a partition c is a set consisting of exactly one element from each block of c . In various contexts, a system of distinct representatives is also called a transversal for c or a set of canonical forms for c . The following theorem sheds considerable light on the concept of an equivalence relation.

Preliminaries

11

Theorem 1.11 1) Let µ be an equivalence relation on a nonempty set W . Then the set of distinct equivalence classes with respect to µ are the blocks of a partition of W . 2) Conversely, if c is a partition of W , the binary relation µ defined by + µ , if + and , lie in the same block of c is an equivalence relation on W , whose equivalence classes are the blocks of c . This establishes a one-to-one correspondence between equivalence relations on W and partitions of W . The most important problem related to equivalence relations is that of finding an efficient way to determine when two elements are equivalent. Unfortunately, in most cases, the definition does not provide an efficient test for equivalence and so we are led to the following concepts. Definition Let µ be an equivalence relation on a nonempty set W . A function 0 À W Ä X , where X is any set, is called an invariant of the equivalence relation if it is constant on the equivalence classes, that is, if +µ,

Ê

0 Ð+Ñ œ 0 Ð,Ñ

A function 0 À W Ä X is called a complete invariant if it is constant and distinct on the equivalence classes, that is, if +µ,

Í

0 Ð+Ñ œ 0 Ð,Ñ

A collection Ö0" ß á ß 08 × of invariants is called a complete system of invariants if +µ,

Í

03 Ð+Ñ œ 03 Ð,Ñ for all 3 œ "ß á ß 8

Definition Let µ be an equivalence relation on a nonempty set W . A subset G © W is said to be a set of canonical forms for the equivalence relation if G is a system of distinct representatives for the partition consisting of the equivalence classes, that is, if for every = − W , there is exactly one - − G such that - µ =. A set of canonical forms determines equivalence since +ß , − W are equivalent if and only if their corresponding canonical forms are equal. Of course, this will be a practical solution to the problem of equivalence only if there is a practical way to identify the canonical form associated with each element of W . Often, canonical forms provide more of a theoretical tool than a practical one.

12


Cardinality Two sets W and X have the same cardinality, written kW k œ kX k

if there is a bijective function (a one-to-one correspondence) between the sets. If W is in one-to-one correspondence with a subset of X , we write kW k Ÿ kX k. If W is in one-to-one correspondence with a proper subset of X but not with X itself, then we write kW k  kX k. The second condition is necessary, since, for instance,  is in one-to-one correspondence with a proper subset of ™ and yet  is also in one-to-one correspondence with ™ itself. Hence, kk œ k™k. This is not the place to enter into a detailed discussion of cardinal numbers. The intention here is that the cardinality of a set, whatever that is, represents the “size” of the set. It is actually easier to talk about two sets having the same, or different, cardinality than it is to define explicitly the cardinality of a given set (a cardinal number is a special kind of ordinal number). For us, it is sufficient simply to associate with each set W a special kind of set known as a cardinal number, denoted by kW k or cardÐWÑ, that is intended to measure the size of the set. In the case of finite sets, the cardinality is the integer that equals the number of elements in the set. Definition 1) A set is finite if it can be put in one-to-one correspondence with a set of the form ™8 œ Ö!ß "ß á ß 8  "×, for some nonnegative integer 8. A set that is not finite is infinite. 2) The cardinal number of the set  of natural numbers is i! (read “aleph nought”), where i is the first letter of the Hebrew alphabet. Hence, kk œ k™k œ kk œ i!

3) Any set with cardinality i! is called a countably infinite set and any finite or countably infinite set is called a countable set. An infinite set that is not countable is said to be uncountable. Theorem 1.12 ¨ –Bernstein Theorem) For any sets W and X , 1) (Schroder

kW k Ÿ kX k and kX k Ÿ kW k Ê kW k œ kX k

2) (Cantor's theorem) If cÐWÑ denotes the power set of W then kW k  kc ÐWÑk

Preliminaries

13

3) If c! ÐWÑ denotes the set of all finite subsets of W and if W is an infinite set, then kW k œ kc! ÐWÑk

Cardinal Arithmetic If W and X are sets, the cartesian product W ‚ X is the set of all ordered pairs W ‚ X œ ÖÐ=ß >Ñ ± = − Wß > − X × If two sets \ and ] are disjoint, their union is called a disjoint union and is denoted by \“] More generally, the disjoint union of two arbitrary sets W and X is the set W “ X œ ÖÐ=ß !Ñ ± = − W× ∪ ÖÐ>ß "Ñ ± > − X × This is just a scheme for taking the union of W and X while at the same time assuring that there is no “collapse” due to the fact that the intersection of W and X may not be empty. Definition Let , and - denote cardinal numbers. Let W and X be sets for which kW k œ , and kX k œ -. 1) The sum ,  - is the cardinal number of the disjoint union W “ X . 2) The product ,- is the cardinal number of W ‚ X . 3) The power ,- is the cardinal number of the set of all functions from X to W. We will not go into the details of why these definitions make sense. (For instance, they seem to depend on the sets W and X , but in fact they do not.) It can be shown, using these definitions, that cardinal addition and multiplication are associative and commutative and that multiplication distributes over addition. Theorem 1.13 Let ,, - and . be cardinal numbers. Then the following properties hold: 1) (Associativity) ,  Ð-  .Ñ œ Ð,  -Ñ  . and ,Ð-.Ñ œ Ð,-Ñ. 2) (Commutativity) ,  - œ -  , and ,- œ -, 3) (Distributivity) ,Ð-  .Ñ œ ,-  ,.

14


4) (Properties of Exponents) a ) , - . œ , - , . b) Ð,- Ñ. œ ,-. c) Ð,-Ñ. œ ,. -. On the other hand, the arithmetic of cardinal numbers can seem a bit strange, as the next theorem shows. Theorem 1.14 Let , and - be cardinal numbers, at least one of which is infinite. Then ,  - œ ,- œ maxÖ,ß -×

It is not hard to see that there is a one-to-one correspondence between the power set cÐWÑ of a set W and the set of all functions from W to Ö!ß "×. This leads to the following theorem. Theorem 1.15 For any cardinal , 1) If kW k œ , then kc ÐWÑk œ #, 2) ,  #, We have already observed that kk œ i! . It can be shown that i! is the smallest infinite cardinal, that is, ,  i0

Ê

, is a natural number

It can also be shown that the set ‘ of real numbers is in one-to-one correspondence with the power set c ÐÑ of the natural numbers. Therefore, k‘k œ #i!

The set of all points on the real line is sometimes called the continuum and so #i! is sometimes called the power of the continuum and denoted by - . The previous theorem shows that cardinal addition and multiplication have a kind of “absorption” quality, which makes it hard to produce larger cardinals from smaller ones. The next theorem demonstrates this more dramatically. Theorem 1.16 1) Addition applied a positive countable number of times or multiplication applied a finite number of times to the cardinal number i! yields i! . Specifically, for any nonzero 8 − , we have i! † i! œ i!

and i8! œ i!

Preliminaries

15

2) Addition and multiplication applied a positive countable number of times to the cardinal number #i! yields #i! . Specifically, i! † #i! œ #i!

and

Ð#i! Ñi! œ #i!

Using this theorem, we can establish other relationships, such as #i! Ÿ Ði! Ñi! Ÿ Ð#i! Ñi! œ #i! which, by the Schro¨der–Bernstein theorem, implies that Ði! Ñi! œ #i! We mention that the problem of evaluating ,- in general is a very difficult one and would take us far beyond the scope of this book. We conclude with the following reasonable-sounding result, whose proof is omitted. Theorem 1.17 Let ÖE5 ± 5 − O× be a collection of sets with an index set of cardinality kO k œ ,. If kE5 k Ÿ - for all 5 − O , then » . E5 » Ÿ -,

5−O

Miscellanea The following section need not be read until it is referenced much later in the book. If : is prime, then we will have occasion to write an integer α satisfying α ´ " mod : in the form α œ "  ,:> where : y± , . However, the case where : œ # and α œ "  #. with . odd is exceptional. In this case, we will need to write α œ "  ,#> , where , is odd and > #. This will ensure that > # when : œ #. Accordingly, it will be useful to introduce the following terminology. Definition Let α ´ " mod :. 1) If : œ # and α œ "  #. where . is odd, that is, if α ´ $ mod %, then the :standard form of α is α œ "  ,:> ß

: y± , and > #

2) In all other cases, the :-standard form of α is α œ "  ,:> ß

: y± ,

Theorem 1.18 Let : be a prime and let . ". Let 9: Ð8Ñ denote the largest exponent / for which :/ divides 8.

16


1) For " Ÿ 5 Ÿ :. , 9: ”Œ

:. • œ .  9: Ð5Ñ 5

In particular, :.5" º Œ

:. 5

and if :  # and 5 # or if : œ # and 5 $, then :.5# º Œ

:. 5

2) If the :-standard form of α is α œ /  ,:> then for any . !, .

.

α: œ /:  A:.> where : y± A. Proof. For part 1), write Œ

:. 5

œ

:. Ð:.  "Ñ Ð:.  ?Ñ :.  Ð5  "Ñ â â 5 " ? 5"

where " Ÿ ? Ÿ 5  ". Now, if " Ÿ 3 Ÿ . , then : 3 ± ? if and only if : 3 ± :.  ? and so :8 ¹ Œ

:. 5

Í

8 Ÿ .  9: Ð5Ñ

The rest follows from the fact that :@ ± 5 implies @ Ÿ 5  " and if :  # and 5 # or if : œ # and 5 $, then :@ ± 5 implies @ Ÿ 5  #. For part 2), if the :-standard form for α is α œ /  ,: > , then α

:.

> :.

œ Ð/  ,: Ñ

:.

œ/ /

:. "

:.

,:

.>

 5œ#

Œ

:. :. 5 5 >5 / , : 5

where the terms in the final sum are ! if . œ !. If :  #, then part 1) implies that the 5 th term in the final sum is divisible by : to the power .  5  #  >5 œ .  >  "  Ò"  >Ð5  "Ñ  5Ó .  >  " If : œ #, then > # and so the 5 th term in the final sum is divisible by : to the power

Preliminaries

.  5  "  >5 œ .  >  "  Ò>Ð5  "Ñ  5Ó .  >  " Hence, in both cases, the final sum is divisible by :.>" and so .

.

α: œ / :  / : where : y± Ð/:

.

"

.

"

,  @:Ñ.

.

,:.>  @:.>" œ /:  :.> Ð/:

.

"

,  @:Ñ

17

Chapter 2

Groups and Subgroups

Operations on Sets We begin with some preliminary definitions before defining our principal object of study. For a nonemtpy set \ , the 8-fold cartesian product is denoted by \ 8 œ ðóóóñóóóò \‚â‚\ 8 factors

Definition Let \ be a nonempty set and let 8 be a natural number. 1) For 8 ", an 8-ary operation on \ is a function 0 À \8 Ä \ 2) A "-ary operation 0 À \ Ä \ is called a unary operation on \ . 3) A #-ary operation 0 À \ ‚ \ Ä \ is called a binary operation on \ . 4) A nullary operation on \ is an element of \ . An 8-ary operation, for any natural number 8, is referred to as a finitary operation. It is often the case that the result of applying a binary operation is denoted by juxtaposition, writing +, in place of 0 Ð+ß ,Ñ. Definition If 0 À \ 8 Ä \ is an 8-ary operation on \ and if ] is a nonempty subset of \ , then the restriction of 0 to ] 8 is a map 0 l] 8 À ] 8 Ä \ . We say that ] is closed under the operation 0 if 0 l] 8 maps ] 8 into ] . For a nullary operation B − \ , this means that B − ] .

Groups We are now ready to define our principal object of study. Definition A group is a nonempty set K, called the underlying set of the group, together with a binary operation on K, generally denoted by juxtaposition, with the following properties: S. Roman, Fundamentals of Group Theory: An Advanced Approach, DOI 10.1007/978-0-8176-8301-6_2, © Springer Science+Business Media, LLC 2012

19

20


1) (Associativity) For all +, , , - − K, Ð+,Ñ- œ +Ð,-Ñ 2) (Identity) There exists an element " − K, called the identity element of the group, for which "+ œ +" œ + for all + − K. 3) (Inverses) For each + − K, there is an element +" − K, called the inverse of +, for which ++" œ +" + œ " Two elements +ß , − K commute if +, œ ,+ A group is abelian, or commutative, if every pair of elements commute. A group is finite if the underlying set K is a finite set; otherwise, it is infinite. The order of a group is the cardinality of the underlying set K , denoted by 9ÐKÑ or kKk. It is customary to use the phrase “K is a group” where K is the underlying set when the group operation under consideration is understood. We leave it to the reader to show that the identity element in a group K is unique, as is the inverse of each element. Moreover, for +ß , − K , Ð+" Ñ" œ +

and Ð+,Ñ" œ , " +"

In a group K, exponentiation is defined for integral exponents as follows: Ú Ý Ý" +â+ î 8 + œÛ 8 factors Ý Ý 8 " Ü Ð+ Ñ

if 8 œ ! if 8  ! if 8  !

When K is abelian, the group operation is often (but not always) denoted by  and is called addition, the identity is denoted by ! and called the zero element of the group and the inverse of an element + − K is denoted by + and is called the negative of +. In this case, exponents are replaced by multiples: Ú Ý Ý! +â+ 8+ œ Û ðóóñóóò 8 terms Ý Ý Ü Ð8+Ñ

if 8 œ ! if 8  ! if 8  !


21

The Order of an Element Definition Let K be a group. 1) If + − K, then any integer 8 for which +8 œ " is called an exponent of +. 2) The smallest positive exponent of + − K, if it exists, is called the order of + and is denoted by 9Ð+Ñ. If + has no exponents, then + is said to have infinite order. An element of finite order is said to be periodic or torsion. 3) An element of order # is called an involution. Theorem 2.1 Let K be a group. If + − K has finite order 9Ð+Ñ, then the exponents of + are precisely the integral multiples of 9Ð+Ñ. Proof. Let 8 œ 9Ð+Ñ. Any integral multiple of 8 is clearly an exponent of +. Conversely, if +7 œ ", then 7 œ ;8  < where ! Ÿ <  8. Hence, " œ +7 œ +;8< œ +;8 +< œ +< and so the minimality of 8 implies that < œ !, whence 7 œ ;8 is an integral multiple of 8. Involutions arise often in the theory of groups. Proof of the following result is left as an exercise. Theorem 2.2 A group in which every nonidentity element is an involution is abelian. As we will see in a moment, a group may have elements of finite order and elements of infinite order. Definition A group K is said to be periodic or torsion if every element of K is periodic. A group that has no periodic elements other than the identity is said to be aperiodic or torsion free. It is not hard to see that every finite group is periodic. On the other hand, as we will see, there are infinite periodic groups, that is, an infinite group need not have any elements of infinite order.

Examples Here are some examples of groups. Example 2.3 The simplest group is the trivial group K œ Ö"×, which contains only the identity element. All other groups are said to be nontrivial.

22


Example 2.4 The integers ™ form an abelian group under addition. The identity is !. The rational numbers  form an abelian group under addition and the nonzero rational numbers ‡ form an abelian group under multiplication. A similar statement holds for the real numbers ‘ and the complex numbers ‚ (and indeed for any field J ). Example 2.5 (Cyclic groups) If + is a formal symbol, we can define a group K to be the set of all integral powers of +: G∞ Ð+Ñ œ Ö+3 ± 3 − ™× where the product is defined by the formal rules of exponents: +3 +4 œ +34 This group is also denoted by G∞ or Ø+Ù and is called the cyclic group generated by +. The identity of Ø+Ù is " œ +! . We can also create a finite group G8 Ð+Ñ of positive order 8 by setting G8 Ð+Ñ œ Ö" œ +! ß +ß +# ß á ß +8" × where the product is defined by addition of exponents, followed by reduction modulo 8: +3 +4 œ +Ð34Ñ mod 8 This defines a group of order 8, called a cyclic group of order 8. The inverse of +5 is +Ð5Ñ mod 8 . The group G8 Ð+Ñ is also denoted by G8 or Ø+Ù. Note that for any integer 5 , the symbol +5 refers to the element of G8 Ð+Ñ obtained by multiplying together 5 copies of +. Hence, +5 œ +5 mod 8 and so for any integers 5 and 4, +5 +4 œ +5 mod 8 +4 mod 8 œ +5 mod 84 mod 8 œ +Ð54Ñ mod 8 œ +54 Thus, we can feel free to represent the elements of G8 Ð+Ñ using all integral powers of + and the rules of exponents will hold, although we must remember that a single element of G8 Ð+Ñ has many representations as powers of +. The groups G8 Ð+Ñ or G∞ Ð+Ñ are called cyclic groups. Example 2.6 The set ™8 œ Ö!ß á ß 8  "× of integers modulo a positive integer 8 is a cyclic group of order 8 under addition modulo 8, generated by the element ", since 5 − ™8 is the sum of 5 ones. The notation ™Î8™ is preferred by some mathematicians since when : is a


23

prime, the notation ™: is also used for the : -adic integers. However, we will not use it in this way. If : is a prime, then the set ™‡: œ Ö"ß á ß :  "× of nonzero elements of ™: is an abelian group under multiplication modulo :. Indeed, by definition, the set J ‡ of nonzero elements of any field J is a group under multiplication. It is possible to prove (and we leave it as an exercise) that J ‡ is cyclic if and only if J is a finite field. More generally, the set V ‡ of units of a commutative ring V with identity is a multiplicative group. In the case of the ring ™8 , this group is ™‡8 œ Ö+ − ™8 ± Ð+ß 8Ñ œ "× To see directly that this is a group, note that for each + − ™‡8 , there exists integers B and C such that B+  C8 œ ". Hence, B+ ´ " mod 8, that is, B − ™‡8 is the inverse of + in ™‡8 . The group ™‡8 is abelian and it is possible to prove (although with some work; see Theorem 4.43) that ™‡8 is cyclic if and only if 8 œ #ß %ß :/ or #:/ , where : is an odd prime. Example 2.7 (Matrix groups) The set `7ß8 ÐJ Ñ of all 8 ‚ 7 matrices over a field J is an abelian group under addition of matrices. The set KPÐ8ß J Ñ of all nonsingular 8 ‚ 8 matrices over J is a nonabelian (for 8  ") group under multiplication, known as the general linear group. The set WPÐ8ß J Ñ of all 8 ‚ 8 matrices over J with determinant equal to " is a group under multiplication, called the special linear group. Example 2.8 (Functions) Let K be a group and let \ be a nonempty set. The set K\ of all functions from \ to K is a group under product of functions, defined by Ð0 1ÑÐBÑ œ 0 ÐBÑ1ÐBÑ for all B − \ . The identity in K\ is the function that sends all elements of \ to the identity element " − K . This map is often referred to as the zero map. (We cannot call it the identity map!) Also, the set Y of all bijective functions on K is a group under composition. Example 2.9 The set K œ ÖÐ/"ß /# ß á Ñ ± /3 − ™# × of all infinite binary sequences, with componentwise addition modulo #, is an infinite abelian group that is periodic, since

24


#Ð/"ß /# ß á Ñ œ Ð!ß !ß á Ñ and so every nonidentity element has order #.

The External Direct Product of Groups One important method for creating a new group from existing groups is as follows. If K" ß á ß K8 are groups, then the cartesian product T œ K" ‚ â ‚ K 8 is a group under componentwise product defined by Ð+" ß á ß +8 ÑÐ," ß á ß ,8 Ñ œ Ð+" ," ß á ß +8,8Ñ where +3 ß ,3 − K3 . The group T is called the external direct product of the groups K" ß á ß K8 . Although the notation ‚ is often used for the external direct product of groups, we will use the notation K" } â } K 8 to distinguish it from the cartesian product as a set. As an example, the direct product Z œ G# Ð+Ñ } G# Ð,Ñ œ ÖÐ"ß "Ñß Ð+ß "Ñß Ð"ß ,Ñß Ð+ß ,Ñ× of two cyclic groups of order # is called the (Klein) %-group (and was called the Vierergruppe by Felix Klein in 1884). We will generalize the direct product construction to arbitrary (finite or infinite) families of groups in a later chapter.

Symmetric Groups Let \ be a nonempty set. A bijective function from \ to itself is called a permutation of \ . The set W\ of all permutations of \ is a group under composition, with order k\ kx when \ is finite. Also, W\ is nonabelian for k\ k $. The group W\ is called the symmetric group or permutation group on the set \ . The group of permutations of the set M8 œ Ö"ß á ß 8× is denoted by W8 and has order 8x. We will study permutation groups in detail in a later chapter, but we want to make a few remarks here for use in subsequent examples. (Proofs will be given later.) If +" ß á ß +5 are distinct elements of \ , the expression Ð+" â+5 Ñ denotes the permutation that sends +3 to +3" for 3 œ "ß á ß 5  " and sends the last element +5 to the first element +" . All other elements of \ are held fixed. This permutation is called a 5 -cycle in W\ . For example, in WÖ"ß#ß$ß%× the permutation Ð" $ %Ñ sends " to $, $ to %, % to " and # to itself. A #-cycle Ð+ ,Ñ is


25

called a transposition, since it simply transposes + and , , leaving all other elements of \ fixed. We can now see why a permutation group with at least three elements +, , and - is nonabelian, since for example Ð+ ,ÑÐ+ -Ñ Á Ð+ -ÑÐ+ ,Ñ (Composition is generally denoted by juxtaposition as above.) The support of a permutation 5 − W\ is the set of elements of \ that are moved by 5, that is, suppÐ5Ñ œ ÖB − \ ± 5B Á B× Two permutations 5ß 7 − W\ are disjoint if their supports are disjoint. In particular, two cycles Ð+" â+5 Ñ and Ð," â,7 Ñ are disjoint if the underlying sets Ö+" ß á ß +5 × and Ö," ß á ß ,7 × are disjoint. It is not hard to see that disjoint permutations commute, that is, if 5 and 7 are disjoint, then 57 œ 75. It is also not hard to see that every permutation 5 is a product (composition) of pairwise disjoint cycles, the product being unique except for the order of factors and the inclusion of "-cycles. In fact, this is a direct result of the fact that the relation B´C

if

55 B œ C for some 5 − ™

is an equivalence relation and thereby induces a partition on M8 . (The reader is invited to write a complete proof at this time or to refer to Theorem 6.1.) This factorization is called the cycle decomposition of 5 . The cycle structure of 5 is the number of cycles of each length in the cycle decomposition of 5 . For example, the permutation 5 œ Ð" # $ÑÐ% & 'ÑÐ( )ÑÐ*Ñ has cycle structure consisting of two cycles of length $, one cycle of length # and one cycle of length ". It is easy to see that for k\ k #, any cycle in W\ is a product of transpositions, since Ð+" â+8 Ñ œ Ð+" +8 ÑÐ+" +8" ÑâÐ+" +$ ÑÐ+" +# Ñ and Ð+Ñ œ Ð+ ,ÑÐ+ ,Ñ Hence, the cycle decomposition implies that every permutation is a product of (not necessarily disjoint) transpositions. Although such a factorization is far from unique, we will show that the parity of the number of transpositions in the factorization is unique. In other words, if a permutation can be written as a product of an even number of transpositions, then all factorizations into a

26


product of transpositions have an even number of transpositions. Such a permutation is called an even permutation. For example, since Ð" $ %Ñ œ Ð" %ÑÐ" $Ñ the permutation Ð" $ %Ñ is even. Similarly, a permutation is odd if it can be written as a product of an odd number of transpositions. For example, the equation above for Ð+" â+8 Ñ shows that a cycle of odd length is an even permutation and a cycle of even length is an odd permutation. One of the most remarkable facts about the permutation groups is that every group has a “copy” that sits inside (is isomorphic to a subgroup of) some permutation group. For example, the Klein %-group sits inside W% as follows: Z œ Ö+ ß Ð" #ÑÐ$ %Ñß Ð" $ÑÐ# %Ñß Ð" %ÑÐ# $Ñ× This is the content of Cayley's theorem, which we will discuss later. Thus, if we knew “everything” about permutation groups, we would know “everything” about all groups!

The Order of a Product One must be very careful not to jump to false conclusions about the order of the product of elements in a group. For example, consider the general linear group KPÐ#ß ‚Ñ of all nonsingular # ‚ # matrices over the complex numbers. Let EœŒ

! "

" !

! and F œ Œ "

" "

We leave it to the reader to show that E and F have finite order but that their product EF has infinite order. On the other extreme, we have 9Ð++" Ñ œ " regardless of the value of 9Ð+Ñ. Thus, the order of a product of two nonidentity elements can be as small as " or as large as infinity. On the other hand, the following key theorem relates the order of a power +5 of an element + − K to the order of +. It also tells us something quite specific about the order of the product of commuting elements. Theorem 2.10 Let K be a group and +ß , − K. 1) If 9Ð+Ñ œ 8, then for " Ÿ 5  8, 8 9Ð+5 Ñ œ gcdÐ8ß 5Ñ In particular, Ø+Ù œ Ø+5 Ù

Í

gcdÐ9Ð+Ñß 5Ñ œ "


27

2) If 9Ð+Ñ œ 8 and . ± 8, then 9Ð+5 Ñ œ .

Í

8 5 œ < , where gcdÐ@ œ +=? +>@ Then Ð=?ß @Ñ œ " implies that 9Ð+=? Ñ œ @ and similarly, 9Ð+>@ Ñ œ ?.

Orders and Exponents Let K be a group. We have defined an exponent of + − K to be any integer 5 for which +5 œ ". Here is the corresponding concept for subsets of a group. Definition If W © K is nonempty, then an integer 5 for which =5 œ " for all = − W is called an exponent of W . If W has an exponent, we say that W has finite exponent. Note that many authors reserve the term exponent for the smallest such positive integer 5 . As with individual elements, if the subset W has an exponent, then all exponents of W are multiples of the smallest positive exponent of W . Theorem 2.12 Let K be a group. If a nonempty set W of K has finite exponent, then the set of all exponents of W is the set of all integer multiples of the smallest positive exponent of W . For a finite group K, the smallest exponent minexpÐKÑ is equal to the least common multiple lcmordersÐKÑ of the orders of the elements of K. Thus, if maxorderÐKÑ denotes the maximum order among the elements of K, then maxorderÐKÑ ± minexpÐKÑ and there are simple examples to show that equality may or may not hold. (The reader is invited to find such examples.) However, in a finite abelian group, equality does hold.


29

Theorem 2.13 1) If K is a finite group, then maxorderÐKÑ ± minexpÐKÑ œ lcmordersÐKÑ 2) If K is a finite abelian group, then all orders divide the maximum order and so maxorderÐKÑ œ minexpÐKÑ œ lcmordersÐKÑ and K is cyclic if and only if minexpÐKÑ œ 9ÐKÑ. Proof. For the proof of part 2), let + − K have maximum order 7. Suppose to the contrary that there is a , − K for which 9Ð,Ñ y± 7. We will find an element of K of order greater than 7, which is a contradiction. Since 9Ð,Ñ y± 7, there is a prime : for which 9Ð+Ñ œ 7 œ : 4 @

and 9Ð,Ñ œ : 3?

where : y± ?, : y± @ and 3  4. Then 4

9Ð+: Ñ œ

7 :4

and

9Ð, ? Ñ œ :3

Since K is abelian and these orders are relatively prime, we have 4

9Ð+: , ? Ñ œ 7:34  7 as promised. The last statement of the theorem follows from the fact that a finite group K is cyclic if and only if it has an element of order 9ÐKÑ.

Conjugation Let K be a group. If +ß , − K, then the element , + œ +,+" is called the conjugate of , by +. The conjugacy relation is the binary relation on K defined by +´,

if , œ +B for some B − K

and if + ´ , , we say that + and , are conjugate. Conjugacy is an equivalence " relation on K, since an element is conjugate to itself and if + œ ,B then , œ +B and finally, if , œ +B and - œ , C , then - œ , C œ +BC Note that some authors define the conjugate of , by + as +" ,+, so care must be taken when reading other literature. The function #+ À K Ä K defined by #+ B œ B+

30


is called conjugation by +. Conjugation is very well-behaved: It is a bijection and preserves the group operation, in the sense that #+ ÐBCÑ œ Ð#+ BÑÐ#+ CÑ Thus, in the language of a later chapter, #+ is a group automorphism. The maps #+ are called inner automorphisms and the set of inner automorphisms is denoted by InnÐKÑ. We will have more to say about InnÐKÑ in later chapters. Theorem 2.14 Let K be a group and let +ß ,ß B − K. Then 1) Conjugation is a bijection that preserves the group operation, that is, ÐB+ Ñ" œ ÐB" Ñ+

and

ÐBCÑ+ œ B+ C+

for all Bß C − K. 2) The conjugation map satisfies ÐB, Ñ+ œ B+, for all B − K. 3) Conjugacy is an equivalence relation on K. The equivalence classes under conjugacy are called conjugacy classes. We can also apply conjugation to subsets of K. If W © K and + − K, we write #+ W œ W + œ Ö=+ ± = − W× The previous rules generalize to conjugation of sets. In fact, for any Wß X © K and +ß , − K, we have ÐW + Ñ, œ W ,+

and W + œ X + Í W œ X

Conjugation in the Symmetric Group In general, it is not always easy to tell when two elements of a group are conjugate. However, in the symmetric group, it is surprisingly easy. Theorem 2.15 Let W8 be the symmetric group. 1) Let 5 − W8 . For any 5 -cycle Ð+" â+5 Ñ, we have Ð+" â+5 Ñ5 œ Ð5+" â5+5 Ñ Hence, if 7 œ -" â-5 is a cycle decomposition of 7 , then 7 5 œ -"5 â-55 is a cycle decomposition of 7 5 . 2) Two permutations are conjugate if and only if they have the same cycle structure.


31

Proof. For part 1), we have Ð+" â+5 Ñ5 Ð5+3 Ñ œ œ

5+3" 5+"

35 3œ5

Also, if , Á 5+3 for any 3, then 5" , Á +3 and so Ð+" â+5 Ñ5 , œ 5Ð+" â+5 ÑÐ5" ,Ñ œ 5Ð5" ,Ñ œ , Hence, Ð+" â+5 Ñ5 is the cycle Ð5+" â5+5 Ñ. For part 2), if 7 œ -" â-7 is a cycle decomposition of 7 , then 5 7 5 œ -"5 â-7

and since -35 is a cycle of the same length as -3 , the cycle structure of 7 5 is the same as that of 7 . For the converse, suppose that 5 and 7 have the same cycle structure. If 5 and 7 are cycles, say 5 œ Ð+" â +8 Ñ

and

7 œ Ð," â ,8 Ñ

then any permutation - that sends +3 to ,3 satisfies 5 - œ 7 . More generally, if 5 œ -" â-7

and

7 œ ." â.7

are the cycle decompositions of 5 and 7 , ordered so that -5 has the same length as .5 for all 5 , we can define a permutation - that sends the element in the 3th position of -5 to the element in the 3th position of .5 . Then 5- œ 7 .

The Set Product It is convenient to extend the group operation on a group K from elements of K to subsets of K. In particular, if W and X are subsets of K, then the set product WX (also called the complex product, since subsets of a group are called complexes in some contexts) is defined by WX œ Ö=> ± = − Wß > − X × As a special case, we write Ö+×W as +W , that is, +W œ Ö+= ± = − W× The set product is associative and distributes over union, but not over intersection. Specifically, for Wß X ß Y © K , WÐX Y Ñ œ ÐWX ÑY WÐX ∪ Y Ñ œ WX ∪ WY and ÐX ∪ Y ÑW œ X W ∪ Y W WÐX ∩ Y Ñ © WX ∩ WY and ÐX ∩ Y ÑW © X W ∩ Y W Also,

32


+W œ +X

Í

WœX

Of course, we may generalize the set product to any nonempty finite collection W" ß á ß W8 of subsets of K by setting W" âW8 œ Ö=" â=8 ± =3 − W3 × On the other hand, if 5 is a positive integer, then it is customary to let W 5 œ Ö=5 ± = − W× Thus, in general, W # is a proper subset of the set product WW .

Subgroups The substructures of a group are defined as follows. Definition A nonempty subset L of a group K is a subgroup of K, denoted by L Ÿ K, if L is a group under the restricted product on K. If L Ÿ K and L Á K, we write L  K and say that L is a proper subgroup of K. If L" ß á ß L8 are subgroups of K, we write L" ß á ß L8 Ÿ K . For example, ™ Ÿ , since ™ is an abelian group under addition. However, ™: is not a subgroup of ™, although it is a subset of ™ and it is a group as well: The issue is that ™: is not a group under ordinary addition of integers. However, if L is a subgroup under the first definition above, then L satisfies the second definition. To see this, multiplying the equation "L "L œ "L by the " inverse of "L in K gives "L œ "K . Thus, for all 2 − L , we have 22L œ " " " " œ 22K and so 2L œ 2K . There is another criterion for subgroups that involves checking only closure. Proof is left to the reader. Theorem 2.16 1) A nonempty subset \ of a group K is a subgroup of K if and only if \ is closed under the operations of taking inverses and products, that is, if and only if B−\

Ê

B" − \

and Bß C − \

Ê

BC − \

2) A nonempty finite subset \ of a group K is a subgroup of K if and only if it is closed under the taking of products.


33

Theorem 2.17 The intersection of any nonempty family of subgroups of a group K is a subgroup of K. Example 2.18 The set E8 of all even permutations in W8 is a subgroup of W8 . To see that E8 is closed under the product, if 5 and 7 are even, then they can each be written as a product of an even number of transpositions. Hence, 57 is also a product of an even number of transpositions and so is in E8 . The subgroup E8 is called the alternating subgroup of W8 . For 8 #, the alternating subgroup E8 has order 8xÎ#, that is, E8 is exactly half the size of W8 . To see this, note that 5 − W8 is odd if and only if Ð" #Ñ5 is even and 7 is even if and only if Ð" #Ñ7 is odd. Hence, the map 5 È Ð" #Ñ5 is a self-inverse bijection between E8 and the set of odd permutations. A group has many important subgroups. One of the most important is the following. Definition The center ^ÐKÑ of a group K is the set of all elements of K that commute with all elements of K, that is, ^ÐKÑ œ Ö+ − K ± +, œ ,+ for all , − K× A group K is centerless if ^ÐKÑ œ Ö"×. A subgroup L of K is central if L is contained in the center of K. Two subgroups of a group K can never be disjoint as sets, since each contains the identity of K . However, it will be very convenient to introduce the following terminology and notation. Definition Two subgroups L and O of a group K are essentially disjoint if L ∩ O œ Ö"× We introduce the notation L ìO to denote the set product of two essentially disjoint subgroups and refer to this as the essentially disjoint product of L and O . Note that if 9ÐLÑ œ 8 and 9ÐOÑ œ 7, then 9ÐL ì OÑ œ 87

The Dedekind Law The following formula involving the intersection and set product is very handy and we will use it often. The simple proof is left to the reader.

34


Theorem 2.19 Let K be a group and let Eß Fß G Ÿ K with E Ÿ F . 1) (Dedekind law) EÐF ∩ GÑ œ F ∩ EG 2) E ∩ G œ F ∩ G and EG œ FG

Ê

EœF

Proof. We leave proof of the Dedekind law to the reader. For part 2), E œ EÐE ∩ GÑ œ EÐF ∩ GÑ œ F ∩ EG œ F ∩ FG œ F

We leave it to the reader to find an example to show that the condition that E Ÿ F is necessary in Dedekind's law, that is, EÐF ∩ GÑ is not necessarily equal to EF ∩ EG unless E Ÿ F .

Subgroup Generated by a Subset If K is a group and + − K, then the cyclic subgroup generated by + is the subgroup Ø+Ù œ Ö+5 ± 5 − ™× If 9Ð+Ñ œ 8  ∞, then +5 œ +5 mod 8 and so Ø+Ù œ Ö"ß +ß á ß +8" × Note that Ø+Ù is the smallest subgroup of K containing +, since any subgroup of K containing + must contain all powers of +. More generally, if \ is a nonempty subset of K, then the subgroup generated by \ , denoted by Ø\Ù, is defined to be the smallest subgroup of K containing \ and \ is called a generating set for Ø\Ù. Such a subgroup must exist; in fact, Theorem 2.17 implies that Ø\Ù is the intersection of all subgroups of K containing \ . The following theorem gives a very useful look at the elements of Ø\Ù. Theorem 2.20 Let \ be a nonempty subset of a group K and let \ " œ ÖB" ± B − \×. Let [ œ Ð\ ∪ \ " Ñ‡ be the set of all words over the alphabet \ ∪ \ " . 1) If we interpret juxtaposition in [ as the group product in K and the empty word as the identity in K , then Ø\Ù œ [ and so Ø\Ù œ ÖB/"" âB/88 ± B3 − \ß /3 − ™ß 8  !× ∪ Ö"×


35

2) If K is abelian, then we can collect like factors and so Ø\Ù œ ÖB/"" âB/88 ± B3 − \ß B3 Á B4 for 3 Á 4ß /3 − ™ß 8  !× ∪ Ö"× Proof. It is clear that [ © Ø\Ù. It is also clear that [ is closed under product. /" 8 As to inverses, if A œ B/"" âB/88 − [ then A" œ B/ − [ . Hence, 8 âB" [ Ÿ Ø\Ù. However, since \ © [ , it follows that Ø\Ù Ÿ [ and so [ œ Ø\Ù. Although the previous description of Ø\Ù is very useful, it does have one drawback: Distinct formal words in the set [ may be the same element of the group Ø\Ù, when juxtaposition in [ is interpreted as the group product. For instance, the distinct words B# B" and B are the same group element of Ø\Ù. We will discuss this issue in detail when we discuss free groups later in the book: The matter need not concern us further until then.

Finitely-Generated Groups A group K is finitely generated if K œ Ø\Ù for some finite set \ . If K has a generating set of size 8, then K is said to be 8-generated or to be an 8generator group.

The Burnside Problem There is a fascinating set of problems revolving around the following question. A finite group K is obviously finitely generated and periodic. In 1902, Burnside [39] asked about the converse: Is a finitely-generated periodic group finite? This is the general Burnside problem. A negative answer to the general Burnside problem took 62 years, when Golod [40] showed in 1964 that there are infinite groups that are $-generated and whose elements each have order a power of a fixed prime : (the power depending upon the element). However, this still leaves open some refinements of the general Burnside problem. For example, the Golod groups have elements of arbitrarily large order :8 , that is, they do not have finite exponent, as do finite groups. The Burnside problem is the problem of deciding, for finite integers 8 and 7, whether every 8-generated group of exponent 7 is finite. This problem has been the subject of a great deal of research since Burnside first formulated it in 1902. For example, it has been shown that there are infinite, finitely-generated groups of every odd exponent 7  ''& (Adjan [38], 1979) and of every exponent of the form #5 7, where 5 %) (Ivanov [42], 1992). The restricted Burnside problem, formulated in the 1930's, asks whether or not, for integers 8 and 7, there are a finite number (up to isomorphism) of finite 8-generated groups of exponent 7. In 1994, Zelmanov answered this question

36


in the affirmative. For more on the Burnside problem, we refer the reader to the references located at the back of the book.

Subgroups of Finitely-Generated Groups A far simpler question related to finitely-generated groups is whether every subgroup of a finitely-generated group is finitely generated. We will show when we discuss free groups that for arbitrary groups this is false: There are finitelygenerated groups with subgroups that are not finitely generated. However, in the abelian case, this cannot happen. Theorem 2.21 Any subgroup of an 8-generated abelian group E is also 8generated. In particular, a subgroup of a cyclic group is cyclic. Proof. Let L Ÿ E. The proof is by induction on 8. If 8 œ ", then E œ Ø+Ù is cyclic. Let 5 be the smallest positive exponent for which +5 − L . Then Ø+5 Ù Ÿ L . However, if +7 − L , then 7 œ ;5  < where ! Ÿ <  5 and so +< œ +7;5 œ +7 Ð+5 Ñ; − L which can only happen if < œ !, whence +7 œ Ð+5 Ñ; − Ø+5 Ù. Thus, L œ Ø+5 Ù is also cyclic and so the result holds for 8 œ ". Assume the result is true for any group generated by fewer than 8 # elements. Let E œ ØB" ß á ß B8 Ù and let E8" œ ØB" ß á ß B8" Ù. By assumption, every subgroup of E8" is Ð8  "Ñ-generated, in particular, there exist 23 − L for which L ∩ E8" œ Ø2" ß á ß 28" Ù Now, every 2 − L has the form 2 œ +B/8 where + − E8" and / − ™. If / œ ! for all 2 − L , then L Ÿ E8" and the inductive hypothesis implies that L is at most Ð8  "Ñ-generated. So let us assume that / Á ! for some 2 − L and let = be the smallest positive integer for which 28 œ ,B=8 − L and , − E8" . For an arbitrary 2 œ +B/8 − L , where + − E8" , write / œ ;=  < where ! Ÿ <  =. Then 28; 2 œ Ð,B=8 Ñ; Ð+B/8 Ñ œ +, ; B/;= œ +, ; B7 × be a left transversal for KÎL , with >" œ ". Let ÖB" ß á ß B8 × be a generating set for K and let " [ œ ÖB" ß á ß B8 × ∪ ÖB" " ß á ß B8 ×

Thus, if + − K, then + œ A: âA" for some A3 − [ . But A" œ >=" for some > − X and =" − L and so + œ A: âA# >=" We are now prompted to consider how the >'s and A's commute. For each > − X and A − [ , there exist unique >w − X and 2 − L for which A> œ >w 2 Let W © L be the finite set of all such 2's, as A varies over [ and > varies over X . Note that =" − W since A" >" œ A" œ >=" . By continually moving the element belonging to X forward in the product expression for +, we get + œ >w =: â=" − >w ØWÙ for some >w − X and =3 − W . But if + − L , then + − >w ØWÙ © >w L implies that >w œ ". Hence, L Ÿ ØWÙ and since W © L , we have L œ ØWÙ.

Cosets, Index and Normal Subgroups

65

Quotient Groups and Normal Subgroups Let K be a group and let L Ÿ K. We have seen that the equivalence relation corresponding to the partition KÎL is equivalence modulo L : + ´ , mod L

if +L œ ,L

Now, there seems to be a natural way to “raise” the group operation from K to KÎL by defining (3.4)

+L ‡ ,L œ +,L

Of course, for this operation to make sense, it must be well defined, that is, we must have for all +ß +" ß ,ß ," − K , +"" + − L

and ,"" , − L

Ê

Ð+" ," Ñ" Ð+,Ñ − L

Taking +" œ " and ," œ , gives the necessary condition +−L

Ê

, " +, − L

(3.5)

However, this condition is also sufficient, since if it holds, then " " " " ,"" +" " +, œ Ð," +" ," ÑÐ," +," ÑÐ," ,Ñ − L

Note that (3.5) is equivalent to each of the following conditions: 1) +L+" © L for all + − K 2) +L+" œ L for all + − K 3) +L œ L+ for all + − K. Definition A subgroup L of K is normal in K, written L ü K, if +L œ L+ for all + − K. If L ü K and L Á K, we write L – K. The family of all normal subgroups of a group K is denoted by norÐKÑ. Thus, the product (3.4) is well defined if and only if L ü K. Moreover, if L ü K, then for any +ß , − K, +,L œ +,LL œ +L,L that is, +L ‡ ,L œ +L,L In particular, the set product of two cosets of L is a coset of L . Moreover, if the set product of cosets is a coset, that is, if +L,L œ -L for some - − K, then +, − -L and so -L œ +,L , that is,

66


+L,L œ +,L Let us refer to this as the coset product rule. Finally, if the coset product rule holds, then L ü K, since +L+" © +L+" L œ L for all + − K. Thus, the following are equivalent: 1) The binary operation +L ‡ ,L œ +,L is well defined on KÎL . 2) L ü K . 3) The set product of cosets is a coset. 4) The coset product rule holds. Moreover, if these conditions hold, then KÎL is actually a group under the set product, for it is easy to verify that the set product is associative, KÎL has identity element L and that the inverse of +L is +" L . Thus, we can add a fifth equivalent condition to the list above: 5) KÎL is a group under set product. Before summarizing, let us note that the following are equivalent: L +L , − +L , − +L + ´ , mod L

üK © L+ for all + − K Ê , − L+ for all +ß , − K Ê , " − +" L for all +ß , − K Ê +" ´ , " mod L for all +ß , − K

Also, the following are equivalent: The coset product rule holds +L,L © +,L for all +ß , − K w w + − +Lß , − ,L Ê +w , w − +,L for all +ß , − L w + ´ + mod Lß , w ´ , mod L Ê +w , w ´ +, mod L Now we can summarize. Theorem 3.6 Let L Ÿ K. The following are equivalent: 1) The set product on KÎL is a well-defined binary operation on KÎL . 2) The coset product rule +L,L œ +,L holds for all +ß , − L .


67

3) L is a normal subgroup of K. 4) KÎL is a group under set product, called the quotient group or factor group of K by L . 5) The inverse preserves equivalence modulo L , that is, + ´ , mod L

+" ´ , " mod L

Ê

for all +ß , − K. 6) The product preserves equivalence modulo L , that is, +w ´ + mod Lß

, w ´ , mod L

Ê

+w , w ´ +, mod L

for all +ß +w ß ,ß , w − K. When we use a phrase such as “the group KÎL ” it is the with the tacit understanding that L is normal in K. Note finally that statements 5) and 6) say that equivalence modulo L is a congruence relation on K. A congruence relation ) on an algebraic structure, such as a group, is an equivalence relation that preserves the (nonnullary) algebraic operations. Thus, a congruence relation ) on a group K must satisfy the conditions +) ,

+" ), "

Ê

and +) ,ß

-).

Ê

Ð+-Ñ)Ð,.Ñ

Theorem 3.6 shows that these two conditions are actually equivalent for groups.

More on Normal Subgroups There are several slight variations on the definition of normality that are often useful. We leave proof of the following to the reader. Theorem 3.7 Let L Ÿ K. The following are equivalent: 1) L ü K 2) L + © L for all + − K 3) L + ª L for all + − K 4) Every right coset of L is a left coset, that is, for all + − Kß there is a , − K such that L+ œ ,L 5) Every left coset is a right coset. 6) For all +ß , − K, +, − L

Ê

,+ − L

7) If + − K and 2 − L , then +2 œ 2 + for some 2 w − L . w

Theorem 3.7 implies that a normal subgroup permutes with all subgroups of K. Hence, the normality of either factor guarantees that the set product LO is a subgroup.

68


Theorem 3.8 Let Lß O Ÿ K. 1) If either L or O is normal in K, then L and O permute and LO œ L ” O In this case, we refer to LO as the seminormal join of L and O . 2) If both L and O are normal in K, then LO is also normal in K and we refer to LO as the normal join of L and O . 3) The fact that LO ü K does not imply that either subgroup need be normal. Proof. For part 3), let K œ W% . Let L œ Ö+, Ð"#ÑÐ$%Ñß Ð"$ÑÐ#%Ñ, Ð"%ÑÐ#$Ñß Ð#%Ñß Ð"$Ñß Ð"#$%Ñß Ð"%$#Ñ× and O œ W$ œ Ö+ß Ð"#Ñß Ð"$Ñß Ð#$Ñß Ð"#$Ñß Ð"$#Ñ×

Then L ∩ O œ Ö+ß Ð"$Ñ× has size # and so kLO k œ Ð) ‚ 'ÑÎ# œ #% œ kW% k, which implies that LO œ W% . But neither subgroup is normal: L is not normal since Ð"%ÑÐ"$ÑÐ"%Ñ œ Ð%$Ñ Â L and O is not normal since Ð"%ÑÐ"#ÑÐ"%Ñ œ Ð%#Ñ Â W$ . Example 3.9 (The normal subgroups of H#8 ) We have seen that for . ± 8, the subgroups of the dihedral group H#8 are 1) the cyclic subgroup Ø38Î. Ù of order . , 2) for each ! Ÿ 5  8Î. , the dihedral subgroup W œ 535 Ø38Î. Ù “ Ø38Î. Ù œ Ø535 ß 38Î. Ù of order #.. Subgroups of type 1) are normal, since conjugation gives 533 Ð3 B" − R . 27. Let K be a group and let L Ÿ K. Prove that if L has finite index in K, then ÐK À LÑ œ ÐK À L B Ñ for any B − K.

100


28. Let K be a nonabelian group. Show that ^ÐKÑ is a proper subgroup of any centralizer GK Ð1Ñ. 29. Let L Ÿ K. Prove that GK ÐLÑ ü RK ÐLÑÞ 30. Let K be a group and let L Ÿ K. a) Prove that R Ÿ K is normal in K if and only if all subsets of K normalize R . b) Prove that if W normalizes the subgroups L and O of K, then W normalizes their join L ” O , that is, RK ÐLÑ ∩ RK ÐOÑ Ÿ RK ÐL ” OÑ c)

Prove that if W and X normalize L , then the set product WX normalizes L. d) Prove that a subgroup L Ÿ K normalizes all supergroups of itself. 31. Prove that if L Ÿ K, then RK ÐL + Ñ œ RK ÐLÑ+ . In particular, GK Ð2Ñ+ œ GK Ð2+ Ñ. 32. Let K be a group and let L Ÿ O Ÿ K with L ü K. Prove that RK Œ

L O

œ

RK ÐLÑ O

33. a)

34.

35. 36. 37.

38.

Show that the subgroup L œ Ö+ß Ð" #ÑÐ$ %Ñß Ð" $ÑÐ# %Ñß Ð" %ÑÐ# $Ñ× of W% is normal. Hint: Show that the conjugate of any transposition is another transposition, in fact, Ð+ ,Ñ5 œ Ð5+ 5,Ñ. b) Show that the subgroup O œ Ö3ß Ð" #ÑÐ$ %Ñ× is normal in L but not normal in W% . c) Conclude that normality is not transitive. Let K be a group. a) Let Q be the intersection of all subgroups of K that have finite index in K. Show that Q is normal in K. b) Let L Ÿ K with finite index. Show that there is a normal subgroup R ü K for which R Ÿ L Ÿ K where R also has finite index in K. Let K be a group generated by two involutions B and C. Show that K has a normal subgroup of index #. Let K be a finite group of odd order. Show that the product of all of the elements of K, taken in any order, is in the commutator subgroup Kw . (P. Hall) Let Eß F Ÿ K. Show that if ÒEß Eß FÓ œ Ö"×, then ÒEß FÓ is an abelian group. Hint: One possible proof is as follows: Show that E commutes with ÒEß FÓ and with ÒFß EÓ and that F commutes with Ew . Then use a direct computation to show that Ò+ß ,ÓÒBß CÓ œ ÒCß B" ÓÒ+ß ,Ó where +ß B − E and ,ß C − F . Finally, use Theorem 3.37. Let : Á # be prime and consider the group K œ ™Ð:∞ Ñ { ™# where ™Ð:∞ Ñ is the :-quasicyclic group. Show that K has a unique


101

maximal subgroup Q , but that Q is not maximum in the lattice of all proper subgroups of K. 39. A group K is said to be metabelian if its commutator subgroup Kw is abelian. (Some authors define a group K to be metabelian if Kw is central in K, which is stronger than our definition.) a) Prove that K is metabelian if and only if K has a normal abelian subgroup E for which KÎE is also abelian. b) Prove that the dihedral group H#8 is metabelian. c) Let J be a finite field, such as ™: where : is prime. Let +ß , − J where + Á !. The map 5+ß, À J Ä J defined by 5+ß, À B È +B  , is called an affine transformation of J . Show that the set affÐJ Ñ of all affine transformations of J is a subgroup of J J . Show that affÐJ Ñ is metabelian. d) Prove that if K œ EF where E and F are abelian, then K is metabelian. Hint: Show that Ò+ß ,Ó," +" œ Ò+ß ,Ó+" ," for +ß +" − E and ,ß ," − F . Use the fact that EF œ FE. 40. Let K be a group of order :8 , where : is prime. Suppose that for each + − K, the centralizer G+ œ GK Ð+Ñ has index " or :. a) Prove that G+ ü K. Hint: Assume ÐK À G+ Ñ œ :. Let M œ , G+1 Ÿ G+ 1−K

Show that M is normal in K. Then show that the elements of KÎM are actually permutations of KÎG+ , where 1MÐBG+ Ñ œ 1BG+ . Show that KÎM is a subgroup of WKÎL . b) Prove that Kw Ÿ ^ÐKÑ. 41. We have seen that the family subÐKÑ of all subgroups of K is a complete lattice. Let norÐKÑ be the family of normal subgroups of K. a) Show that the join of two normal subgroups E and F is the set product EF. b) Show that norÐKÑ is a complete sublattice of subÐKÑ. c) Prove that the distributive laws E ” ÐF ∩ GÑ œ ÐE ” FÑ ∩ ÐE ” GÑ E ∩ ÐF ” GÑ œ ÐE ∩ FÑ ” ÐE ∩ GÑ for Eß Fß G Ÿ K imply the modular law: For Eß Fß G Ÿ K with E Ÿ F, E ” ÐF ∩ GÑ œ F ∩ ÐE ” GÑ A lattice that satisfies the modular law is said to be a modular lattice. d) Prove that norÐKÑ is a modular lattice.

102

Fundamentals of Group Theory e)

Is norÐKÑ necessarily distributive, that is, do the distributive laws necessarily hold? Hint: Consider the %-group Z œ Ö"ß +ß ,ß +,×. f) Prove that subÐKÑ need not be modular. Hint: Consider the alternating group E% of order "#. Let E œ ØÐ" #ÑÐ$ %ÑÙß F œ ØÐ" #ÑÐ$ %Ñß Ð" $ÑÐ# %ÑÙ and G œ ØÐ" # $ÑÙ. g) Prove the Dedekind law: For Eß Fß G Ÿ K with E Ÿ F , EÐF ∩ GÑ œ F ∩ ÐEGÑ Find an example to show that for arbitrary subgroups Eß Fß G of K, EÐF ∩ GÑ is not necessarily equal to EF ∩ EG and so the condition that E Ÿ F is necessary. h) Let Eß F and G be subgroups of K with E Ÿ F . Prove that if E∩G œF∩G

and

EG œ FG

then E œ F . 42. Let K be a group with subgroups R and L and suppose that R L Ÿ K. Prove that if ÐK À LÑ and kR k are finite and relatively prime, then R Ÿ L . 43. Let K be a group and L Ÿ K. a) Let L © \ © K. Show that the relation B ´ C if B" C − L is an equivalence relation on \ . What do the equivalence classes of this relation look like? Let Ð\ À LÑ be the cardinality of the set of equivalence classes. b) If L © \ © ] © K, where \L © \ . Show that if Ð\ À LÑ œ Ð] À LÑ  ∞, then \ œ ] . 44. Prove that R ü K and KÎR both have the ACC on subgroups if and only if K has the ACC on subgroups. 45. Sometimes it is useful to relate the commutator subgroup ÒLß OÓ of subgroups L and O to the commutator subgroup Ò\ß ] Ó of generating sets for L and O . Let \ and ] be nonempty subsets of a group K. a) Show that Ò\ß Ø] ÙÓ œ Ò\ß Ø] ÙÓØ] Ù œ Ò\ß ] ÓØ] Ù b) Show that ÒØ\Ùß Ø] ÙÓ œ Ò\ß ] ÓØ\ÙØ] Ù œ Ò\ß ] ÓØ] ÙØ\Ù

Complex Groups Let K be a group. Let Z be a nonempty family of subsets of K that forms a group under set product. Such a group has been called a complex group based on K (see Allen [1]). Let -Z be the union of the subsets in Z . We denote the identity of Z by I . Also, we denote the inverse of E − Z by E" . In the following exercises, let Z be an arbitrary complex group based on the group K.


103

46. Let R ü K. Show that KÎR is a complex group in which the members form a partition of K . Thus, quotient groups are complex groups. 47. Show that if  is the multiplicative group of all positive rational numbers and if d œ ÖÐ" ÐB 5Ñ)>" â)7 where )>" â)7 does not involve 5 . Note that > #, since otherwise the only appearance of 5 is in )" and so 7 Á +. However, we can easily move this rightmost occurrence of 5 one transposition to the left by using the following substitutions. Suppose that )>" œ Ð+ ,Ñ. Note that Ð+ ,Ñ Á ÐB 5Ñ since otherwise the two transpositions would cancel, contradicting the definition of 7. 1) If Ð+ ,Ñ and ÐB 5Ñ are disjoint, then they commute Ð+ ,ÑÐB 5Ñ œ ÐB 5ÑÐ+ ,Ñ 2) If Ð+ ,Ñ œ ÐB ,Ñ for , Á 5 , then ÐB ,ÑÐB 5Ñ œ ÐB 5 ,Ñ œ Ð, B 5Ñ œ Ð, 5ÑÐ, BÑ 3) If Ð+ ,Ñ œ Ð5 ,Ñ where + Á 5 , then write Ð5 ,ÑÐB 5Ñ œ Ð5 B ,Ñ œ ÐB , 5Ñ œ ÐB 5ÑÐB ,Ñ Thus, we can move the rightmost occurrence of 5 to the left, contradicting the construction of 7 and proving part 1). For part 2), a cycle of length 7 # can be written as a product of 7  " transpositions as above. Now suppose that the cycle decomposition of 5 is 5 œ -" â-< ." â.= where lenÐ-3 Ñ œ 73 # and lenÐ.3 Ñ œ ". Then 5 can be written as a product of the following number of transpositions:
is in :-standard form, then Lemma 1.18 implies that for any ? !, ?

?

α: œ /:  A:?> where : y± A. In particular, ."

."

5) :.>" ± α:  /: . . . 6) :.>" y± α:  /: œ α:  " From 3) and 6), we see that 7 Ÿ .  >. ."

Now, if /: œ " then 4) and 5) imply that .  > Ÿ 7 and so 7 œ >  . . This implies that 7  . œ > ". Also, α œ /  -:7. and so for " Ÿ 5 Ÿ :. , α5 œ Ð/  -:7. Ñ5 œ /5  :7. A5 Hence, 5

5

+α œ + /

:7. A5

where no two distinct A5 's are congruent modulo :. , since otherwise we would not get :. distinct conjugates. Thus, we can assume that A5 ranges over the set Ö"ß á ß :. × and so +Ø,Ù œ Ö+/5 5:

7.

± 5 œ "ß á ß :. ×

Now, if :  # or α ´ y $ mod %, then / œ " and so /5 œ " for all 5 . If : œ # and

222


.

α ´ $ mod % but .  ", then we still have /: œ " but since / œ ", as 5 ranges from " to :. , the term /5 alternates between " and " and so half of the terms /5 are " and half are ". Also, as 5 ranges from " to :. , the exponents /5  5:7. range from „"  : to „"  :7 . But + is conjugate to itself and so one of these exponents must be conjugate to " modulo :7 . Therefore, the last exponent is "  :7 and since no other exponent is conjugate to " modulo :7 , it follows that +" Â +Ø,Ù . ."

The case /: œ " occurs precisely when : œ #, α ´ $ mod % and . œ ", in which case + has exactly two conjugates and α œ "  -#> with - odd and > #. >

If > 7, then +α œ +"-# œ +" and so +Ø,Ù œ Ö+ß +" ×. If >  7, then 7 Ÿ "  > implies that > œ 7  " and so α œ "  -#7" where - is odd, that is, - œ ". Hence, the two conjugates of + are + itself and +α œ +"#

7"

Note finally that the case +Ø,Ù œ Ö+ß +" × does occur in the dihedral group H#7" œ Ø3ß 5Ù where 9Ð3Ñ œ #7 and 9Ð5Ñ œ # and 35 œ 3" . Also, the case +Ø,Ù œ Ö+ß +"#

7"

×

occurs in the semidihedral group WH7 œ Øαß 0Ùß

9ÐαÑ œ #7 ß

9Ð0Ñ œ #ß

0α œ α#

7"

"

0

For part 3), if 9Ð+Ñ œ :, then the number of conjugates of + is both a power of : and at most :  ", whence it must equal ".

*Unique Subgroups in a :-Group A cyclic :-group has a unique subgroup of order :. If :  #, then the converse of this is true: A :-group that has a unique subgroup of order : is cyclic. We begin with a definition. Definition A generalized quaternion group of order #8 , 8 # is a group U8 with the following properties: U8 œ Ø+ß ,Ùß 9Ð+Ñ œ #8" ß 9Ð,Ñ œ %ß , # œ +#

8#

ß ,+, " œ +"

If 8 œ $, then U8 is a quaternion group. We will show later in the book that such a group exists: It is a special case of the dicyclic group. We leave it as an exercise to show that Ø,# Ù is the only subgroup of order # in U7 but that for any #  #=  #8 , the group U8 has at least two subgroups of order #= . Also, any B − U8 Ï Ø+Ù has the form B œ +5 , , where

Group Actions; The Structure of :-Groups

Ð+5 ,Ñ# œ +5 Ð,+5 Ñ, œ , # œ +#

223

7"

and so 9Ð+5 ,Ñ œ %. Thus, any element of U8 Ï Ø+Ù has order %. It follows that if 8 %, then Ø+Ù is the unique cyclic subgroup of U8 of order #8" and so Ø+Ù « U8 . We will prove that if a : -group K has a unique subgroup of order :, then K is cyclic if :  # and K is either cyclic or generalized quaternion if : œ #. First, let us show that if K has a unique subgroup L= of any order := , where : Ÿ :=  9ÐKÑ, then K must have a unique subgroup of order :. Since a :-group has subgroups of all orders dividing the order of the group, we have for any subgroup O Ÿ K , 9ÐL= Ñ Ÿ 9ÐOÑ

Ê

L= Ÿ O

Also, since any subgroup O Ÿ K of order less than := is contained in some subgroup of order := , we have 9ÐOÑ Ÿ 9ÐL= Ñ

Ê

O Ÿ L=

Thus, all subgroups of K either contain L= or are contained in L= . In this sense, L= forms a bottleneck in the lattice of subgroups of K. It follows that L= is cyclic, for if + Â L= , then Ø+Ù Ÿ y L= and so L= Ÿ Ø+Ù, whence L= is cyclic. Since L= is cyclic, the subgroup lattice of K has the form shown in Figure 7.1 and so K contains exactly one subgroup of each order :. with ! Ÿ . Ÿ =. In particular, K has a unique subgroup of order :. Thus, we have shown that K has a unique subgroup of some order := , where : Ÿ :=  9ÐKÑ if and only if K has a unique subgroup of order :.

G ...

...

Hs

{1} Figure 7.1

224


Let K be a noncyclic group of order :8  ". To show that K has more than one subgroup of order :, it suffices to show K contains a nontrivial subgroup E as well as an element of order : that is not in E. We first consider the case :  #. Theorem 7.16 Let :  # be prime and let 9ÐKÑ œ :8  ". 1) If K is noncyclic and if + − K is an element of maximum order, then K has an element of order : that is not contained in Ø+Ù. 2) If K has a unique subgroup of order := for some " Ÿ =  8, then K is cyclic. Proof. We have already seen that part 2) follows from part 1). To prove part 1), assume that K is noncyclic. Let E œ Ø+Ù, where + − K has maximum order :7 . If 7 œ ", then all nonidentity elements of K have order : and so we may assume that # Ÿ 7  8. Let E – F where 9ÐFÑ œ :7" . If , − F Ï E, then , : − E and so , : œ +> for some > !. If > œ !, then 9Ð,Ñ œ : and we are done, so let us assume that >  !. Since +> œ , : does not have maximum order in K , it follows that : ± > and so , : œ +?: for some !  ?  :7" . Now, if , commutes with +, then ,+? Â E has order :. Assume now that no , − F Ï E commutes with +. We wish to show that there is still an integer B for which 9Ð,+B Ñ œ :. If , − F Ï E, then to get a formula for Ð,+B Ñ: , we need a commutativity rule for + and , . But since E – F , there is an α Á " for which +, œ +α and since , : commutes with +, Theorem 7.15 implies that +Ø,Ù œ Ö+"5:

7"

± 5 œ "ß á ß :×

Moreover, , Â E implies that , 3 − F Ï E for all " Ÿ 3  : and so we may replace , by an appropriate power of , so that α œ "  : 7" . Now, ,+ œ +α , and so ,+B œ +αB , Then an easy induction shows that for 5 ", #

5

Ð,+B Ñ5 œ +BÐαα âα Ñ , 5 and so #

:

#

:

Ð,+B Ñ: œ +BÐαα âα Ñ , : œ +?:BÐαα âα Ñ Hence, we want an integer B for which


Bα

"  α: ´ ?: "α

225

Ðmod :7 Ñ

Since α œ "  :7" , Theorem 1.18 implies that α: œ Ð"  :7" Ñ: œ "  A:7 where : y± A and so Bα

"  α: œ BÐ"  :7" ÑA: ´ BA: "α

Ðmod :7 Ñ

Hence, we want an integer B for which BA ´ ?

Ðmod :7" Ñ

But A is invertible in ™‡:7" and so we may take B œ ?A" . Now we turn to the case : œ #. (The reader may wish to skip the proof upon first reading.) Theorem 7.17 Let K be a nontrivial group of order #8 . 1) K has a unique subgroup of order # if and only if K is cyclic or a generalized quaternion group. 2) If K has a unique subgroup of order #= for some "  =  8, then K is cyclic. Proof. We have already seen that part 2) follows from part 1). Let K be a noncyclic group of order #8 with a unique involution. We will show that K is a generalized quaternion group. Let + − K be an element of maximum order #7 and let E œ Ø+Ù. Clearly, we may assume that 7 #. In one case we will need to be a bit more specific about the choice of the cyclic subgroup E. Namely, if 7 œ #, then since the unique subgroup of order # is normal in K, it has a #-cover R of order %, which is cyclic since the %-group has two involutions. In this case, we let E œ R and so E ü K. Thus, if 7 œ #, we may assume that E – K. If F œ Ø,ß EÙ is a #-cover of E, then 9ÐFÎEÑ œ # and so , # − E , which implies that , # œ +5 for some 5  !. But 9Ð+5 Ñ œ 9Ð, # Ñ  9Ð,Ñ Ÿ 9Ð+Ñ and so # ± 5 , that is, >

, # œ +# ? for > " and ? odd. Since Ø+? Ù œ E, we can rename +? to + to get , # œ +# >

>

for > ". Since 9Ð, # Ñ œ 9Ð+# Ñ œ #7> , we also have

226


9Ð,Ñ œ #7>" where > Ÿ 7  " since 9Ð,Ñ  #. >"

If , commutes with +, then ,+# Â E is an involution, contrary to assumption. Thus, , does not commute with +; in fact, E is not properly contained in any abelian subgroup of K. However, since E – F and , # − E, Theorem 7.15 implies that +Ø,Ù œ Ö+ß +"#

7"

or +Ø,Ù œ Ö+ß +" ×

×

and so for any #-cover F œ Ø,ß EÙ of E, there is an + − E for which ,+, " œ +α

>

and , # œ +#

where either α œ "  #7" or α œ ". Conjugating the second equation by , and using the first equation gives >

>

,# œ ,+# , " œ +α# œ +# >

>

>

>+1

and so +# œ +# , that is, +# œ ". Hence, #7 ± #>" , which implies that 7 Ÿ >  " Ÿ 7, that is, > œ 7  ". Now, if for any #-cover F œ Ø,ß EÙ, we have α œ "  #7" , then α œ "  #> and so >

,+, " œ +"# œ , # +" which implies that +, " Â E is an involution, a contradiction. Hence, for all #covers F œ Ø,ß EÙ of E, we have α œ " and ,+, " œ +"

and

7"

, # œ +#

It follows that 9Ð,Ñ œ % and so F can be described as follows: F œ Ø+ß ,Ùß 9Ð+Ñ œ #7 ß 9Ð,Ñ œ %ß , # œ +#

7"

ß ,+, " œ +"

that is, F œ U7" is generalized quaternion and so every element of F Ï E has order % and if 7 $, then E « F . We want to show that F œ K. If not, then F has a #-cover G , that is, E–F–G ŸK where 9ÐGÑ œ :72 . Recall that if 7 œ #, then we have chosen E so that E ü K and if 7 $, then E « F and so in either case, E – G .


227

The quotient group GÎE is either G% Ð-EÑ or G# Ð-EÑ  G# Ð.EÑ. In the latter case, the subgroups Ø-ß EÙ and Ø.ß EÙ are #-covers of E and so +- œ +" œ +. Hence, +-. œ + which implies that E  ØEß -.Ù is abelian, a contradiction. Hence, GÎE œ G% Ð-EÑ. But then Ø- # ß EÙ is a #-cover of E and so #

+- œ +" However, since E is not properly contained in an abelian subgroup of K, the smallest power of - that commutes with + is - % − E and so Theorem 7.0 (where . œ #) implies that +" Â +Ø-Ù , a contradiction. Thus K œ F œ U7" is generalized quaternion of order #7 .

*Groups of Order : 8 With an Element of Order :8" We can use the previous result to take a close look at nonabelian groups K of order :8 that have an element of order : 8" . We will restrict attention to the case :  #. Let 9Ð+Ñ œ :8" and E œ Ø+Ù. Theorem 7.16 implies that there is a , − K Ï E with 9Ð,Ñ œ :. Then K œ Ø+Ù z Ø,Ù and it remains to see how + and , interact. Since E ü K, we have , " +, œ +5 for some 5  " and Theorem 7.15 implies that the conjugates of + by Ø,Ù are +Ø,Ù œ Ö+"5:

8#

± 5 œ "ß á ß :×

Since any nonidentity element of Ø,Ù generates Ø,Ù, we can take 5 œ " and write ,+, " œ +":

8#

Theorem 7.18 Let :  # be a prime. Let K be a nonabelian group of order :8 with an element + of order :8" . Then K œ Ø+Ù z Ø,Ù where 9Ð,Ñ œ : and ,+, " œ +":

8#

To see that such a group exists, recall from Example 5.30 that there is a semidirect product

228


G:8" Ð+Ñ z ) G: Ð,Ñ where ), Ð+Ñ œ +":

8#

*Groups of Order : $ We have seen that groups of order : are cyclic and that groups of order :# are either cyclic or the direct product of two cyclic subgroups of order : (Corollary 7.9). Theorem 7.18 gives us insight into groups of order :$ . If : œ #, we have seen that, up to isomorphism, the groups of order :$ œ ) are 1) 2) 3) 4) 5)

G) G% } G# G# } G# } G# U, the (nonabelian) quaternion group H) , the (nonabelian) dihedral group

More generally, we will show that for any prime :, the groups of order :$ are (up to isomorphism): 1) 2) 3) 4) 5)

G:$ G:# } G: G: } G: } G: Y X Ð$ß ™: Ñ, the unitriangular matrix group (described below) The group K œ Ø+ß ,Ù where K œ Ø+ß ,Ùß 9Ð+Ñ œ : # ß 9Ð,Ñ œ :ß ,+, " œ +":

Thus, there are only two nonabelian groups of order :$ (up to isomorphism). We will leave analysis of the abelian groups of order :$ to a later chapter, where we will prove that any finite abelian group is the direct product of cyclic groups. So let :  # be prime and let K be a nonabelian group of order :$ . If K has an element + of order :# , then Theorem 7.18 implies that K œ Ø+Ù z Ø,Ù where 9Ð,Ñ œ : and , " +, œ +": It remains to consider the case where K has exponent :. The center ^ œ ^ÐKÑ is nontrivial but cannot have order :# , since then KÎ^ is cyclic and so K is abelian. Hence, 9Ð^Ñ œ : and so 9ÐKÎ^Ñ œ :# . Hence, KÎ^ is abelian with exponent : and so


229

KÎ^ œ Ø+^Ù  Ø,^Ù Moreover, since D ³ Ò,ß +Ó − ^ , we have ^ œ ØDÙ. Hence, K œ Ø+ß ,ß DÙß 9Ð+Ñ œ 9Ð,Ñ œ 9ÐDÑ œ "ß D − ^ÐKÑß ,+ œ D+, To see that this does describe a group, we have the following. Definition Let V be a commutative ring with identity. A matrix Q − KPÐ8ß VÑ is unitriangular if it is upper triangular (has !'s below the main diagonal) and has "'s on the main diagonal. We denote the set of all unitriangular matrices by Y X Ð8ß VÑ. We will leave it as an exercise to show that

kY X Ð8ß ™: Ñk œ :Ð8 8ÑÎ# #

and that for :  #, the group Y X Ð$ß ™: Ñ has order :$ and exponent :. Also, Y X Ð$ß ™# Ñ ¸ U.

Exercises 1.

2.

3. 4. 5. 6.

A left action of K on \ is sometimes defined as a map from the cartesian product K ‚ \ to \ , sending Ð+ß BÑ to an element +B − \ , satisfying a) "B œ B for all B − \ b) Ð+,ÑB œ +Ð,BÑ for all B − \ , +, , − K . A right action of K on \ is a map from the cartesian product \ ‚ K to \ , sending ÐBß +Ñ to an element B+ − \ , satisfying c) B" œ B for all B − \ d) BÐ+,Ñ œ ÐB+Ñ, for all B − \ , +, , − K . Given a left action, show that the map ÐBß 1Ñ œ 1" B is a right action. What about ÐBß 1Ñ œ 1B? Let -À K Ä W\ be an action of K on \ . a) Prove that - is regular if and only if it is transitive and stabÐBÑ œ Ö"× for some B − \ . b) Prove that - is regular if and only if it is transitive and for all distinct 1ß 2 − K, we have 1B Á 2B for all B − \ . c) Prove that if - is faithful and transitive and if K is abelian, then the action is regular. Let K be a finite group and let : be the smallest prime dividing 9ÐKÑ. Prove that any normal subgroup of order : is central. Let K be an infinite group. Use normal interiors (not Poincaré's theorem) to prove that if L and O have finite index in K, then so does L ∩ O . Show that the condition that K be finitely generated cannot be removed from the hypotheses of Theorem 7.5. Let K be a finite simple group and let L Ÿ K have prime index ÐK À LÑ œ :. Prove that : must be the largest prime dividing 9ÐKÑ and that :# does not divide 9ÐKÑ.

230


Let K be a finite group. Prove that a transitive action of K on \ is regular if and only if kKk œ k\ k. 8. Let 9ÐKÑ œ #8 where 8 " is odd. Let + − K have order #. Show that under the left regular representation of K on itself, the element + corresponds to an odd permutation. Show that K is not simple. 9. a) Prove that if K is a finitely generated infinite group and L is a subgroup of finite index in K, then K has a characteristic subgroup O of finite index for which O Ÿ L . b) Show that the condition that K be finitely generated is necessary. 10. The action of a group K on a set \ is #-transitive if for any pairs ÐBß CÑß Ð?ß @Ñ − \ ‚ \ where B Á C and ? Á @, there is an + − K for which +B œ ? and +C œ @. Prove that for a #-transitive action, the stabilizer stabÐBÑ is a maximal subgroup of K for all B − \ . 7.

Equivalence of Actions Two group actions -À K Ä W\ and .À L Ä W] are equivalent if there is a pair Ðαß 0 Ñ where αÀ K Ä L is a group isomorphism and 0 À \ Ä ] is a bijection satisfying the condition 0 Ð1BÑ œ Ðα1ÑÐ0 BÑ In this case, we refer to Ðαß 0 Ñ as an equivalence from - to .. 11. a) Show that the inverse of an equivalence is an equivalence. b) Show that the (coordinatewise) composition of two “compatible” equivalences is an equivalence. 12. Let -À K Ä W\ be a transitive action and let B − \ . Show that - is equivalent to the action of left-translation by K on KÎstabÐBÑ. 13. Suppose that -À K Ä W\ and .À L Ä W] are equivalent transitive actions, under the equivalence Ðαß 0 Ñ. Prove that stabÐBÑ ¸ LC for any B − \ , where C œ 0 B.

Conjugacy 14. Let K be a group and let 1 − K. Show that Ø1K Ù is a normal subgroup of K. 15. Let K be a finite group and let 1 − K . Show that kGK Ð1Ñk kKÎK w k where Kw is the commutator subgroup of K. 16. Let K be a finite group and let L Ÿ K with ÒK À LÓ œ #. Suppose that GK Ð2Ñ Ÿ L for all 2 − L . Prove that K Ï L is a conjugacy class of K. 17. Let K be a :-group and let L Ÿ K be a nonnormal subgroup of K and let + − K. Show that the number of conjugates of L that are fixed by every element of L + is positive and divisible by :. 18. a) Let K be a finite group and let L ü K. Show that 5ÐKÎLÑ œ 5ÐKÑ  5K ÐLÑ  " where 5K ÐLÑ is the number of K-conjugacy classes of L .


231

b) Let K be a finite nonabelian group such that KÎ^ÐKÑ is abelian. Show that 5ÐKÑ kKÎ^ÐKÑk  k^ÐKÑk  "

19. a)

Find all finite groups (up to isomorphism) that have exactly one conjugacy class. b) Find all finite groups (up to isomorphism) that have exactly two conjugacy classes. c) Find all finite groups (up to isomorphism) that have exactly three conjugacy classes. 20. a) If ; −  and 8  !, show that there are only finitely many solutions 5" ß á ß 58 in positive integers to the equation ;œ

" " â 5" 58

Hint: Use induction on 8. Look at the smallest denominator first. b) Show that for any integer 8  !, there are only finitely many finite groups (up to isomorphism) that have exactly 8 conjugacy classes. Hint: Use the class equation. 21. a) Let L be a proper subgroup of a finite group K. Show that the set W œ . L1 1−K

is a proper subset of K. b) If L is a proper subgroup of a group K and ÐK À LÑ  ∞, then the set W œ . L1 1−K

is a proper subset of K. 22. Let \ be a conjugacy class of K and let \ " œ ÖB" ± B − \×. a) Show that \ " is also a conjugacy class of K. b) Show that if K has odd order, then \ œ Ö"× is the only conjugacy class for which \ œ \ " . c) Show that if K has even order, then there is a conjugacy class \ other than Ö"× for which \ œ \ " . d) Show that if K is finite and 5ÐKÑ is even, then 9ÐKÑ is even. Show by example that the converse does not hold. 23. Let K be a group of order #7. Suppose that K has a conjugacy class of size 7. Prove that 7 is odd, and that K has an abelian normal subgroup of size 7. 24. Let L be normal in K and suppose that ÐK À LÑ œ : is a prime. Let B − L have the property that there is a 1 − K Ï L such that 1B œ B1. a) Show that kGK ÐBÑk œ :kGL ÐBÑk. b) Show that BL œ BK .

232


:-Groups and : -Subgroups 25. a)

26. 27. 28. 29. 30. 31.

32.

Let 0 À K Ä L be a group homomorphism. If K is a :-group, under what conditions, if any, is L a :-group? b) Let 0 À K Ä L be a group homomorphism. If L is a :-group, under what conditions, if any, is L a :-group? c) Let L ü K. If L and KÎL are both :-groups, under what conditions, if any, is K a :-group? Let K be a finite :-group. a) Prove that any cover of L  K has index :. b) Prove that a cover of the center ^ÐKÑ is abelian. Let L Ÿ K. Prove that K is a :-group if and only if L and KÎL are :groups. Let K be a finite simple nonabelian group. Show that 9ÐKÑ is divisible by at least two distinct prime numbers. Prove that the derived group Kw of a :-group K is a proper subgroup of K. Let K be a :-group. Show that if L Ÿ K and ÐK À LÑ  ∞, then ÐK À LÑ is a power of :. Let K œ  K: be a direct product of :-subgroups for distinct primes :. Show that if L Ÿ K, then L œ  ÐL ∩ K: Ñ. What if the primes are not distinct? Show that the generalized quaternion group U7 œ Ø+ß ,Ùß 9Ð+Ñ œ #7" ß 9Ð,Ñ œ %ß , # œ +#

7#

ß ,+, " œ +"

has only is single involution. 33. Prove that a finite :-group has the normalizer condition using the action of RK ÐLÑ on the conjugates conjK ÐLÑ of L by conjugation. 34. Let K be a nonabelian group of order :$ , where : is a prime. Determine the number 5ÐKÑ of conjugacy classes of K.

Additional Problems 35. Let : be a prime. a) Show that

kKPÐ8ß ™: Ñk œ Ð:8  "ÑÐ: 8  :ÑâÐ: 8  : 8" Ñ

b) Show that Y X Ð8ß ™: Ñ is a :-group, in fact,

kY X Ð8ß ™: Ñk œ :Ð8 8ÑÎ# #

c) Show that Y X Ð8ß ™: Ñ is a Sylow :-subgroup of KPÐ8ß ™: Ñ. d) For 8 œ # or 8  " Ÿ :, show that Y X Ð8ß ™: Ñ has exponent :. e) Show that Y X Ð$ß ™# Ñ ¸ U. 36. Let J be a finite field of size ; and let Z be an 8-dimensional vector space over J . Show that the number of subspaces of Z of dimension 5 is


233

8 Ð; 8  "ÑÐ; 8  ;ÑâÐ; 8  ; 5" Ñ Š ‹ ³ 5 5 ; Ð;  "ÑÐ; 5  ;ÑâÐ; 5  ; 5" Ñ The expressions Ð 85 Ñ; are called Gaussian coefficients. Hint: Show that the number of 5 -tuples of linearly independent vectors in Z is Ð; 8  "ÑÐ; 8  ;ÑâÐ; 8  ; 5" Ñ 37. Let Lß O Ÿ K. a) Show that the distinct double cosets E1F , where + − K , form a partition of K. b) Show that kE1F k œ ÐE À F 1 ∩ EÑ.

Chapter 8

Sylow Theory

In 1872, the Norwegian mathematician Peter Ludwig Mejdell Sylow [32] published a set of theorems which are now known as the Sylow theorems. These important theorems describe the nature of maximal : -subgroups of a finite group, which are now called Sylow :-subgroups. (For convenience, we will collect the Sylow theorems into a single theorem.)

Sylow Subgroups We begin with the definition of a Sylow subgroup. Definition Let K be a group and let : be a prime. A Sylow :-subgroup of K is a maximal :-subgroup of K (under set inclusion). The set of all Sylow :subgroups of K is denoted by Syl: ÐKÑ. The number of Sylow :-subgroups of a group K is denoted by 8: ÐKÑ, or just 8: when the context is clear. Of course, if a prime : divides 9ÐKÑ, then K contains a Sylow :-subgroup; in fact, every :-subgroup of K is contained in a Sylow :-subgroup. Also, if K is an infinite group and if L is a :-subgroup of K, then an appeal to Zorn's lemma shows that K has a Sylow :-subgroup containing L . Since conjugation is an order isomorphism and also preserves the group order of elements, it follows that if W is a Sylow : -subgroup of K, then so is every conjugate W + of W . Note also that if K is finite and 9ÐKÑ œ :8 7 where Ð:ß 7Ñ œ ", then any subgroup of order :8 is a Sylow :-subgroup. We will prove the converse of this a bit later: Any Sylow subgroup of K has order :8 .

The Normalizer of a Sylow Subgroup Let K be a finite group. If a Sylow :-subgroup W of K happens to be normal in K, then KÎW has no nonidentity :-elements. Hence, : y± ÐK À WÑ and so W is the

S. Roman, Fundamentals of Group Theory: An Advanced Approach, DOI 10.1007/978-0-8176-8301-6_8, © Springer Science+Business Media, LLC 2012

235

236


set of all :-elements of K . It also follows that W « K , since automorphisms preserve order. Of course, W is always normal in its normalizer RK ÐWÑ. Theorem 8.1 Let K be a finite group and let W − Syl: ÐKÑ. 1) W is the set of all :-elements of RK ÐWÑ. 2) Any :-element + − K Ï W moves W by conjugation, that is, W + Á W . 3) W is the only Sylow :-subgroup of RK ÐWÑ. 4) : y± ÐRK ÐWÑ À WÑ. 5) W « RK ÐWÑ. If W − Syl: ÐKÑ, then W + Ÿ RK ÐWÑ+ for any + − K. Hence, if + normalizes RK ÐWÑ, then W + Ÿ RK ÐWÑ and since W + is also a Sylow :-subgroup of RK ÐWÑ, Theorem 8.1 implies that W + œ W . In other words, if + normalizes RK ÐWÑ, then + also normalizes W and so RK ÐRK ÐWÑÑ œ RK ÐWÑ Theorem 8.2 The normalizer RK ÐWÑ of a Sylow subgroup of K is selfnormalizing, that is, RK ÐRK ÐWÑÑ œ RK ÐWÑ

Soon we will be able to prove that not only is RK ÐWÑ self-normalizing, but so is any subgroup of K containing RK ÐWÑ.

The Sylow Theorems Let K be a finite group and let W − Syl: ÐKÑ. The fact that any : -element + Â W moves W by conjugation prompts us to look at the action of a :-subgroup O of K by conjugation on the set conjK ÐWÑ œ ÖW + ± + − K× of conjugates of W in K. As to the stabilizer of W + , we have stabÐW + Ñ œ RK ÐW + Ñ ∩ O œ W + ∩ O and so

korbO ÐW + Ñk œ ÐO À W + ∩ OÑ

which is divisible by : unless O Ÿ W + , in which case the orbit has size ".

Sylow Theory

237

Hence, FixconjK ÐWÑ ÐOÑ œ ÖW + ± O Ÿ W + × and so

kconjK ÐWÑk ´ kÖW + ± O Ÿ W + ×k mod :

Now if O is a Sylow :-subgroup of K, then O Ÿ W + if and only if O œ W + and so kconjK ÐWÑk ´ œ

" mod : ! mod :

if O − conjK ÐWÑ if O Â conjK ÐWÑ

It follows that O Â conjK ÐWÑ is impossible and so Syl: ÐKÑ œ conjK ÐWÑ is a conjugacy class and 8: ´ " mod : Note also that

8: œ kconjK ÐWÑk œ ÐK À RK ÐWÑÑ ± 9ÐKÑ

Finally, we can determine the order of a Sylow : -subgroup W , since ÐK À WÑ œ ÐK À RK ÐWÑÑÐRK ÐWÑ À WÑ and neither of the factors on the right is divisible by :. Hence, the order of W is the largest power of : dividing 9ÐKÑ. We have proved the Sylow theorems. Theorem 8.3 (The Sylow theorems [32], 1872) Let K be a finite group and let 9ÐKÑ œ :8 7, where : is a prime and : y± 7. 1) The Sylow :-subgroups of K are the subgroups of K of order :8 . 2) Syl: ÐKÑ is a conjugacy class in subÐKÑ. 3) The number 8: of Sylow :-subgroups satisfies 8: ´ " mod :

and

8: œ ÐK À RK ÐWÑÑ ± 9ÐKÑ

where W − Syl: ÐKÑ. 4) Let W − Syl: ÐKÑ. a) W is normal if and only if 8: œ ". b) W is self-normalizing if and only if 8: œ ÐK À WÑ œ 7, in which case all Sylow :-subgroups of K are self-normalizing. 5) If O is a :-subgroup of K, then kÖW − Syl: ÐKÑ ± O Ÿ W×k ´ " mod :

We will prove later in the chapter that every normal Sylow : -subgroup of a finite group is complemented. This is implied by the famous Schur–Zassenhaus theorem. However, we also have the following simple consequence of Theorem 3.1 concerning supplements of Sylow subgroups.


238

Theorem 8.4 Let K be a finite group. Then any Sylow : -subgroup of K and any subgroup whose index is a power of : are supplements.

Sylow Subgroups of Subgroups Let K be a finite group and let L Ÿ K. We wish to explore the relationship between Syl: ÐKÑ and Syl: ÐLÑ. On the one hand, every W − Syl: ÐLÑ is contained in a X − Syl: ÐKÑ and so the set Syl: ÐWà KÑ œ ÖX − Syl: ÐKÑ ± W Ÿ X × is nonempty. Moreover, since E − Syl: ÐWà KÑ implies that E ∩ L is a :subgroup of L containing W , we have E − Syl: ÐWà KÑ

Ê

E∩L œW

In particular, the families Syl: ÐWà KÑ are disjoint, that is, W Á X − Syl: ÐLÑ

Ê

Syl: ÐWà KÑ ∩ Syl: ÐX à KÑ œ g

and so 8: ÐLÑ Ÿ 8: ÐKÑ On the other hand, if W − Syl: ÐKÑ, then the intersection W ∩ L need not be a Sylow :-subgroup of L , as can be seen by taking L and W to be distinct Sylow :-subgroups of K. However, if LW is a subgroup of K, then 9ÐWÑ ± 9ÐLWÑ and so kL ∩ W k

kLW k œ kL k kW k

where kLW kÎkW k is not divisible by :. Hence, kL ∩ W k and kL k are divisible by the same powers of : and so L ∩ W − Syl: ÐLÑ. Theorem 8.5 Let K be a finite group and let L Ÿ K . 1) If W − Syl: ÐLÑ, then E − Syl: ÐWà KÑ

Ê

E∩L œW

and W Á X − Syl: ÐLÑ

Ê

Syl: ÐWà KÑ ∩ Syl: ÐX à KÑ œ g

and so 8: ÐLÑ Ÿ 8: ÐKÑ

Sylow Theory

239

2) If W − Syl: ÐKÑ and LW Ÿ K, then W ∩ L − Syl: ÐLÑ

Some Consequences of the Sylow Theorems Let us consider some of the more-or-less direct consequences of the Sylow theorems.

A Partial Converse of Lagrange's Theorem A Sylow :-subgroup W of a group K has subgroups of all orders dividing 9ÐWÑ. This gives a partial converse to Lagrange's theorem. Theorem 8.6 Let K be a finite group and let : be a prime. If :5 ± 9ÐKÑ, then K has a subgroup of order :5 .

More on the Normalizer of a Sylow Subgroup Recall that the normalizer RK ÐWÑ of a Sylow subgroup W of K is selfnormalizing. Now we can say more. Theorem 8.7 Let K be a finite group and let W − Syl: ÐKÑ. If W Ÿ RK ÐWÑ Ÿ L Ÿ K then L is self-normalizing. In particular, if L  K , then L is not normal in K . Proof. Conjugating by any + − RK ÐLÑ gives W + Ÿ RK ÐWÑ+ Ÿ L Ÿ K and so both W and W + are Sylow :-subgroups of L . It follows that W and W + are conjugate in L . Hence, there is an 2 − L for which W 2+ œ W , that is, 2+ − RK ÐWÑ Ÿ L . Thus, + − L and so RK ÐLÑ œ L . The normalizer of a Sylow :-subgroup has a somewhat stronger property than is expressed in Theorem 8.7. In the exercises, we ask the reader to prove that RK ÐWÑ is abnormal.

Counting Subgroups in a Finite Group In an earlier chapter, we proved that if K is a :-group and :5 ± 9ÐKÑ, then the number 8:ß5 ÐKÑ of subgroups of K of order :5 satisfies 8:ß5 ÐKÑ ´ " mod : We have just proved that for any finite group K for which : ± 9ÐKÑ, 8: ÐKÑ ´ " mod : To see that (8.8) holds for all finite groups, we count the size of the set

(8.8)


240

Y5 œ ÖÐLß WÑ ± L Ÿ Wß W − Syl: ÐKÑß 9ÐLÑ œ :5 × modulo :. On the one hand, for each W − Syl: ÐKÑ, there are 8:ß5 ÐWÑ ´ " subgroups of W of order :5 and so kY5 k ´ 8: ÐKÑ † " ´ "

On the other hand, for each L Ÿ K of order :5 , Theorem 8.3 implies that kÖW − Syl: ÐKÑ ± L Ÿ W×k ´ "

and so

kY5 k ´ 8:ß5 ÐKÑ † " ´ 8:ß5 ÐKÑ

Hence, 8:ß5 ÐKÑ ´ " and we have proved an important theorem of Frobenius. Theorem 8.9 (Frobenius [13], 1895) Let K be a group with 9ÐKÑ œ :8 7 where Ð7ß :Ñ œ ". Then for each " Ÿ 5 Ÿ 8, the number 8:ß5 ÐKÑ of subgroups of K of order :5 satisfies

8:ß5 ÐKÑ ´ " mod :

When All Sylow Subgroups Are Normal Several good things happen when all of the Sylow subgroups of a group K are normal. In particular, let K be a finite group. In an earlier chapter (see Theorem 4.22 and Theorem 4.35), we showed that among the conditions: 1) 2) 3) 4)

Every subgroup of K is subnormal K has the normalizer condition Every maximal subgroup of K is normal KÎFÐKÑ is abelian

the following implications hold: 1) Í 2) Ê 3) Í 4) We also promised to show that these four conditions are equivalent, which we can do now, adding several additional equivalent conditions into the bargin. First, let us speak about arbitrary (possibly infinite) groups. If K is a group, let Ktor denote the set of all torsion (periodic) elements of K. If K is abelian, then Ktor is a subgroup of K. However, in the nonabelian general linear group KPÐ#ß ‚Ñ, the elements EœŒ

! "

" !

and F œ Œ

! "

" "

Sylow Theory

241

are torsion but their product is not. Hence, Ktor is not always a subgroup of K. Note that when Ktor is a subgroup of K, then Ktor « K since automorphisms preserve order. Theorem 8.10 Let K be a group in which every Sylow subgroup is normal. Let the Sylow subgroups of K be Ö]: ± : − c ×. Then Ktor œ  ]: :−c

and so Ktor Ÿ K. Thus, the product of two elements of finite order has finite order. Proof. Since the Sylow :-subgroups are normal and pairwise essentially disjoint, they commute elementwise. In particular, if +" ß á ß +8 come from distinct Sylow subgroups, then 9Ð+" â+8 Ñ œ 9Ð+" Ñâ9Ð+8 Ñ and so the family of Sylow subgroups is strongly disjoint and ] ³  ]: © Ktor :−c

/7 For the reverse inclusion, if + − Ktor has order 8 œ :"/" â:7 where the primes :3 are distinct, then Corollary 2.11 implies that + œ +" â+7 , where 9Ð+5 Ñ œ :5/5 and so +5 − ]:5 , whence + − ] .

Now we turn to finite groups in which all Sylow subgroups are normal. Theorem 8.11 Let K be a finite group, with Sylow subgroups Ö]: ± : − c × . The following are equivalent: 1) Every Sylow subgroup of K is normal. 2) K is the direct product of its Sylow :-subgroups K œ  ]: :−c

3) If L Ÿ K, then L œ  ÐL ∩ ]: Ñ :−c

4) K is the direct product of :-subgroups. 5) (Strong converse of Lagrange's theorem) If 8 ± 9ÐKÑ, then K has a normal subgroup of order 8. 6) Every subgroup of K is subnormal. 7) K has the normalizer condition. 8) Every maximal subgroup of K is normal. 9) KÎFÐKÑ is abelian. If these conditions hold, then K has the center-intersection property. In particular, ^ÐKÑ is nontrivial.

242


Proof. Theorem 8.10 shows that 1) implies 2) and the converse is clear. If 1) holds, then L]: Ÿ K and so the Sylow :-subgroups of L are ÖL ∩ ]: ± : − c ×. Moreover, since L ∩ ]: ü L , it follows that L is the direct product of its Sylow :-subgroups and so 3) holds. It is clear that 3) implies 2) and so 1)–3) are equivalent. Also, it is clear that 2) Ê 4). If 4) holds and : is a prime dividing 9ÐKÑ, then we can isolate the factors in the direct product of K that have exponent :, say KœT U where T is a direct product of :-subgroups and U is a direct product of ; subgroups for various primes ; Á :. Then T is the set of all :-elements of K and so T a Sylow :-subgroup of K. But T ü K and so 1) holds. Thus, 1)–4) are equivalent. It is clear that 5) Ê 1). To see that 2) Ê 5), any divisor 8 of 9ÐKÑ is a product 8 œ #.: where .: ± 9Ð]: Ñ and since ]: has a normal subgroup of order .: , the direct product of these subgroups is a normal subgroup of K of order 8. Thus, 1)–5) are equivalent and we have already remarked that 6) Í 7) Ê 8) Í 9) To see that 3) Ê 6), if L  K, then one of the factors L ∩ ]: is proper in ]: and so there is a subgroup R: for which L ∩ ]: – R: Ÿ ] : Hence, L –  ÐL ∩ ]; Ñ  R: ;Á:

Since L is an arbitrary proper subgroup of K, it follows that all subgroup of K are subnormal. To see that 6) Ê 1), if ]: is not normal in K, then since RK Ð]: Ñ  K is subnormal, there is a subgroup L: for which RK Ð]: Ñ – L: Ÿ K. But this contradicts the fact that RK Ð]: Ñ is self-normalizing. Hence, ]: – K. Thus, 1)–7) are equivalent and imply 8). Similarly, if 8) holds but ]: is not normal in K, then there is a maximal subgroup Q Ÿ K for which ]: Ÿ RK Ð]: Ñ Ÿ Q – K which contradicts Theorem 8.7. Hence, ]: ü K and 1) holds. Finally, if 3) holds and R Ÿ K is nontrivial, then R ∩ ]; ü ]; is nontrivial for some ; − c and R ∩ ^ÐKÑ œ  ÐR ∩ ^ÐKÑ ∩ ]: Ñ :−c

Sylow Theory But R ∩ ^ÐKÑ ∩ ]; œ R ∩ ^Ð]; Ñ R ∩ ^ÐKÑ.

is

nontrivial

and

therefore

243 so

is

The hypotheses of the previous theorems hold for all abelian groups. Corollary 8.12 Let K be an abelian group. 1) (Primary decomposition) Then Ktor is the direct product of its Sylow :subgroups. 2) (Converse of Lagrange's theorem) If K is finite and 8 ± 9ÐKÑ, then K has a subgroup of order 8. We will add one additional characterization (nilpotence) to Theorem 8.11 in a later chapter (see Theorem 11.8).

When a Subgroup Acts Transitively; The Frattini Argument The Frattini argument (Theorem 7.2) shows that if a group K acts on a nonempty set \ and if L Ÿ K is transitive on \ , then K œ L stabK ÐBÑ and if L is regular on \ , then K œ L ì stabK ÐBÑ and so stabK ÐBÑ is a complement of L in K. To apply this idea, let K be a finite group and let WŸLüK where W − Syl: ÐLÑ. Let K act on conjK ÐWÑ by conjugation. Since W + − Syl: ÐLÑ for any + − K, it follows that L acts transitively on conjK ÐWÑ. Hence, K œ L stabK ÐWÑ œ LRK ÐWÑ This specific argument is also referred to as the Frattini argument. Theorem 8.13 Let K be a finite group and let L Ÿ K . If W − Syl: ÐLÑ , then K œ LRK ÐWÑ and if the action of L by conjugation on conjK ÐWÑ is regular, then K œ L ì RK ÐWÑ

This theorem can be used to show that the Frattini subgroup of a finite group K has the property that all of its Sylow subgroups are normal in K.

244


Theorem 8.14 (Frattini [12], 1885) If K is a finite group, then the Frattini subgroup FÐKÑ has the property that all of its Sylow subgroups are normal in K. Proof. If W − Syl: ÐFÑ, then W Ÿ F ü K and the Frattini argument shows that K œ FRK ÐWÑ But if RK ÐWÑ  K, then there is a maximal subgroup Q of K for which RK ÐWÑ Ÿ Q and so K Ÿ Q , a contradiction. Hence, RK ÐWÑ œ K and W ü K.

The Search for Simplicity The Sylow theorems, along with group actions and counting arguments, provide powerful tools for the analysis of finite groups. A key issue with respect to finite groups is the question of simplicity. As we will discuss in a later chapter, the issue of which finite groups (up to isomorphism) are simple appears to be resolved, but the resolution is so complex that some mathematicians may still have questions regarding its completeness and its accuracy. We have seen that a group of prime-power order :8 has a normal subgroup of each order :5 ± :8 . Accordingly, we will do no further direct analysis of :groups in this chapter. Throughout our discussion, : will denote a prime, ]: will denote an arbitrary Sylow :-subgroup and, as always, 8: denotes the number of Sylow :-subgroups of K. Recall that 1) 8: œ "  5: for some integer 5 !. 2) 8: œ ÐK À RK Ð]: ÑÑ ± 9ÐKÑ. Note that if 9ÐKÑ œ :8 7, where : y± 7, then 8: ± 9ÐKÑ if and only if 8: ± 7. The following facts (among others) are useful in showing that a group is not simple: 3) 4) 5) 6)

]: is normal if and only if 8: œ ". If ÐK À LÑ is equal to the smallest prime dividing 9ÐKÑ, then L – K. The kernel of any action -À K Ä W\ is a normal subgroup of K. If L  K, then ÐK À L ‰ Ñ ± ÐK À LÑx. Hence, if 9ÐKÑ y± ÐK À LÑx, then L ‰ is a nontrivial proper normal subgroup of K.

We will also have use for the fact that if : is prime and " Ÿ /  : , then /: is the smallest integer for which :/ ± Ð/:Ñx.

Sylow Theory

245

The 8: -Argument It happens quite often that for some odd prime : ± 9ÐKÑ, the integers "  5: do not divide 9ÐKÑ unless 5 œ !, in which case 8: œ " and ]: ü K . Let us refer to the argument 8: œ "  5: ± 9ÐKÑ

Ê

5œ!

as the 8: -argument. Note that the 8: -argument does not hold if : œ #, unless 9ÐKÑ is a power of #, since "  #5 ± 9ÐKÑ for some 5  ". Example 8.15 If 9ÐKÑ œ **)# œ # † ( † #$ † $", then routine calculation shows that the 8( argument holds: "  (5 ± 9ÐKÑ

Ê

5œ!

and so ]( – K and K is not simple. Example 8.16 If 9ÐKÑ œ :8 7 for 8 ", 7  " and : y± 7, then 8: œ Ð"  5:Ñ ± 7 and so if 7  :, then 5 œ !, whence ]: – K . Thus, groups of order :8 ß #:8 ß á ß Ð:  "Ñ:8 for : prime and 8 " have ]: – K and so are not simple. A little programming shows that among the orders up to "!!!! (not including prime powers) there are only &'* orders (less than '%) that are not succeptible to the 8: -argument for some :. Thus, the vast majority of orders up to "!!!! are either prime powers or have the property that groups of that order have a normal Sylow :-subgroup.

Counting Elements of Prime Order If : is a prime and : ± 9ÐKÑ but :# y± 9ÐKÑ, then each of the 8: distinct Sylow : subgroups of K has order : and so the subgroups are pairwise essentially disjoint. Hence, K contains exactly 8: † Ð:  "Ñ distinct elements of order : . Sometimes this simple counting of elements (for different primes :) is enough to show than one of the Sylow subgroups is normal. Example 8.17 Let 9ÐKÑ œ $! œ # † $ † &. Then based on the fact that 8: œ "  5: ± 9ÐKÑ, we can conclude only that 8$ − Ö"ß "!× and 8& − Ö"ß '×. However, if 8$ œ "! and 8& œ ', then K contains at least 8$ † Ð$  "Ñ œ #! elements of order $ and #% elements of order &, totalling %% elements. Hence, one of ]$ or ]& must be normal in K.

246


Index Equal to Smallest Prime Divisor If 9ÐKÑ œ :; 5 where :  ; are primes, then ]; – K, because ÐK À ]; Ñ œ : is the smallest prime dividing 9ÐKÑ. Moreover, it is clear that K œ ]; z ]: Example 8.18 If 9ÐKÑ œ $ † &# œ (&, then ]& – K and K œ ]$ z ]& Also, "  $5 ± #& holds only for 5 œ ! or 5 œ ) and so 8$ œ " or 8$ œ #&. Note that if 8$ œ ", then K œ ]$  ]& is abelian. When 9ÐKÑ œ :; , we can give a fairly complete analysis as follows. Theorem 8.19 Let 9ÐKÑ œ :; , with :  ; primes. Then K œ G; Ð,Ñ z G: Ð+Ñ where ,+ œ ,5 for some " Ÿ 5  ; and 5 : ´ " mod ;. Moreover, K is cyclic if and only if : y± ;  ". Proof. We have seen that K œ ]; z ]: œ G; Ð,Ñ z G: Ð+Ñ Thus, +,+" œ , 5 for some " Ÿ 5  ; and repeated conjugation by + gives , œ +: ,+: œ , 5

:

which implies that 5 : ´ " mod ; . Moreover, 8: ± ; and so 8: œ " or 8: œ ; . But 8: œ " if and only if ]: ü K, that is, if and only if K is cyclic and 8: œ ; if and only if "  5: œ ; , that is, if and only if : y± ;  ". Example 8.20 Let us return to the case 9ÐKÑ œ $! œ # † $ † &. We saw in Example 8.17 that one of ]$ or ]& must be normal in K. It follows that ]$ ]& ü K has order "& and so is cyclic. Hence, ]$ ß ]& « ]$ ]& ü K and so both ]$ and ]& are normal in K.

Using the Kernel of an Action The kernel of an action -À K Ä W\ is normal in K and this can be a useful technique for finding normal subgroups, although they need not be Sylow subgroups. For example, if K acts on Syl: ÐKÑ by conjugation, then the representation map -À K Ä W5:" has kernel

Sylow Theory ,

Oœ

247

RK Ð] Ñ

] −Syl: ÐKÑ

which is a normal subgroup of K. The problem is that it may be either trivial or equal to K. Let 9ÐKÑ œ :7 ? and 9ÐOÑ œ := @, where 7 "ß ?  " and : y± ?. Also, let 8: œ 5:  ". It is clear that O œ K is equivalent to 8: œ " and implies that = œ 7. Conversely, if = œ 7, then O contains a Sylow : -subgroup W of K. But the only Sylow :-subgroup of K in RK Ð] Ñ is ] itself and so 8: œ " and O œ K. Thus, OœK

Í

]: – K

Í

=œ7

As to the nontriviality of O , the induced embedding of KÎO into W5:" implies that :7=

? ¹ Ð5:  "Ñx @

and so :7= ± Ð5:Ñx. Hence, if 5  :, then 7  5 Ÿ =. It follows that if 5  7, then =  ! and O is nontrivial. Thus, 5  minÖ7ß :×

Ê

O Á Ö"×

We note finally that O has a somewhat simpler form if 8: œ ?, since then each ]: is self-normalizing and ,

Oœ

]

] −Syl: ÐKÑ

Theorem 8.21 Let 9ÐKÑ œ :7 ? where : is prime, 7 ", ?  " and : y± ?. Let 8: œ "  5:. 1) If 5 œ !, then ]: – K. 2) If !  5  minÖ7ß :×, then Oœ

,

RK Ð] Ñ

] −Syl: ÐKÑ

is a nontrivial proper normal subgroup of K of order := @, where 7  5 Ÿ = Ÿ 7  " and @ ± ?. In addition, if 5 œ Ð?  "ÑÎ:, then Oœ

,

]

] −Syl: ÐKÑ

has order := . Example 8.22 If 9ÐKÑ œ "!) œ $$ † %, then "  $5 ± % and so 5 œ ! or 5 œ ". Thus, this case is not amenable to the 8: -argument. However, if 5 œ " then

248


Theorem 8.21 implies that

,

Oœ

]

] −Syl$ ÐKÑ

is a nontrivial proper normal subgroup of K of order *. Thus, K is not simple. If 9ÐKÑ œ ")* œ $$ † (, then "  $5 ± ( and so 5 œ ! or 5 œ #. If 5 œ #, then Theorem 8.21 implies that ,

Oœ

]

] −Syl$ ÐKÑ

is a nontrivial proper normal subgroup of K of order $ or *. If 9ÐKÑ œ $!! œ ## † $ † &# , then 8& œ "  &5 ± "# and so 5 œ ! or 5 œ " (and 8& œ '). But if 5 œ ", then Theorem 8.21 implies that K is not simple. Even when O is trivial and the previous theorem does not apply, we learn that K ä W5:" , which can sometimes be useful. Example 8.23 If 9ÐKÑ œ :Ð:  "Ñ, where : is prime. Then 8: œ " or 8: œ :  ". While the previous theorem does not apply, if 8: œ :  ", then Oœ

,

] œ Ö"×

] −Syl: ÐKÑ

Hence, K ä W:" . As an example, if 9ÐKÑ œ "# œ $ † %, then either ]$ – K or y E% K ä W% . But 9ÐKÑ œ 9ÐE% Ñ and so in the latter case, K ¸ E% . Thus, if K ¸ then ]$ – K. We will use this fact later to determine all groups of order "#.

The Normal Interior If L  K, we have seen that ÐK À L ‰ Ñ ± ÐK À LÑx and so if 9ÐKÑ y± ÐK À LÑx, then L ‰ is a nontrivial proper normal subgroup of K . Hence, if /7 9ÐKÑ œ :"/" â:7

where :"  â  :7 are primes and 7 #, then for any 5 , /5  :5 ß ÐK À LÑ  /5 :5

Ê

:5/5 y± ÐK À LÑx

Ê

9ÐKÑ y± ÐK À LÑx

and so L ‰ – K is nontrivial. /7 Theorem 8.24 Let 9ÐKÑ œ :"/" â:7 where :"  â  :7 are primes and 7 #. Suppose that /5  :5 for some " Ÿ 5 Ÿ 7.

Sylow Theory

249

1) If L  K has index ÐK À LÑ  /5 :5 , then L ‰ is a nontrivial proper normal subgroup of K. 2) In particular, if "  8:3  /5 :5 , then RK Ð]:3 Ñ‰ is a nontrivial proper normal subgroup of K. Example 8.25 Let 9ÐKÑ œ '#!" œ $# † "$ † &$. Then 8$ œ "  $5 ± "$ † &$, which implies that 8$ − Ö"ß "$×. If 8$ œ ", then ]$ – K . If 8$ œ "$  &$, then Theorem 8.24 implies that RK Ð]$ Ñ‰ is a nontrivial proper normal subgroup of K. Thus K is not simple.

Using the Normalizer of a Sylow Subgroup Let : Á ; be primes dividing 9ÐKÑ and let ]; − Syl; ÐKÑ. Under the assumption that 8;  " and so RK Ð]; Ñ  K, suppose that : y± 8; , that is, : ± 9ÐRK Ð]; ÑÑ and that T − Syl: ÐRK Ð]; ÑÑ. There are various things we can say about 9ÐRK ÐT ÑÑ. First, if T – RK Ð]; Ñ, then RK Ð]; Ñ Ÿ RK ÐT Ñ and so 9ÐRK Ð]; ÑÑ ± 9ÐRK ÐT ÑÑ On the other hand, even if T is not normal in RK Ð]; Ñ, the fact that ]; ü RK Ð]; Ñ implies that T ]; is a subgroup of K. Hence, if T ]; is abelian, then ]; Ÿ RK ÐT Ñ and so 9Ð]; Ñ ± 9ÐRK ÐT ÑÑ In either case, if T is not a Sylow : -subgroup of K but T – T ‡ − Syl: ÐKÑ, then T ‡ Ÿ RK ÐT Ñ, whence 9ÐT ‡ Ñ ± 9ÐRK ÐT ÑÑ These conditions tend to make RK ÐT Ñ large. Example 8.26 If 9ÐKÑ œ $'(& œ $ † &# † (# , then it is easy to see that 8( − Ö"ß "&×. If 8( œ "&, then 9ÐRK Ð]( ÑÑ œ & † (# . Let T be a Sylow &subgroup of RK Ð]( Ñ. The number of such subgroups is "  &5 ± (# and so T – RK Ð]( Ñ. Hence, & † (# ± 9ÐRK ÐT ÑÑ Also, T has index & in T ‡ − Syl& ÐKÑ and so T – T ‡ , whence &# ± 9ÐRK ÐT ÑÑ and so &# † (# ± 9ÐRK ÐT ÑÑ Hence, either RK ÐT Ñ œ K , in which case T – K or else RK ÐT Ñ has index $ in K and so is normal in K.

250


Suppose that 9ÐKÑ œ :;?, where : Á ; are primes that do not divide ?. If : y± 8; , that is, if : ± 9ÐR Ð]; ÑÑ, then ]: Ÿ R Ð]; Ñ and so ]: ]; Ÿ K has order :; . Hence, if : y± Ð;  "Ñ, then ]: ]; is abelian (cyclic) and so ]; Ÿ R Ð]: Ñ. Thus, ; ± 9ÐR Ð]: ÑÑ and so 8: ± 9ÐKÑÎ:; . Theorem 8.27 If 9ÐKÑ œ :;?, where :  ; are primes that do not divide ?, then : y± Ð;  "Ñ and

: y± 8;

Ê

8: ¹

9ÐKÑ :;

Example 8.28 If 9ÐKÑ œ "()& œ $ † & † ( † "(, then a routine calculation gives 8$ − Ö"ß (ß )&ß &*&×

and 8"( − Ö"ß $&×

But $  "(ß

$ y± Ð"(  "Ñß

$ y± 8"(

Ê

8$ ¹

9ÐKÑ œ $& $ † "(

and so 8$ œ " or 8$ œ (. Hence, Theorem 8.24 now implies that one of ]$ or R Ð]$ Ñ‰ is a proper nontrivial normal subgroup of K.

Groups of Small Order We have already examined the groups of order %, ' and ). Let us now look at all groups of order "& or less. Of course, all groups of prime order are cyclic. We will again denote an arbitrary Sylow : -subgroup of K by ]: .

Groups of Order % The groups of order % are (up to isomorphism): 1) G% , the cyclic group 2) Z ¸ G# } G# , the Klein 4-group.

Groups of Order ' The groups of order ' are (up to isomorphism): 1) G' , the cyclic group 2) H' ¸ W$ , the nonabelian dihedral (and symmetric) group.

Groups of Order ) The groups of order ) are (up to isomorphism): 1) G) , the cyclic group 2) G% } G# , abelian but not cyclic 3) G# } G# } G# , abelian but not cyclic

Sylow Theory

251

4) H) , the (nonabelian) dihedral group 5) U, the (nonabelian) quaternion group.

Groups of Order * Theorem 7.9 implies that the groups of order * are (up to isomorphism): 1) G* , the cyclic group 2) G$ } G$ , abelian but not cyclic.

Groups of Order "! If 9ÐKÑ œ "! œ # † &, then Theorem 8.19 implies that K œ Ø+ß ,Ùß 9Ð+Ñ œ #ß 9Ð,Ñ œ &ß +,+" œ , 5 where 5 # ´ " mod &, that is, 5 œ " or %. In the former case, + and , commute and K is cyclic. In the latter case, K œ H"! . Thus, the groups of order "! are (up to isomorphism): 1) G"! ¸ G& } G# , the cyclic group 2) H"! , the nonabelian dihedral group.

Groups of Order "# We have accounted for three nonabelian groups of order "#: the alternating group E% , the dihedral group H"# and the semidirect product X œ G$ Ð+Ñ z G% Ð,Ñ, where ,+, " œ +# We wish to show that this completes the list of nonabelian groups of order "#. y E% . Then Example 8.23 shows that ]$ œ G$ Ð+Ñ ü K and so Assume that K ¸ K œ G$ Ð+Ñ z ]# , where ]# œ G% Ð,Ñ or ]# œ G# ÐBÑ  G# ÐCÑ. If K œ G$ Ð+Ñ z G% Ð,Ñ, then ,+, " œ +

or ,+, " œ +#

In the former case K is abelian and in the latter case K œ X . If K œ G$ Ð+Ñ z ÐG# ÐBÑ  G# ÐCÑÑ then +B œ + or +#

and +C œ + or +#

We consider three cases. If +B œ + and +C œ +, then K is abelian. If +B œ + and +C œ +# , then 9Ð+BÑ œ 9Ð+Ñ9ÐBÑ œ ' and Ð+BCÑ# œ +BC+BC œ +CÐB+BÑC œ +C+C œ +$ œ "

252


Hence, Ø+BCß CÙ is generated by two distinct involutions whose product has order ' and so Theorem 2.36 implies that K œ H"# . Finally, if +B œ +# œ +C , then Ð+BÑ# œ +B+B œ +$ œ " and +BC œ Ð+# ÑC œ +% œ + and so 9Ð+BCÑ œ 9Ð+Ñ9ÐBCÑ œ '. Thus, Ø+Bß CÙ is dihedral and K œ H"# . Thus, the groups of order "# are (up to isomorphism) 1) 2) 3) 4) 5)

G"# ¸ G% } G$ , the cyclic group G# } G# } G$ , abelian but not cyclic E% , the nonabelian alternating group H"# , the nonabelian dihedral group X , the nonabelian group described above.

Groups of Order "% If 9ÐKÑ œ "% œ # † (, then Theorem 8.19 implies that K œ Ø+ß ,Ùß 9Ð+Ñ œ #, 9Ð,Ñ œ (ß +,+" œ , 5 where 5 # ´ " mod (, that is, 5 œ " or '. In the former case, K is cyclic. In the latter case, K is dihedral. Thus, the groups of order "% are (up to isomorphism): 1) G"% ¸ G( } G# , the cyclic group 2) H"% , the nonabelian dihedral group.

Groups of Order "& Theorem 8.19 implies that all groups of order "& are cyclic.

On the Existence of Complements: The Schur–Zassenhaus Theorem In this section, we use group actions to prove the Schur–Zassenhaus Theorem, which gives a simple sufficient (but not necessary) condition under which a normal subgroup L of a group K has a complement O , thus giving a semidirect decomposition K œ L z O . Definition Let K be a finite group. A Hall subgroup L of K is a subgroup with the property that its order 9ÐLÑ and index ÐK À LÑ are relatively prime. The Schur–Zassenhaus Theorem states that a normal Hall subgroup L has a complement. The tool that we will use in proving the Schur–Zassenhaus Theorem is the Frattini argument (Theorem 7.2). In particular, we consider the action of left translation by K on the set e of all right transversals of L and show that this action is regular, whence K œ L z stabK ÐVÑ for any V − e. So let us take a closer look at transversals and their actions.

Sylow Theory

253

Transversals and Their Actions Let K be a finite group and let L be a normal Hall subgroup of K, with right cosets LÏK œ ÖL œ L" ß á ß L7 × Let e be the set of all right transversals of L . If V œ Ö, we have > œ =Ð=>Ñ œ =B where B œ => has the property that B= œ B" This property of B is very important. Definition Let K be a group. 1) An element B − K is real by + − K if B+ œ B" Also, B is real if it is real by some element +. Let e be the set of real elements of K. 2) An element B − K is strongly real by = if B= œ B" where = is an involution. Also, B is strongly real if it is strongly real by some involution =. Let f denote the set of strongly real elements of K. Thus, every pair of involutions is related by a strongly real element. It is not hard to see that \ , f and e are each closed under conjugation and so each set is a union of conjugacy classes. Associated with the equation B+ œ B" are some important sets. Definition Let K be a group. 1) For B − K, let G w ÐBÑ œ Ö+ − K ± B+ œ B" × be the set of all elements by which B is real. The extended centralizer of B − K is G ‡ ÐBÑ œ GÐBÑ ∪ G w ÐBÑ where GÐBÑ is the centralizer of B.

The Classification Property for Groups 2

267

a) For B − f , let EÐBÑ œ Ö= − \ ± B= œ B" × be the set of involutions by which B is strongly real. b) For = − \ , let fÐ=Ñ œ ÖB − K ± B= œ B" × be the set of strongly real elements by the involution =.

Let us take a look at these sets. Note first that B − K is real if and only if G w ÐBÑ is nonempty and B − K is strongly real if and only if EÐBÑ is nonempty. G ‡ ÐBÑ If B − e and +ß , − G w ÐBÑ, then B+ œ B" œ B, and so +" , − GÐBÑ. Thus, G w ÐBÑ is a coset of GÐBÑ, G w ÐBÑ œ +GÐBÑ and so

kG w ÐBÑk œ kGÐBÑk

Thus, for B − e and + − G w ÐBÑ, G ‡ ÐBÑ œ GÐBÑ ∪ +GÐBÑ œ œ

GÐBÑ GÐBÑ “ +GÐBÑ

if B − \ w if B − e Ï \ w

where if B − e Ï \ w , then ÐG ‡ ÐBÑ À GÐBÑÑ œ # EÐBÑ If B − f , then EÐBÑ © G w ÐBÑ and so

kEÐBÑk Ÿ kGÐBÑk

But we can do a bit better in some cases. Since EÐ"Ñ œ \ , we have kEÐ"Ñk œ 8

and if B − \ , then EÐBÑ © GÐBÑ Ï Ö"× and so

kEÐBÑk Ÿ kGÐBÑk  "

Also, for any 1 − K, it is easy to see that EÐB1 Ñ œ EÐBÑ1

268


and so

kEÐB1 Ñk œ kEÐBÑk

that is, kEÐBÑk is constant on conjugacy classes of K. WÐ=Ñ If = − \ , then we have seen that \ w © =f Ð=Ñ Conversely, if B − f Ð=Ñ for = − \ , then Ð=BÑ# œ =B=B œ B" B œ " and so > œ =B − \ w . Hence, \ w œ =f Ð=Ñ and so

kf Ð=Ñk œ k=f Ð=Ñk œ 8  "

The Fundamental Relation Now we can count the size of the set Y œ ÖÐ=ß BÑ − \ ‚ f ± B= œ B" × in two ways. From the point of view of an = − \ , kY k œ

=−\

kf Ð=Ñk œ 8Ð8  "Ñ

From the point of view of an B − f ,

kY k œ

B−f

kEÐBÑk

Thus, 8#  8 œ B−f

kEÐBÑk

(9.1)

To split up the sum on the right, we choose a system of distinct representatives (SDR) Q œ ÖB! ß B" ß á ß B> × for the conjugacy classes of K, where 1) ÖB! œ "× 2) ÖB" ß á ß B? × is an SDR for the conjugacy classes in \

The Classification Property for Groups

269

3) ÖB?" ß á ß B@ × is an SDR for the conjugacy classes in f Ï \ w Then since kEÐBÑk is constant on conjugacy ¸BK ¸ œ ÐK À GÐBÑÑ, equation (9.1) can be written @

8#  8 œ 3œ!

classes

and

since

kEÐB3 ÑkÐK À GÐB3 ÑÑ

Splitting this sum further gives ?

8#  8 œ 8  3œ" ?

Ÿ8 3œ"

kEÐB3 ÑkÐK À GÐB3 ÑÑ 

œ 8  @kKk  and so

3œ?"

kEÐB3 ÑkÐK À GÐB3 ÑÑ

ÐkGÐB3 Ñk  "ÑÐK À GÐB3 ÑÑ 

Ÿ 8  ?kKk  œ @kKk

@

? 3œ" ? 3œ"

@ 3œ?"

kGÐB3ÑkÐK À GÐB3ÑÑ

ÐK À GÐB3 ÑÑ  Ð@  ?ÑkKk

¸BK ¸ 3

8#  8 Ÿ @kK k

(9.2)

To get further estimates, note that 8 œ k\ k œ

?

ÐK À GÐB3 ÑÑ

(9.3)

3œ"

Now, if we assume that the center ^ÐKÑ of K has odd order, then it contains no involutions and so GÐB3 Ñ  K for B3 − \ . Hence, if 7 is the smallest index among all proper subgroups of K, we have 8 œ k\ k 7?

We can make a similar estimate for kf k œ "  8 

@

ÐK À GÐB3 ÑÑ 3œ?"

and since B3 − f Ï \ w , the terms in the final sum satisfy ÐK À GÐB3 ÑÑ œ ÐK À G ‡ ÐB3 ÑÑÐG ‡ ÐB3Ñ À GÐB3ÑÑ œ #ÐK À G ‡ÐB 3ÑÑ Therefore, if we assume that G ‡ ÐB3 Ñ is also proper in K , then

(9.4)

270


ÐK À GÐB3 ÑÑ #7 and so

kf k "  8  #7Ð@  ?Ñ

(9.5)

The condition that G ‡ ÐB3 Ñ is proper in K is a bit awkward, but is satisfied if we assume that K has no subgroups of index #. The inequalities (9.4) and (9.5) together imply that @Ÿ

kf k  "  8 #7

and so (9.2) implies that 8#  8 Ÿ

kf k  "  8 kKk #7

Some elementary algebra, using the fact that kf k Ÿ kKk, gives 7Ÿ

" kKk kK k " Œ Œ # 8 8

œŒ

kKkÎ8  " #

We have proved a key theorem. Theorem 9.6 (R. Brauer and K. A. Fowler [4], 1955) Let K be a group of even order with exactly 8 " involutions. Assume that ^ÐKÑ has odd order. Then either K has a subgroup of index # (which must be normal) or K has a proper subgroup L with ÐK À LÑ Ÿ Œ

kKkÎ8  " #

Equation (9.3) implies that for any involution , , which we can assume is B" , we have 8 œ kK k that is,

? 3œ"

" " kGÐB3 Ñk kGÐ,Ñk

kKk Ÿ kGÐ,Ñk 8

Hence, if K fails to have a (normal) subgroup of index #, then it has a subgroup L for which

The Classification Property for Groups

ÐK À LÑ Ÿ Œ

271

kGÐ,Ñk  " #

Since L is a proper subgroup of K, it follows that the normal interior L ‰ is a proper normal subgroup with ÐK À L ‰ Ñ ± ÐK À LÑx. In particular, if K is simple, then 9ÐKÑ ± ÐK À LÑx and so 9ÐKÑ º Œ

kGÐ,Ñk  " x #

Thus, we arrive at our final goal. Theorem 9.7 Let K be a finite nonabelian simple group and let , − K be an involution. Then the centralizer GÐ,Ñ is a proper subgroup of K and 9ÐKÑ º Œ

kGÐ,Ñk  " x #

This is the result that we promised at the beginning of this section and is as far as we propose to take our discussion of the classification problem for nonabelian finite simple groups.

Exercises 1.

2. 3. 4. 5. 6. 7.

Let K be a group. a) Under what conditions does the set W œ \ ∪ Ö"× of elements of K of exponent # form a subgroup of K? b) Under what conditions does the set W form a normal subgroup of K? c) If W is a subgroup of K, what can you say about the strongly real elements of the group? Prove that f is closed under conjugation. Prove that if a finite group K has a nontrivial real element, then K has even order. Find the real elements in the symmetric group W8 . Find the strongly real elements. In the alternating group E8 , show that any permutation that is a product of disjoint cycles of length congruent to " modulo % is real. Find the real elements of the dihedral group H#8 . Find the strongly real elements. Find the real elements of the quaternion group U. Find the strongly real elements.

Chapter 10

Finiteness Conditions

There are many forms of finiteness that a group can possess, the most obvious of which is being a finite set. However, as we have observed, chain conditions are also a form of finiteness condition. Another type of finiteness condition on a group K is the condition that K has a finite direct sum decomposition K œ H"  â  H 8 that cannot be further refined by decomposing any of the factors H3 into a direct sum; that is, for which each H3 is indecomposable. In this chapter, we explore these finiteness conditions. First, however, we generalize the notion of a group.

Groups with Operators As we have seen, a group K has several important families of subgroups, in particular, the families of all subgroups, all normal subgroups, all characteristic subgroups and all fully-invariant subgroups. Each of these families can be characterized as being the family of all subgroups that are invariant under a certain subset of EndÐKÑ. In particular, a subgroup L of K is 1) normal if and only if it is invariant under InnÐKÑ, 2) characteristic if and only if it is invariant under AutÐKÑ, 3) fully invariant if and only if it is invariant under EndÐKÑ. We can also say that the subgroups of K are invariant under the empty subset of EndÐKÑ. This point of view leads us to define the concept of groups with operators, which will include all of these special cases. Intuitively speaking, a group with operators is a group K with a distinguished family X of endomorphisms of K. Rather than associate a subgroup of EndÐKÑ with K directly, we use a function 0 À H Ä EndÐKÑ from a set H into EndÐKÑ. Here is the formal definition.


273

274


Definition Let H be a set. An H-group is a pair ÐKß 0 À H Ä EndÐKÑÑ where K is a group and 0 À H Ä EndÐKÑ is a function. It is customary to denote the endomorphism 0 Ð=Ñ simply by = and thus write 0 Ð=Ñ+ as =+ (some authors write += ). An H-group is called a group with operators and H is called the operator domain. Let K be an H-group. 1) An H-subgroup L of K is an H-invariant subgroup L of K. We use the notations L Ÿ H K and L ü H K to denote an H-subgroup and a normal Hsubgroup of K, respectively. We also use the notations H-subÐKÑ and

H-norÐKÑ

to denote the set of all H-subgroups of K and the set of all normal Hsubgroups of K, respectively. 2) If K and L are H-groups, an H-homomorphism from K to L is a homomorphism 5À K Ä L that is compatible with the group operators, that is, 5Ð=+Ñ œ =Ð5+Ñ for all + − K. A bijective H-homomorphism is an H-isomorphism, and similarly for the other types of homomorphisms. We write 5À K ¸ H L to denote an H-isomorphism from K to L . The existence of an H-isomorphism from K to L is denoted by K ¸ H L . 3) If L − H-norÐKÑ, then the H-quotient group (or H-factor group) is the quotient group KÎL with operators = − H defined by =Ð+LÑ œ Ð=+ÑL for all + − K, that is, defined so that the canonical projection 1L is an Hhomomorphism. Let ÐKß 0 À H Ä EndÐKÑÑ be an H-group. Then a subgroup L Ÿ K is an Hsubgroup of K if and only if the restricted operators HlL œ Ö0 Ð=ÑlL ± = − H× are operators on L , that is, if and only if L is an HlL -group. We will always think of an H-subgroup L of K as an HlL -group, although we will use notation such as O Ÿ H L in place of O Ÿ HlL L . Thus, H-subgroupness is transitive, that is, L Ÿ H Kß

O ŸH L

Ê

O ŸH K

Note that an H-subgroup L of K may be an operator group in other related ways. To illustrate, if H œ InnÐKÑ, then L is an HlL -group as well as an operator group under its own family InnÐLÑ of inner automorphisms. But in


275

general, InnÐLÑ is a proper subset of InnÐKÑlL , since conjugation by + − K Ï L need not be an inner automorphism of L . If H is the empty set, then an H-group is nothing more than an ordinary group and it is customary to drop the prefix “H-”. Also, in the most important cases, H is a subset of EndÐKÑ and 0 À H Ä EndÐKÑ is the inclusion map, in which case 0 is suppressed. This applies to the cases H œ InnÐKÑ, H œ AutÐKÑ and H œ EndÐKÑ. Example 10.1 Let Z be a vector space over a field J . Each α − J defines an endomorphism of the abelian group Z by scalar multiplication. Thus, a vector space over J is a group with operators J . An J -subgroup is a subspace and an J -homomorphism is a linear transformation.

The Lattice of H-Subgroups of an H-Group Let K be an H-group. Then the intersection and the join of any family Y œ ÖL3 ± 3 − M× of H-subgroups of K is also an H-subgroup of K. Hence, the meet and join in H-subÐKÑ is the same as the meet and join in subÐKÑ. In other words, H-subÐKÑ is a complete sublattice of subÐKÑ. We leave it to the reader to show that the H-subgroup Ø\ÙH generated by a nonempty subset \ of K is Ø\ÙH œ Ø=B ± B − \ß = − HÙ An H-subgroup L of an H-group K is finitely H-generated if L œ Ø\ÙH for some finite set \ . This generalizes the normal closure of a subset \ of K.

The H-Isomorphism and H-Correspondence Theorems The concept of universality given in Theorem 4.5 and the consequent isomorphism theorems have direct generalizations to groups with operators. Let K be a H-group and let O Ÿ H K. Let YH ÐKà OÑ be the family of all pairs ÐLß 5À K Ä LÑ, where 5À K Ä L is an H-homomorphism and O © kerÐ5Ñ. Then the pair ÐKÎOß 1O À K Ä KÎOÑ is universal in YH ÐKà OÑ, in the sense that for any pair ÐLß 5À K Ä LÑ in YH ÐKà OÑ, there is a unique mediating H-homomorphism 7 À W Ä L for which 7 ‰ 1O œ 5 To see this, note that Theorem 4.5 guarantees the existence of a mediating homomorphism 7 . But if 1O and 5 are H-homomorphisms, then for any = − H and + − K, 7 Ð=Ð+OÑÑ œ 7 ÐÐ=+ÑOÑ œ 5Ð=+Ñ œ =Ð5+Ñ œ =7 Ð+OÑ and so 7 is also an H-homomorphism. Also, H-universality enjoys the same

276


uniqueness up to isomorphism as ordinary universality (H œ g). The Hisomorphism theorems now follow in the same manner as before. Theorem 10.2 (The H-isomorphism theorems) Let K be an H-group. 1) (First H-isomorphism theorem) Every H-homomorphism 5À K Ä L induces an H-embedding 5À KÎkerÐ5Ñ ä L defined by 5Ð1kerÐ5ÑÑ œ 5Ð1Ñ and so K ¸ H imÐ5Ñ kerÐ5Ñ 2) (Second H-isomorphism theorem) If Lß O − H-subÐKÑ with O ü K, then L ∩ O − H-norÐLÑ and LO L ¸H O L ∩O 3) (Third H-isomorphism theorem) If L Ÿ O Ÿ K with Lß O − H-norÐKÑ, then OÎL − H-norÐKÎLÑ and K O K „ ¸H L L O 4) (The H-correspondence theorem) Let R − H-norÐKÑ. If H-subÐR à KÑ denotes the lattice of all H-subgroups of K that contain R , then the map 1À subÐR à KÑ Ä subÐKÎR Ñ defined by 1ÐLÑ œ LÎR preserves H-invariance in both directions and so maps H-subÐR à KÑ bijectively onto H-subÐKÎR Ñ.

H-Series and H-Subnormality We can now generalize the notion of series and subnormality to groups with operators. This will provide considerable time savings in our later work. Definition Let K be an H-group. 1) An H-series in K is a series Z À K! ü K" ü â ü K8 in which every term is an H-subgroup of K . 2) A refinement of an H-series Z is an H-series [ obtained from Z by including zero or more additional H-subgroups between the endpoints. A proper refinement is a refinement that includes at least one new subgroup.


277

We review the usual suspects in the context of H-series: a) If H œ g, an H-series is just a series. b) If H œ InnÐKÑ, an H-series is a normal series. c) If H œ AutÐKÑ, an H-series is a characteristic series. d) If H œ EndÐKÑ, an H-series is a fully invariant series. Of course, if ÖK3 × is a nonproper H-series for K, we can dedup the series by removing any duplicate subgroups to obtain a proper series. Definition Let K be an H-group. Two equal-length H-series Z À K! ü K" ü â ü K8" ü K8 and [À L! ü L" ü â ü L8" ü L8 in K with common endpoints K! œ L! and K8 œ L8 are H-isomorphic (also called H-equivalent) if there is a bijection 0 of the index set Ö!ß á ß 8  "× for which K3" ÎK3 ¸ H L0 Ð3Ñ" ÎL0 Ð3Ñ

As usual, when H œ g, we use the term isomorphic. Thus, for example, the series Ö"× – G: – ÐG:  G; Ñ and Ö"× – G; – ÐG:  G; Ñ are isomorphic.

H-Subnormality H-subnormality of a subgroup L Ÿ H K requires not just that L be subnormal and H-invariant, but that the entire series that witnesses subnormality be an Hseries. Definition Let K be an H-group. An H-subgroup L of K is H-subnormal in K , denoted by L ü ü H K, if there is an H-series from L to K. We use the notations subnH ÐKÑ

and subnH ÐR à KÑ

to denote the family of all H-subnormal subgroups of K and the family of all Hsubnormal subgroups of K that contain R , respectively. We can now generalize Theorem 4.24.


278

Theorem 10.3 Let K be an H-group and let Lß O − H-subÐKÑ. 1) (Transitivity) L ü ü HO

and O ü ü H K

Ê

L ü ü HK

2) (Intersection) If P − H-subÐKÑ, then L ü ü HO

Ê

L ∩ P ü ü HO ∩ P

In particular, L Ÿ O Ÿ Kß

L ü ü HK

Ê

L ü ü HO

and L ü ü H Kß

O ü ü HK

Ê

L ∩ O ü ü HK

3) (Normal lifting) If R − H-norÐKÑ, then L ü ü HO

Ê

LR ü ü H OR

4) (Quotient/unquotient) If R − H-norÐOÑ and R Ÿ L Ÿ O , then L ü ü HO

Í

LÎR ü ü H OÎR

Composition Series If K! ß K8 Ÿ H K, then refinement is a partial order on the set of all proper Hseries from K! to K8 (assuming that this set is nonempty). Maximal proper Hseries are particularly important. Definition Let K be an H-group and let K! ß K8 Ÿ H K . An H-composition series from K! to K8 is a proper H-series K! – K" – â – K8 that is maximal in the family of all proper H-series from K! to K8 , under refinement. If there is an H-composition series from K! to K8 , we will write bCompSerH ÐK! à K8 Ñ

or bCompSerH ÐK8 Ñ when K! œ Ö"×

The factor groups of an H-composition series are called H-composition factors. 1) A maximal series (H œ g) is simply called a composition series and the factor groups are called composition factors. 2) A maximal normal series (H œ InnÐKÑ) in K is called a chief series or principal series and the factor groups are called chief factors or principal factors. When the endpoints of a series are clear from the description of the series, we will drop the “from-to” terminology. To characterize maximal series, we use the following concept.


279

Definition A nontrivial H-group K is H-simple if K has no nontrivial proper normal H-subgroups. The H-correspondence theorem implies the following. Theorem 10.4 A proper H-series is an H-composition series if and only if its factor groups are H-simple. Thus, a series ZÀ K! – K" – â – K8 in K is a composition series if and only if its factor groups K5" ÎK5 are simple and Z is a chief series if and only if each factor group K5" ÎK5 is a minimal normal subgroup of KÎK5 . It is clear from Theorem 10.4 that any H-series that is H-isomorphic to an Hcomposition series is also an H-composition series. Also, if we remove an endpoint from an H-composition series, the result is also an H-composition series (with different endpoints, of course). Finally, if K! – â – K5

and K5 – â – K8

are H-composition series in K, then so is their concatenation K! – â – K8

The Extension Problem An extension of a pair ÐR ß UÑ of groups is a group K that has a normal subgroup R w isomorphic to R and for which KÎR w is isomorphic to U. The extension problem for the pair ÐR ß UÑ is the problem of determining (up to isomorphism) all possible extensions of ÐR ß UÑ. Note that any external semidirect product K œ R z ) U is an extension of ÐR ß UÑ. However, semidirect products alone do not solve the extension problem. For example, ™ is an extension of Ð#™ß ™# Ñ but ™ is not a semidirect of any of its nontrivial proper subgroups. The importance of the extension problem can be clearly seen in the light of composition series. Suppose that we can solve the extension problem and that we can determine (up to isomorphism) all simple groups. The simple groups are precisely the groups that have a composition series of length ": Ö"× œ K! – K" Next, for each K" , we solve the extension problem for all pairs of the form ÐK" ß L" Ñ, where L" ranges over the simple groups. The solutions K# are precisely the groups that have composition series of length #:

280


K! – K" – K# where K# ÎK" ¸ L" . Continuing to extend by all possible simple groups produces all possible groups that have composition series, and this includes all finite groups. Thus, in particular, if we can solve the extension problem and if we can determine all finite simple groups, we can determine all finite groups. Unfortunately, a practical solution to the extension problem does not exist at this time.

The Zassenhaus Lemma and the Schreier Refinement Theorem Let K be an H-group. Our next goal is to show that any two H-series in K with the same endpoints have refinements that are H-isomorphic. This result is called the Schreier refinement theorem and has two extremely important consequences, as we will see. First, let us recall that the projection ÐE ü FÑ Ä ÐL ü OÑ of E ü F into L ü O is the extension LÐE ∩ OÑ ü LÐF ∩ OÑ Moreover, when E, F , L and O are H-subgroups, then so are LÐE ∩ OÑ and LÐF ∩ OÑ and the Zassenhaus lemma (Theorem 4.12) generalizes directly to H-subgroups. Theorem 10.5 (Zassenhaus lemma [37], 1934) Let K be an H-group and let EüF

and

LüO

where Eß Fß Lß O − H-subÐKÑ. Then the reverse projections ÐE ü FÑ Ä ÐL ü OÑ and

ÐL ü OÑ Ä ÐE ü FÑ

have H-isomorphic factor groups, that is, LÐF ∩ OÑ EÐO ∩ FÑ ¸H LÐE ∩ OÑ EÐL ∩ FÑ

Now we can prove the Schreier refinement theorem. Let L Ÿ O be Hsubgroups of an H-group K and consider a pair of H-series from L to O : Z À L œ K! ü K " ü â ü K 8 œ O and [À L œ L! ü L" ü â ü L7 œ O Projecting each of the 7 steps of [ into each of the 8 steps of Z creates a new H-series with 78 steps, some of which may be trivial. In view of the preceeding


281

remarks, the new H-series is a refinement of Z . Similarly, projecting each of the 8 steps of K into each of the 7 steps of [ creates a new H-series with 78 steps. Moreover, since the two sets of projections consist of inverse pairs, the Zassenhaus lemma implies that the resulting series are H-isomorphic. Theorem 10.6 (Schreier refinement theorem, Schreier [29], 1928) Let K be an H-group. Then any two H-series in K with the same endpoints have Hisomorphic refinements.

Consequences of the Schreier Refinement Theorem The Schreier refinement theorem has two very important consequences. First, suppose that there is an H-composition series V in K from L to O . Then any proper H-series [À L! – L" – â – L8 with H-subnormal endpoints between L and O can be refined to an Hcomposition series. To see this, note that since L! and L8 are H-subnormal, the series [ can be expanded to an H-series ^ from L to O and the Schreier refinement theorem implies that V and ^ have H-isomorphic refinements to proper series Vw and ^w , respectively. But Vw œ V is an H-composition series and therefore so is ^w , which contains a refinement of [. The second consequence of the Schreier refinement theorem is that any two Hcomposition series with the same endpoints are H-isomorphic. Theorem 10.7 Let K be an H-group and let L Ÿ O be H-subgroups of K. 1) Suppose that there is an H-composition series from L to O of length 8. If L! Ÿ L8 are H-subnormal subgroups of K between L and O , then any Hseries from L! to L8 can be refined to an H-composition series and so has length at most 8. 2) (The Jordan–Hölder Theorem) Every two H-composition series from L to O are H-isomorphic. In particular, they have the same length. The Jordan–Hölder Theorem allows us to make the following definition. Definition Let K be an H-group and let L and O be H-subgroups of K . If bCompSerH ÐLà OÑ, then the H-composition distance .ÐLß OÑ from L to O is the length of any H-composition series from L to O . The distance .ÐOÑ œ .ÐÖ"×ß OÑ is called the H-composition length of O . For chief series, the terms chief distance and chief length are also employed. Of course, the composition distance is positive definite (when it is defined), that is, if bCompSerH ÐL à OÑ, then .ÐLß OÑ ! and .ÐLß OÑ œ !

Í

L œO


282

The Existence of H-Composition Series Any finite group has an H-composition series. For abelian groups, composition series and chief series coincide and an abelian group has a composition series if and only if it is finite, since each factor group must have prime order. Thus, not all groups have composition or chief series. The existence of an H-composition series between H-subnormal subgroups L and O of an H-group K can be characterized in terms of chain conditions on Hsubnormal subgroups. If bCompSerH ÐLà OÑ, then any H-series from L to O has length at most .ÐLß OÑ. Hence, Theorem 1.5 gives the following. Theorem 10.8 Let K be an H-group and let Lß O − subnH ÐKÑ. Then bCompSerH ÐLà OÑ

subnH ÐLà OÑ has BCC

Í

Proof of the following is left to the reader. Theorem 10.9 Let K be an H-group. 1) (Subgroup) If O − subnH ÐKÑ and L Ÿ O , then bCompSerH ÐL à KÑ

bCompSerH ÐLà OÑ and bCompSerH ÐOà KÑ

Ê

and .ÐLß KÑ œ .ÐLß OÑ  .ÐOß KÑ 2) (Quotients) If R − H-norÐKÑ, then bCompSerH ÐR à KÑ

Í

bCompSerH ÐKÎR Ñ

and .ÐR ß KÑ œ .ÐKÎR Ñ 3) (Extensions) If R − H-norÐKÑ, then bCompSerH ÐR Ñß

bCompSerH ÐKÎR Ñ

Ê

bCompSerH ÐKÑ

4) (Direct products) If K and L are H-groups, then bCompSerH ÐKÑ and bCompSerH ÐLÑ

Í

bCompSerH ÐK } LÑ

and .ÐK { LÑ œ .ÐKÑ  .ÐLÑ

The Remak Decomposition Let us recall Theorem 5.12.

283


Theorem 10.10 (Remak) If a group K has either (and therefore both) chain condition on direct summands, then K has a Remak decomposition K œ V"  â  V8 that is, each V5 is indecomposible. The question of uniqueness of a Remak decomposition is rather more complicated than the question of existence. Recall that if K œ L"  â  L8 then the 5 th projection map 3L5 À K Ä L5 is defined by 3L5 Ð+Ñ œ Ò+ÓL5 where Ò+ÓL5 is the 5 th coordinate of +. Moreover, 3L3 is idempotent and normal as an endomorphism of K. Note also that the sum 3L3"  â  3L35 is projection onto L3"  â  L354 and so is an endomorphism of K.

The Krull–Remak–Schmidt Theorem Suppose now that K œ L"  â  L8

(10.11)

with projection maps 1L" ß á ß 1L8 and K œ O"  â  O7

(10.12)

with projection maps ,O" ß á ß ,O7 , where the factors L5 and O5 are indecomposable and 8 Ÿ 7. In searching for possible isomorphisms between the L -factors and the O factors, we recall Theorem 5.25 as it applies to the restricted projection maps 1L3 lO4 À O4 Ä L3

and

,O4 lL3 À L3 Ä O4

Namely, if Ð1L3 lO4 ÑÐ,O4 lL3 Ñ − AutÐL3 Ñ and imÐ,O4 lL3 Ñ ü O4 , then 1L3 lO4 and ,O4 lL3 are isomorphisms. Note however that imÐ,O4 lL3 Ñ ü O4 , since if 2 − L3 and 5 − O4 , then 5Ò,O4 lL3 Ð2ÑÓ5 " œ ,O4 Ð5Ñ,O4 Ð2Ñ,O4 Ð5Ñ" œ ,O4 Ð525 " Ñ − imÐ,O4 lL3 Ñ Thus, we have the following. Theorem 10.13 Let K œ L"  â  L8 œ O"  â  O7 where the factors L5 and O5 are indecomposable. If the composition

284


α3ß4 œ Ð1L3 lO4 Ñ ‰ Ð,O4 lL3 ÑÀ L3 Ä L3 of the restricted projection maps is an automorphism of L3 , then the maps 1L3 lO4 À O4 ¸ L3

and ,O4 lL3 À L3 ¸ O4

are isomorphisms. In attempting to show that a composition is an automorphism, we are reminded of Fitting's lemma. We have assumed that L3 is indecomposable. Also, since the restriction and composition of normal maps is normal, α3ß4 is normal and so has normal higher images. Thus, if we assume that K has BCC on normal subgroups, then Fitting's lemma implies that α3ß4 is either nilpotent or an automorphism for all 3ß 4. To see that for each 3, not all of the maps α3ß4 can be nilpotent, note that for 4 Á 5, α3ß4  α3ß5 œ Ð1L3 ,O4  1L3 ,O5 ÑlL3 œ 1L3 Ð,O4  ,O5 ÑlL3 which is an endomorphism of L3 and so the images imÐα3ß4 Ñ and imÐα3ß5 Ñ commute elementwise. Also, 1L3 ,O"  â  1L3 ,O7 œ 1L3 Ð,O"  â  ,O7 Ñ œ 1L3 and so α3ß"  â  α3ß7 œ 1L3 lL3 œ +L3 which is not nilpotent. Hence, Theorem 4.3 implies that α3ß4 is not nilpotent for some 4. It follows from Fitting's lemma that for each 3, there is a 4 for which α3ß4 − AutÐL3 Ñ and so 1L3 lO4 À O4 ¸ L3


Now, we can make a significant improvement to this by noticing that if O4 œ L5 for some 5 Á 3, then α3ß4 œ Ð1L3 lO4 ÑÐ,O4 lL3 Ñ œ Ð1L3 lL5 ÑÐ,L5 lL3 Ñ œ ! and so if we delete from the sum 4 α3ß4 all terms indexed by a 4 for which O4 − ÖL" ß á ß L8 × Ï ÖL3 ×, the sum remains unchanged and so is not nilpotent. Hence, 1L3 lO4 À O4 ¸ L3


for some 4 for which O4 Â ÖL" ß á ß L8 × Ï ÖL3 ×. Now suppose that after possible reindexing of the O 's, there is a 5 " for which


K œ O"  â  O5"  L5  â  L8

285

(10.14)

with projections ." ß á ß .8 , where -3 ³ ,O3 lL3 À L3 ¸ O3 for all " Ÿ 3 Ÿ 5  ". We may also assume that L5 Á O4 for all 4 5 or else we can replace L5 by O4 (reindexed to O5 Ñ. This certainly holds for 5 œ ". Then there is a 4 5 and we may assume that 4 œ 5 , for which 1L5 lO5 À O5 ¸ L5

and

-5 ³ ,O5 lL5 À L5 ¸ O5

Moreover, since 1L5 maps O5 isomorphically onto L5 and maps the complement LÐ5Ñ œ L"  â  L5"  L5"  â  L8 to Ö"×, it follows that LÐ5Ñ ∩ O5 œ Ö"× Thus, if we replace L5 by O5 in (10.14), the result is a direct sum K" œ O"  â  O5  L5"  â  L8 and our goal is to show that K" œ K. To this end, if .Ð5Ñ œ ."  â  .5"  .5"  â  .8 then .5  .Ð5Ñ œ +K and the map ) œ ,O5 .5  .Ð5Ñ is a normal endomorphism of K with imÐ) Ñ © O5 LÐ5Ñ . To show that ) is surjective and so K œ O5 LÐ5Ñ , it is sufficient to show that ) is injective. Now, any + − K has the form + œ B2 for B − LÐ5Ñ and 2 − L5 . But for any B − LÐ5Ñ , ) ÐBÑ œ ,O5 .5 ÐBÑ.Ð5Ñ ÐBÑ œ B and any 2 − L5 , ) Ð2Ñ œ ,O5 .5 Ð2Ñ.Ð5Ñ Ð2Ñ œ ,O5 Ð2Ñ and so ) ÐB2Ñ œ B,O5 Ð2Ñ and since LÐ5Ñ ∩ O5 œ Ö"×, it follows that )ÐB2Ñ implies B œ " and ,O5 Ð2Ñ œ ". But ,O5 lL5 À L5 ¸ O5 and so 2 œ "Þ Thus, ) is injective.

286


It follows that K œ O"  â  O5  L5"  â  L8 for all " Ÿ 5 Ÿ 8 and -5 œ ,O5 lL5 À L5 ¸ O5 and so this holds for all 5 œ "ß á ß 8. In particular, 8 œ 7. Note also that since ,O3 ÐL3 Ñ œ O3 , the map - œ -" 1L"  â  -8 1L8 œ ,O" 1L"  â  ,O8 1L8 is a surjective normal endomorphism of K and so - − AutÐKÑ. Moreover, -ÐL5 Ñ œ O5 We can now summarize. Theorem 10.15 (The Krull–Remak–Schmidt Theorem) Let K be a group that has BCC on normal subgroups. Suppose that K œ L"  â  L8 œ O"  â  O7 where all factors L3 and O4 are indecomposable. Then 8 œ 7 and there is a reindexing of the O3 's and a normal automorphism - of K for which -À L3 ¸ O3 for all 3 œ "ß á ß 8 and for each " Ÿ 5 Ÿ 8, K œ O"  â  O5  L5"  â  L8

True Uniqueness The Krull–Remak–Schmidt Theorem gives uniqueness of the terms of a Remak decomposition up to isomorphism. Let us now consider the question of when a group K has an essentially unique Remak decomposition, that is, a Remak decomposition that is unique up to the order of the factors. First suppose that K œ L"  â  L8 œ O"  â  O 8 are Remak decompositions of K (where 8  "). Then the Krull–Remak– Schmidt Theorem implies that there is a normal automorphism -À K Ä K for which -L5 œ O5 (after reindexing). Hence, if L5 is invariant under every normal automorphism of K, then O5 œ -L5 Ÿ L5 and so O5 œ L5 for all 5 and K has an essentially unique Remak decomposition. By way of converse, suppose that K has an essentially unique Remak decomposition K œ L"  â  L8 with projections Ö1" ß á ß 18 ×, but that there is a normal endomorphism - of K


287

for which -L" Ÿ y L" . Then there is a 5  " for which 15 - Á ! on L" , that is, there is an B − L" for which " Á 15 -ÐBÑ − L5 . The set O" œ Ö2 † 15 -Ð2Ñ ± 2 − L" × is easily seen to be a normal subgroup of K and O" Á L" , since B † 15 -ÐBÑ − O" Ï L" But it is easy to see that K œ O"  L#  â  L 8 Moreover, if O" is decomposable, then there would be a Remak decomposition of K consisting of more than 8 terms, which is false. Hence, this is a Remak decomposition of K that is distinct from the previous decomposition. We have proved the following. Theorem 10.16 Let K have BCC on normal subgroups and let K œ L"  â  L8 be a Remak decomposition of K. The following are equivalent: 1) This Remak decomposition of K is essentially unique . 2) L5 is invariant under all normal endomorphisms of K. 3) L5 is invariant under all normal automorphisms of K. If αÀ L3 Ä ^ÐKÑ is a nonzero homomorphism, then we can build a normal endomorphism -À K Ä K by specifying that -lL5 œ œ

if 5 œ 3 if 5 Á 3

α !

The map - is normal since for any + − K, 23 − L3 , 25 − L5 where 5 Á 3, -Ð23+ Ñ œ -23 œ Ð-23 Ñ+

and

-Ð25+ Ñ œ " œ Ð-25 Ñ+

Thus, if L3 is not --invariant, then Theorem 10.16 implies that the Remak decomposition of K is not unique. Conversely, suppose that the Remak decomposition of K is not unique and so there is a normal endomorphism - of K for which 14 -lL3 Á ! for some 4 Á 3. Then for 23 − L3 and 24 − L4 , 2

Ò-Ð23 ÑÓ24 œ -Ð23 4 Ñ œ -Ð23 Ñ which shows that -lL3 À L3 Ä ^ÐKÑ. Hence, 14 -lL3 À L3 Ä ^ÐL4 Ñ is a nonzero homomorphism.


288

Theorem 10.17 Let K have BCC on normal subgroups and let K œ L"  â  L8 be a Remak decomposition of K. 1) The following are equivalent: a) This Remak decomposition of K is essentially unique. b) Every homomorphism αÀ L3 Ä ^ÐKÑ satisfies αÀ L3 Ä ^ÐL3 Ñ. c) There are no nonzero homomorphisms -À L3 Ä ^ÐL4 Ñ for 3 Á 4. 2) If K is either perfect or centerless, then K has an essentially unique Remak decomposition. Proof. For part 2), if K œ K w , then L3 œ L3w for all 3 and so if -À L3 Ä ^ÐL4 Ñ for 4 Á 3, then for +ß , − L3 , -ÐÒ+ß ,ÓÑ œ Ò-+ß -,Ó œ " and so -lL3 œ !. If K is centerless, the result is immediate.

Exercises 1. 2.

Give an example of an infinite group with a composition series. Find isomorphic refinements of the two series Ö!× – :™ – ™ and Ö!× – ; ™ – ™

3.

where : and ; are distinct primes. Prove that if K and L are groups and if ZÀ Ö"× œ K! – K" – â – K8 œ K and [À Ö"× œ L! – L" – â – L7 œ L are composition series for K and L , respectively, then the series ÐÖ"× { Ö"×Ñ – ÐK" { Ö"×Ñ – â – ÐK8 { Ö"×Ñ – ÐK8 { L" Ñ – â – ÐK8 { L7 Ñ

4. 5. 6. 7.

is a composition series for K { L . How many composition series does a cyclic group of order :8 have? Prove that the multiset of composition factors of a group is an invariant under isomorphism, but not a complete invariant. Prove the uniqueness part of the fundamental theorem of arithmetic using the Jordan–Hölder Theorem. Let K œ G:8 be the direct product of 8 cyclic groups of prime order :. How many compositions series does K have? Hint: K is a vector space.

Finiteness Conditions 8.

289

Prove that a subgroup of a group with a composition series need not have a composition series as follows. Let M œ Ö"ß #ß á × and let K œ WÐMÑ be the restricted symmetric group on M . (Recall from an earlier exercise that this is the set of all permutations of M that have finite support.) For each 8 ", let K8 œ Ö5 − K ± 5B œ B for B  8× and let L8 œ Ö5 − K8 ± 5 lÖ"ßáß8× is even×

9.

a) Show that K œ -8" K8 . b) Show that L œ -8" L8 has index # in K and so L ü K. c) Show that L is simple. d) Show that L contains an infinite abelian subgroup E. e) Show that E has no composition series but that L does. Let c be an isomorphic-invariant property of finite groups. Let K be a group that has c and for which if R ü K, then KÎR has c . Prove that the following are equivalent: a) K has a normal series Ö"× œ K! ü K" ü â ü K8 œ K

whose factor groups have c . b) K has the property that any nontrivial quotient group of K has a nontrivial normal subgroup that has c . Such a group K is said to be hyper-c . 10. Prove the following facts: a) For any +ß , − K, Ò+,ÓL3 œ Ò+ÓL3 Ò,ÓL3 and so the projection map 35 À K Ä L5 is a homomorphism (and an endomorphism of K). b) For any + − K, + œ Ò+ÓL" âÒ+ÓL= c)

The projection map commutes with any inner automorphism #1 of K and therefore preserves normality. 11. Let K have BCC on normal subgroups and suppose that L is a group for which K } K ¸ L } L . Show that K ¸ L .

Chapter 11

Solvable and Nilpotent Groups

Classes of Groups By a class ^ of groups, we mean a subclass of the class of all groups with the following two properties: 1) ^ contains a trivial group 2) ^ is closed under isomorphism, that is, K−^

and

L ¸K

Ê

L −^

For example, the abelian groups form a class of groups. A group of class ^ is called a ^-group and ^-group L that is a subgroup of a group K is called a ^subgroup of K. A class ^ is a trivial class if it contains only one-element groups.

Closure Properties We will be interested in the following closure properties for a class ^ of groups: 1) (Subgroup) K − ^ß

L ŸK

Ê

L−^

2) (Intersection and Cointersection) For Lß O Ÿ K , Lß O − ^ K K ß −^ L O

Ê Ê

L ∩O −^ K −^ L ∩O

3) (Quotient and Extension) For R ü K, K−^ R ß KÎR − ^

Ê Ê

KÎR − ^ K−^


291

292


4) (Seminormal Join, Normal Join and Cojoin) For Lß O Ÿ K, Lß O − ^ß one normal in K Lß O − ^ß both normal in K K K ß −^ L O

Ê Ê Ê

LO − ^ LO − ^ K −^ LO

5) (Direct product) Lß O − ^

Ê

L }O − ^

These properties are not independent. Theorem 11.1 The following implications hold for a class ^ of groups: 1) subgroup Ê intersection 2) quotient Ê cojoin 3) seminormal join Ê normal join Ê direct product 4) subgroup and direct product Ê cointersection Thus, a class that is closed under subgroup, quotient, seminormal join, extension is closed under all nine properties above. Proof. Part 1) is clear. For part 2), we have K K LO ¸ ‚ −^ LO L L For part 3), the direct product L { O is the seminormal join of L { Ö"× and Ö"× { O , each of which is in ^ if Lß O − ^. For part 4), if KÎLß KÎO − ^, then K K K ä } −^ L ∩O L O via the map 5À 1ÐL ∩ OÑ È Ð1Lß 1OÑ. The following definition will prove very convenient. Definition Let ^ be a class of groups. 1) A ^-series is a series whose factor groups belong to the class ^. 2) A ^= -group is a group that has a ^-series and a ^8 -group is a group that has a normal ^-series. 3) The ^= -class is the class of all ^= -groups and the ^8 -class is the class of all ^8 -groups. Our main interest is in the ^= and ^8 classes in which ^ is either the class of cyclic groups or the class of abelian groups. However, we are also interested in a class of groups that is not a ^= or ^8 class, namely, the nilpotent groups.


293

Definition 1) a) A cyclic series is a series that has cyclic factor groups. b) An abelian series is a series that has abelian factor groups. c) A central series for a group K is a normal series Ö"× œ K! ü K" ü â ü K8 œ K for which K5" ÎK5 Ÿ ^ÐKÎK5 Ñ for all 5 . 2) As shown in the table below, we have the following definitions.

Abelian Cyclic Central a) b) c) d)

Series Solvable Polycyclic

Normal Series œ Solvable Supersolvable Nilpotent

A group is solvable if it has an abelian series. A group is polycyclic if it has a cyclic series. A group is supersolvable if it has a normal cyclic series. A group K is nilpotent if it has a central series.

Note that in previous chapters (see Theorem 7.10), we have found it convenient to use the term central series even when the series does not start at the trivial group, which requires that we distinguish carefully between series in K and series for K. As the title of this chapter suggests, our primary interest is in nilpotent and solvable groups. It is clear that nilpotent groups are solvable. Also, all abelian groups are nilpotent and Theorem 7.10 shows that all finite :-groups are nilpotent: finite : -group or abelian

Ê nilpotent Ê solvable

Note, however, that W$ is solvable but not nilpotent. Also, the symmetric groups W8 are solvable if and only if 8  &. In fact, if 8 &, then W8 has only one nontrivial proper normal subgroup E8 , which is not abelian.

Operations on Series In order to study the closure properties of series-based classes of groups and of the nilpotent class, we must consider various operations on series. Indeed, the operations of intersection, normal lifting, quotient and unquotient as defined in

294


Theorem 4.10 can be applied to each step in a series. Specifically, we have the following operations on series. Let Z À K! ü â ü K 8 be a series in K. 1) The intersection of Z with L Ÿ K is Z ∩ L À K! ∩ L ü â ü K8 ∩ L œ L 2) The normal lifting of Z by R ü K is Z R À K! R ü â ü K8 R 3) The quotient of Z by R ü K is Z R K! R K8 R À üâü R R R 4) The unquotient of the series ZÀ

K! K8 üâü R R

where R Ÿ K! is Z Å R À K! ü â ü K8 5) For L3 Ÿ K, the concatenation of the series [ À L! ü â ü L7 and ^ À L7 ü â ü L78 is the series [ ‡ ^ À L! ü â ü L7 ü â ü L78 6) For the series [À L! ü â ü L 8 and Z À K! ü â ü K7 in K, the interleaved series [ µ Z is formed as follows. First, if = œ maxÖ8ß 7×, then we extend whichever series is shorter by repeating the upper endpoint (L8 or K7 ) an appropriate number of times to make the


295

resulting series of equal length =. Then [ µ Z À ÐL! { K! Ñ ü ÐL" { K! Ñ ü ÐL" { K" Ñ ü ÐL# { K" Ñ ü ÐL# { K# Ñ ü â ü ÐL= { K= Ñ Note that the intersection, normal lifting, quotient, unquotient and interleave of normal series is normal. However, the concatenation of two normal series need not be normal. These operations are used as follows. Theorem 11.2 Let K be a group and let L Ÿ K and R ü K. Let a) Z be a series for K b) [ be a series for L c) a be a series for R d) d be a series for KÎR . Then 1) (Subgroup) Z ∩ L is a series for L 2) (Seminormal join) a ‡ [R is a series for LR 3) (Quotient) ZR ÎR is a series for KÎR 4) (Extension) a ‡ Ðd Å R Ñ is a series for K 5) (Direct product) If Z3 is a series for K3 for 3 œ "ß #, then Z" µ Z# is a series for K" { K# .

Closure Properties of Groups Defined by Series Theorem 4.10 and the previous theorems provide the following facts about closure of ^= -classes and ^8 -classes. Theorem 11.3 Let ^ be a class of groups. 1) (Subgroup) If ^ is closed under subgroup, then the ^= and ^8 classes are closed under subgroup. 2) (Seminormal join) If ^ is closed under quotient, then the ^= -class is closed under seminormal join. 3) (Quotient) If ^ is closed under quotient, then the ^= and ^8 classes are closed under quotient. 4) (Extension) The ^= -class is closed under extension. 5) (Direct product) The ^= and ^8 classes are closed under direct product. In particular, if ^ is closed under subgroup and quotient, then the ^= -class is closed under the following eight operations: 6) subgroup 7) quotient 8) intersection 9) cointersection 10) cojoin 11) direct product

296


12) seminormal join 13) extension and the ^8 -class is closed under all of these operations except seminormal join and extension. Proof. For 1), if ^ is closed under subgroup, then (normal) ^-series are closed under intersection and so the ^= and ^8 classes are closed under subgroup. For 2) and 3), if ^ is closed under quotient, then ^-series are closed under normal lifting. Hence, a ‡ [R and Z R ÎR are ^-series. It follows that the ^= -class is closed under seminormal join and the ^= and ^8 classes are closed under quotient. For 5), if Z3 is a (normal) ^-series for K3 , then Z" µ Z# is a (normal) ^-series for K" { K# . Thus, since the classes of cyclic groups and abelian groups are closed under subgroup and quotient, we have the following. Theorem 11.4 The polycyclic, solvable and supersolvable classes are closed under the following eight operations (except where noted): 1) subgroup 2) quotient 3) intersection 4) cointersection 5) cojoin 6) direct product 7) seminormal join (except for supersolvable) 8) extension (except for supersolvable). Let us now turn to the closure properties of nilpotent groups. Theorem 11.5 1) Central series are closed under intersection, normal lifting, quotient and unquotient. 2) The class of nilpotent groups has the following seven closure properties: a) subgroup b) quotient c) normal join (this is Fitting's theorem, to be proved a bit later) d) direct product e) intersection f) cointersection g) cojoin but not extension. Proof. Part 1) follows from Theorem 4.10 and Theorem 4.11. The statement about extensions in part 3) can be verified by looking at W$ .


297

Nilpotent Groups We now undertake a closer look at nilpotent groups. We will prove that a finite group K is nilpotent if and only if it is the direct product of :-groups and so Theorem 8.11 shows that finite nilpotent groups have many other interesting characterizations.

Higher Centers An extension L ü O in K is central in K if and only if O K Ÿ ^Œ L L and so the largest subgroup O of K for which L ü O is central in K is the subgroup O for which O K œ ^Œ L L With this in mind, we define a function ' œ 'K on norÐKÑ by ^Œ

K R

œ

'K ÐR Ñ R

for R ü K. Note also that 'K ÐR Ñ K œ ^Œ R R

«

K R

and so 'K ÐR Ñ « K. Writing ' 5 ÐÖ"×Ñ as ' 5 Ð"Ñ, the proper series ' ! Ð"Ñ ö ' " Ð"Ñ ö ' # Ð"Ñ ö â is called the upper central series for K and each ' 5 Ð"Ñ is called a higher center of K. The first higher center is the center ^ÐKÑ of K . To see that the upper central series ascends more rapidly than any other central series of the form Ö"× œ K! ü K" ü K# ü â we have 'ÐK5 Ñ K5" for all 5 and it is clear that K5 Ÿ ' 5 Ð"Ñ holds for 5 œ !. If this holds for a particular value of 5 , then the monotonicity of ' implies that

298


K5" Ÿ ' ÐK5 Ñ Ÿ ' Ð' 5 Ð"ÑÑ œ ' 5" Ð"Ñ Hence, K5 Ÿ ' 5 Ð"Ñ for all 5 . Theorem 11.6 Let K be a nilpotent group. The upper central series ' ! Ð"Ñ ö ' " Ð"Ñ ö ' # Ð"Ñ ö â for K is characteristic and ascends more rapidly than any other central series for K, that is, if Ö"× œ K! ü K" ü K# ü â is central in K, then K5 Ÿ ' 5 Ð"Ñ for all 5 !. 1) K is nilpotent if and only if the upper central series reaches K. 2) If K is nilpotent, then all central series for K have length greater than or equal to the length of the upper central series for K. The higher centers have some important applications. Theorem 11.7 Let K be nilpotent, with higher centers ' 5 Ð"Ñ. 1) If L Ÿ K, then L ' 5 Ð"Ñ ü L ' 5" Ð"Ñ 2) If R ü K, then R ∩ ' 5 Ð"Ñ œ Ö"×

Ê

R ∩ ' 5" Ð"Ñ Ÿ ^ÐKÑ

As a consequence, 3) K has the property that every subgroup is subnormal 4) K has the center-intersection property 5) Every chief series for K is central. Proof. For part 1), since ÒKß ' 5" Ð"ÑÓ Ÿ ' 5 Ð"Ñ, Theorem 3.41 implies that ÒL ' 5 Ð"Ñß L ' 5" Ð"ÑÓ œ ÒL ' 5 Ð"Ñß LÓÒL ' 5 Ð"Ñß ' 5" Ð"ÑÓL But each factor on the right is contained in L ' 5 Ð"Ñ and so L ' 5 Ð"Ñ ü L ' 5" Ð"Ñ. Part 3) follows from part 1), since we may lift the upper central series for K by L to get a series from L to K, whence L is subnormal in K. For part 2), since Ò' 5" Ð"Ñß KÓ Ÿ ' 5 Ð"Ñ and ÒR ß KÓ Ÿ R , it follows that ÒR ∩ ' 5" Ð"Ñß KÓ Ÿ R ∩ ' 5 Ð"Ñ œ Ö"× and so R ∩ ' 5" Ð"Ñ Ÿ ^ÐKÑ. For part 4), there is a largest 5 for which


299

R ∩ ' 5 Ð"Ñ œ Ö"× and so R ∩ ' 5" Ð"Ñ is a nontrivial subgroup of ^ÐKÑ. For part 5), the factor group K5" ÎK5 of a chief series Z for K is a minimal normal subgroup of the nilpotent group KÎK5 and so the center-intersection property implies that K5" ÎK5 is central. Thus, Z is central. We can now augment Theorem 8.11 by adding the nilpotent condition. Theorem 11.8 The following are equivalent for a finite group K: 1) K is nilpotent. 2) Every Sylow subgroup of K is normal. 3) K is the direct product of its Sylow :-subgroups K œ  ]: :−c

4) If L Ÿ K, then L œ  ÐL ∩ ]: Ñ :−c

5) (Strong converse of Lagrange's theorem) If 8 ± 9ÐKÑ, then K has a normal subgroup of order 8. 6) K is the direct product of :-subgroups. 7) Every subgroup of K is subnormal. 8) K has the normalizer condition. 9) Every maximal subgroup of K is normal. 10) KÎFÐKÑ is abelian. Proof. Theorem 8.11 states that 2)–10) are equivalent. Moreover, Theorem 7.10 implies that a finite :-group is nilpotent and therefore so is any direct product of finite :-groups. Hence, 6) implies 1). Theorem 11.7 shows that 1) implies 7).

Lower Centers In terms of commutators, an extension L ü O in K is central in K if and only if ÒOß KÓ Ÿ L and so L œ ÒOß KÓ is the smallest subgroup O of K for which L ü O is central in K; in fact, ÒOß KÓ « O . Thus, if we define the “commutator with K” function > œ >K on subÐKÑ by >K ÐOÑ œ ÒOß KÓ then >K ÐOÑ is the smallest subgroup of K for which >K ÐOÑ Ÿ O is central in K. The (possibly infinite) descending central series â ö ># ÐKÑ ö >" ÐKÑ ö >! ÐKÑ œ K is called the lower central series for K and each >5 ÐKÑ is called a lower center of K. The first lower center is the commutator subgroup >ÐKÑ œ Kw .

300


An induction argument shows that the lower central series descends more rapidly than any other central series, in the sense that if â ü K# ü K" ü K! œ K is central in K, then >5 ÐKÑ Ÿ K5 for all 5 !. For if >5 ÐKÑ Ÿ K5 , then the monotonicity of > implies that >5" ÐKÑ œ >Ð>5 KÑ Ÿ >ÐK5 Ñ Ÿ K5" Theorem 11.9 Let K be a group. The lower central series â ö ># ÐKÑ ö >" ÐKÑ ö >! ÐKÑ œ K for K is characteristic and descends more rapidly than any central series for K, that is, if â ü K# ü K" ü K! œ K is central in K, then >5 ÐKÑ Ÿ K5 for all 5 !. 1) K is nilpotent if and only if the lower central series reaches Ö"×. 2) If K is nilpotent, then all central series for K have length greater than or equal to the length of the lower central series for K. The commutator function >K has some simple but useful properties that are consequences of Theorems 3.40 and 3.41. Theorem 11.10 Let K be a group and let L ß O ß P ü K. 1) >L ÐOÑ ü K 2) >L ÐOÑ œ >O ÐLÑ 3) >L ÐOPÑ œ >L ÐOÑ>L ÐPÑ and >LO ÐPÑ œ >L ÐPÑ>O ÐPÑ 4) >L is deflationary, that is, >L ÐOÑ Ÿ L In fact, >L ÐOÑ Ÿ L ∩ O Hence, for all 5 ", 5 >5L ÐLÑ Ÿ >K ÐKÑ


301

5) If R ü K and R Ÿ L ∩ O , then for any 5 ", >5OÎR Œ

L R

6) For 5 ",

œ

>5O ÐLÑR R

$

>5LO ÐLOÑ œ

>E" â>E5 ÐFÑ

E" ßáßE5 ßF−ÖLßO×

Proof. Part 5) holds for 5 œ " since >OÎR Œ

L R

œ”

L O ÒLß OÓR >O ÐLÑR ß •œ œ R R R R

and if it holds for a particular value of 5 , then >5" OÎR Œ

L R

œ ”>5OÎR Œ

L O ß • R R >5O ÐLÑR O œ” ß • R R Ò>5O ÐLÑß OÓR œ R >5" O ÐLÑR œ R

and so this holds for all 5 ". For part 6), we have >LO ÐLOÑ œ >L ÐLÑ>O ÐLÑ>L ÐOÑ>O ÐOÑ œ

$

>E ÐFÑ

EßF−ÖLßO×

and an easy induction proves the general formula.

Nilpotency Class Theorem 11.9 and Theorem 11.6 imply that for a nilpotent group, the upper and lower central series have the same length. Definition Let K be a nilpotent group. The common length of the upper and lower central series is called the nilpotency class of K, which we denote by nilclassÐKÑ. Moreover, if K is nilpotent and Z À Ö"× œ K! – â – K7 œ K is a central series for K of length 7, then

302


5 >75 K ÐKÑ Ÿ K5 Ÿ 'K Ð"Ñ

for all 5 œ !ß á ß 7, where >3 ÐKÑ œ Ö"× and ' 3 Ð"Ñ œ K if 3 nilclassÐKÑ. The nilpotent groups of class ! are the trivial groups and the nilpotent groups of class " have ^ÐKÑ œ K or equivalently, Kw œ Ö"× and so are the nontrivial abelian groups. A group K has nilpotency class # if and only if either of the following conditions holds: 1) Ö"×  Kw  K and ÒKw ß KÓ œ Ö"×, or equivalently, Ö"×  Kw Ÿ ^ÐKÑ  K 2) Ö"×  ^ÐKÑ  K and ^Œ

K ^ÐKÑ

œ

K ^ÐKÑ

or equivalently, K is not abelian but KÎ^ÐKÑ is abelian. Theorem 3.42 implies that Kw Ÿ ^ÐKÑ if and only if ÒÒ+ß ,Óß -Ó œ Ò+ß Ò,ß -ÓÓ for all +ß ,ß - − K . Hence, for nonabelian groups, this condition is equivalent to being nilpotent of class #. Theorem 11.11 Let K be nilpotent. 1) If L Ÿ K, then nilclassÐLÑ Ÿ nilclassÐKÑ 2) If R ü K, then nilclassÐKÎR Ñ Ÿ nilclassÐKÑ 3) (Fitting's theorem) The join of two normal nilpotent subgroups of a group K is nilpotent. In fact, if Lß O ü K , then nilclassÐLOÑ Ÿ nilclassÐLÑ  nilclassÐOÑ Proof. The first two parts follow from Theorem 11.10. For part 3), Theorem 11.10 implies that >5LO ÐLOÑ œ

$

>E" â>E5 ÐFÑ

E" ßáßE5 ßF−ÖLßO×

for all 5 ". Now, suppose that nilclassÐLÑ œ - and nilclassÐOÑ œ . and let 5 œ -  .. Then each factor on the right above has the form


303

P œ >E" â>E-. ÐFÑ Among the subgroups E" ß á ß E-. ß F , suppose there are 2 L 's and 5 O 's, where 2  5 œ -  .  ". Then either 2 -  " or 5 .  " and we may assume without loss of generality that 2 -  ". Since >O is deflationary, removing all >O 's from the expression for P results in a possibly larger subgroup Q œœ

>2L ÐOÑ >2" L ÐLÑ

if F œ O if F œ L

However, in the former case, 2" 2" >2L ÐOÑ œ >2" L >L ÐOÑ œ >L >O ÐLÑ Ÿ >L ÐLÑ

and so P Ÿ Q Ÿ >2" L ÐLÑ Ÿ >L ÐLÑ œ Ö"×

Hence, >-. LO ÐLOÑ œ Ö"× which implies that LO is nilpotent of class at most -  . .

An Example We now describe a family of groups showing that for any - !, there are nilpotent groups of nilpotency class - . Let `8 ÐVÑ be the family of all 8 ‚ 8 matrices over a commutative ring V with identity and let Y œ Y X Ð8ß VÑ be the unitriangular matrices over V . (Recall that a matrix is unitriangular if it is upper triangular, with "'s on the main diagonal.) Denote the Ð3ß 4Ñth entry in Q by Q3ß4 . For 5 !, the 5 th superdiagonal Q Ð5Ñ of a matrix Q are the elements of the form Q3ß35 . For ! Ÿ 5 Ÿ 8  ", let R5 be the set of all 8 ‚ 8 matrices over V with !'s on or below the 5 th superdiagonal, that is, for E − `8 ÐVÑ, E − R5

Í

E3ß4 œ ! for all 4 Ÿ 3  5

It is routine to confirm that R5 R7 © R57" In particular, R5 R5 © R5 For 5 !, let Y5 œ ÖM×  R5 œ ÖM  E ± E − R5 ×

304


To see that Y5 is a subgroup of Y , we have Y5 Y5 œ ÐÖM×  R5 ÑÐÖM×  R5 Ñ © ÖM×  R5  R5  R5 R5 © Y5 and if E − R5 , then since E? œ ! for some ? !, it follows that ÐM  EÑ" œ M  E  E#  â „ E?" − Y5 As to commutators, if E − R5 and F − R7 , then ÒM  Eß M  FÓ œ ÐM  EÑÐM  FÑÐÐM  FÑÐM  EÑÑ" œ ÐM  E  F  EFÑÐM  F  E  FEÑ" œ ÐM  \ÑÐM  ] Ñ" œ M  Ð\  ] ÑÐM  ] Ñ" where \ œ E  F  EF

and ] œ F  E  FE

Moreover, \  ] œ EF  FE − R57" implies that ÒM  Eß M  FÓ − Y57" and so ÒY5 ß Y7 Ó Ÿ Y57" Taking 7 œ ! gives ÒY5 ß Y Ó Ÿ Y5" Ÿ Y5 which shows both that Y5 is normal in Y and that Y5" – Y5 is central in Y . Thus, the series ÖM× œ Y8" – â – Y" – Y! œ Y is central and so Y is nilpotent of class at most 8  ". On the other hand, let I3ß4 denote the matrix with all !s except for a " in the Ð3ß 4Ñthe position. Then for 3 Á 4  ", an easy calculation shows that ÒM  I3ß4 ß M  I4ß4" Ó œ M  I3ß4" In particular, if 8 $, then ÒM  I"ß# ß M  I#ß$ Ó œ M  I"ß$ and so >ÐKÑ œ ÒKß KÓ Á ÖM×. Also, if 8 %, then ÒM  I"ß$ ß M  I$ß% Ó œ M  I"ß% and so ># ÐKÑ œ Ò>ÐKÑß KÓ Á ÖM×. More generally,


305

ÒM  I"ß8" ß M  I8"ß8 Ó œ M  I"ß8 Á M and so >8# ÐKÑ Á ÖM×, which shows that Y is nilpotent of class 8  ".

Solvability We now turn to a discussion of solvable groups.

Perspective on Solvability Solvable groups have played an extremely important role in the study of the location of the roots of polynomials over a field J . Let us pause to describe this role in general terms. For more details, we refer to reader to Roman, Field Theory [27]. If J is a subfield of a field I , we say that J Ÿ I is a field extension. Associated to each field extension J Ÿ I is a group KJ ÐIÑ, called the Galois group of the extension and defined as the group of all (ring) automorphisms 5 of I that fix the elements of J , that is, for which 5+ œ + for all + − J . It turns out that the properties of the “simpler” Galois group can often shed considerable light on properties of the “more complex” field extension. To illustrate, one of the principal motivations for the development of abstract algebra since, oh say 3000 B.C., has been the desire (expressed in one form or another) to find the roots of a polynomial :ÐBÑ with coefficients from a given base field J . In fact, we now know that for any nonconstant polynomial :ÐBÑ of degree ., there is an extension J of J , called an algebraic closure of J , that contains a full set of . roots for :ÐBÑ. Moreover, lying between the fields J and J is the smallest field I containing these roots of :ÐBÑ, called a splitting field for :ÐBÑ. The desire to express the roots of :ÐBÑ by arithmetic formula (similar to the quadratic formula) or to show that this could not be done is what motivated Galois to first define some version of what we now know as a group. The idea of expressing the roots of a polynomial :ÐBÑ by formula means that, starting with the elements of the base field J , we can “capture” all of the roots of :ÐBÑ through a finite number of special types of extensions of J . In particular, for each extension, we are allowed to include an 8th root of an existing element and only whatever else is required in order to make a field. Specifically, for the first extension, we may choose any ?! − J and any root 8!

307

However, W8 is not solvable for 8 &, since E8 is simple and so the only nontrivial series for W8 is Ö+× – E8 – W8 , which is not abelian. Thus, the roots of the polynomials described above cannot be captured within a radical series, that is, these polynomials are not solvable by radicals. Note that this shows that there are individual polynomials that are not solvable by radicals. Thus, not only is there no general formula, similar to the quadratic, cubic and quartic formulas, for the solutions of arbitrary quintic equations, but there are even individual quintic equations whose solutions are not obtainable by formula! Galois used his remarkable theory in his paper Memoir on the Conditions for Solvability of Equations by Radicals of 1831 (but not published until 1846!), to show that the general equation of degree & or larger is not solvable by radicals. (Proofs that the &th degree equation is not solvable by radicals were offered earlier: An incomplete proof by Ruffini in 1799 and a complete proof by Abel in 1826.) Thus, the notion of solvability arose through the desire to settle the question of whether we could solve all polynomial equations by simple formula. Of course, solvable groups are important for other reasons. In fact, we will see that the class of solvable groups has a sort of super-Sylow theorem, to wit, if K is solvable of order 78 where Ð7ß 8Ñ œ ", then K has a (Hall) subgroup of order 7 and all subgroups of order 7 are conjugate.

The Derived Series For any solvable group, there is an abelian series that descends more rapidly than any other abelian series. Moreover, this series is also characteristic in K. A series Ö"× œ K! ü K" ü â ü K7 œ K is abelian if and only if w K5" Ÿ K5

for all 5 œ !ß á ß 7  ", where the prime denotes commutator subgroup. Definition Let K be a group. The subgroups defined by KÐ!Ñ œ K,

KÐ"Ñ œ Kw

and, in general for 8 ", KÐ8Ñ œ ÐKÐ8"Ñ Ñw are called the higher commutators of K. The group KÐ8Ñ is called the 8th commutator subgroup of K. The series of higher commutators:

308


â ö KÐ$Ñ ö KÐ#Ñ ö KÐ"Ñ ö K is called the derived series for K. The monotonicity of the commutator operation implies that the derived series descends from K more rapidly than any other abelian series. Moreover, since KÐ5"Ñ « KÐ5Ñ , the derived series is characteristic. Theorem 11.12 Let K be a group. 1) The derived series â ö KÐ$Ñ ö KÐ#Ñ ö KÐ"Ñ ö K is the abelian series of steepest descent, in the sense that if the series âK$ ü K# ü K" ü K! œ K is abelian, then KÐ5Ñ Ÿ K5 for all 5 . 2) K is solvable if and only if its derived series reaches the trivial group, that is, if and only if there is a 5 " for which KÐ5Ñ œ Ö"×. The smallest integer 8 for which KÐ8Ñ œ Ö"× is called the derived length of K, which we denote by derlenÐKÑ. 3) A group K is solvable if and only if it has a normal abelian series. 4) The length of any abelian series for K is greater than or equal to the derived length of K. We will have use for the following fact about higher commutators of quotient groups. Theorem 11.13 Let K be a group and let R ü K. Then K Œ R

Ð8Ñ

œ

K Ð8Ñ R R

for all 8 !. Proof. For any R Ÿ L ü K, we have Œ

L R

w

œ”

L L ÒLß LÓR L wR ß •œ œ R R R R

In particular, for L œ K, we have Œ

K R

w

œ”

K K ÒKß KÓR Kw R ß •œ œ R R R R


309

and so the result holds for 8 œ ". Assuming that the result holds for an arbitrary 5 , we have with L œ KÐ8Ñ R , K Œ R

Ð8"Ñ

K œ –Œ R

Ð8Ñ w

— œ

KÐ8Ñ R R

w

œ

ÐKÐ8Ñ R Ñw R R

Finally, Theorem 3.41 implies that ÐKÐ8Ñ R Ñw R œ ÒKÐ8Ñ R ß K Ð8Ñ R ÓR œ ÒK Ð8Ñ ß K Ð8Ñ ÓR œ K Ð8"ÑR and so the result follows. Theorem 11.14 If K is solvable, Eß Fß L Ÿ K and R ß O ü K, then 1) derlenÐLÑ Ÿ derlenÐKÑ 2) derlenÐKÎR Ñ Ÿ derlenÐKÑ 3) derlenÐKÑ Ÿ derlenÐR Ñ  derlenÐKÎR Ñ 4) derlenÐE  FÑ Ÿ maxÖderlenÐEÑß derlenÐFÑ×. Proof. For part 1), since L Ð5Ñ Ÿ KÐ5Ñ for all 5 !, if 8 œ derlenÐKÑ then L Ð8Ñ Ÿ KÐ8Ñ œ Ö"×. Thus, derlenÐLÑ Ÿ 8. For part 2), Theorem 11.13 implies that if KÐ8Ñ œ Ö"×, then ÐKÎR ÑÐ8Ñ œ ÖR ×. For part 3), if KÎR has derived length ., then KÐ.Ñ R ÎR œ ÖR × and so KÐ.Ñ Ÿ R . If the derived length of R is /, then KÐ./Ñ Ÿ R Ð/Ñ œ Ö"× and so the derived length of K is at most .  /. Part 4) follows from the fact that ÐE  FÑw œ Ew  F w .

Properties of Solvable Groups If R is a minimal normal subgroup of a group K and if Z À Ö"× œ K. ü â ü K! œ K is any normal series in K , then R Ÿ K3 or else R ∩ K3 œ Ö"× for all 3 and so there is an index 5 for which R Ÿ K5

and R ∩ K5" œ Ö"×

Therefore, if K is solvable and if Z is the derived series for K , then R Ÿ KÐ5Ñ

and

R ∩ KÐ5"Ñ œ Ö"×

and so R w Ÿ R ∩ KÐ5"Ñ œ Ö"×, whence R is abelian. Theorem 11.15 Let K be solvable. Any minimal normal subgroup R of K is abelian. Moreover, if R contains a nontrivial element of finite order, then R is elementary abelian.

310


Proof. For the final statement, R has an element of prime order : and since R: ³ ÖB − R ± B: œ "× ü R it follows that R œ R: is an elementary abelian group. If K is solvable and has a composition series, then the factor groups of the composition series are both simple and solvable and therefore cyclic of prime order. Theorem 11.16 The following are equivalent for a group K that has a composition series. 1) K is solvable. 2) Every composition series for K has prime order factor groups. 3) K has a cyclic series in which each factor group has prime order. 4) K has a cyclic series, that is, K is polycyclic. 5) Every chief series for K has factor groups that are elementary abelian. Proof. We have seen that 1) implies 2) and it is clear that 2) implies 3), that 3) implies 4) and that 4) implies 1). Thus, 1)–4) are equivalent. It is clear that 5) implies 1). If K is solvable, the factor groups K5" ÎK5 of a chief series are minimal normal in the solvable group KÎK5 and so are elementary abelian by Theorem 11.15. The following theorem contains some sufficient (but not necessary) conditions for solvability. The proof of the Feit–Thompson Theorem is quite involved, running almost 300 pages. For a proof of the Burnside result, we refer the reader to Robinson [26]. Theorem 11.17 1) (Feit–Thompson Theorem) Any group of odd order is solvable; equivalently, every finite nonabelian simple group has even order. 2) (Burnside :; Theorem) Every group of order :7 ; 8 where : and ; are primes, is solvable. Proof. The equivalence in part 1) is left as an exercise.

Hall's Theorem on Solvable Groups Let K be a finite group. Recall that a Hall subgroup L of K is a subgroup with the property that its order 9ÐLÑ and index ÐK À LÑ are relatively prime. The Schur–Zassenhaus Theorem tells us that every normal Hall subgroup has a complement and that all such complements are conjugate. As to the existence of Hall subgroups, the Sylow theorems tell us that if 9ÐKÑ œ :5 7 with : prime and Ð:5 ß 7Ñ œ ", then K has a Hall (Sylow) subgroup of order :5 and that all such subgroups are conjugate. In 1928, Philip


311

Hall showed that for a finite solvable group, this result applies not just to prime power factors. Theorem 11.18 (Hall's theorem, 1928) Let K be a finite solvable group with 9ÐKÑ œ +, , where Ð+ß ,Ñ œ ". Then K has a Hall subgroup of order + and all subgroups of order + are conjugate. Proof. We may assume that +ß ,  ". The proof is by induction on 9ÐKÑ. If 9ÐKÑ œ ", the result holds trivially. Assume that it holds for all groups of order less than 9ÐKÑ. If K does not have a minimal normal subgroup, then K is simple and solvable and therefore cyclic of prime order and so +ß ,  " is false. Thus, K has a minimal normal subgroup R , which as we have seen, is an elementary abelian group of prime power order :7 . There are cases to consider, based on whether :7 ± + or :7 ± , . Case 1: 9ÐR Ñ œ :7 œ , In this case, R is a normal Hall subgroup of K. Hence, the Schur–Zassenhaus Theorem implies that R has a complement and all such complements are conjugate. But the complements of R are precisely the subgroups of order +. Case 2: 9ÐR Ñ œ :7 ± , and :7  , If :7 ± , but :7 Á , , then 9ÐKÎR Ñ œ +Ð,Î:7 Ñ  +, and so the inductive hypothesis implies that KÎR has a subgroup OÎR of order +. Hence, 9ÐOÑ œ +:7  +, and the inductive hypothesis applied to O shows that O (and hence K) has a subgroup L of order +. As to conjugation, if 9ÐL" Ñ œ 9ÐL# Ñ œ +, then L3 ∩ R œ Ö"× and so 9ÐL3 R ÎR Ñ œ +. Hence, the inductive hypothesis implies that Œ

L# R R

BR

œ

L" R R

for some B − K and so L#B R œ L" R But 9ÐL" R Ñ œ 9ÐKÑ and L" and L#B are Hall subgroups of L" R of order +. Hence, the induction hypothesis implies that L" and L#B are conjugate in L" R , whence in K. Case 3: 9ÐR Ñ œ :7 ± + If 9ÐR Ñ œ :7 ± +, then 9ÐKÎR Ñ œ Ð+Î:7 Ñ,  +, and the inductive hypothesis implies that KÎR has a subgroup OÎR of order +Î:7 , whence 9ÐOÑ œ +. As to conjugacy, if 9ÐLÑ œ +, then R Ÿ L , for if not, then 9ÐLR Ñ ± 9ÐLÑ9ÐR Ñ œ +: 7 and so R L is a subgroup of K of order greater than + but relatively prime to , ,

312


which contradicts Lagrange's theorem. Therefore, if 9ÐL" Ñ œ 9ÐL# Ñ œ +, then L" ÎR and L# ÎR are Hall subgroups of KÎR of order +Î:7 and so L" ÎR and L# ÎR are conjugate in KÎR , whence L" and L# are conjugate in K. A sort of converse of the previous theorem also holds. The proof uses the Burnside :; theorem (Theorem 11.16). Definition Let K be a finite group and let : be a prime for which 8 œ :5 7, where Ð7ß :Ñ œ " and 5 ". Then a Hall :w -subgroup of K is a subgroup L of order 7. Theorem 11.19 If a finite group K has a Hall :w -subgroup for every prime : dividing 9ÐKÑ, then K is solvable. Proof. Assume that the theorem is false and let K be a counterexample of smallest order. If K is not simple, then let R be a nontrivial proper normal subgroup of K. We leave it as an exercise to show that R ∩ L is a Hall :w subgroup of R and LR ÎR is a Hall :w -subgroup of KÎR . Hence, R and KÎR are solvable and therefore so is K, a contradiction. Hence, K is simple. Now suppose that 9ÐKÑ œ :"/" â:8/8 , where the :3 's are distinct primes and /3 ". The Burnside :; theorem implies that 5 $. If K3 is a Hall :3w -subgroup of K, then 9ÐK3 Ñ œ 9ÐKÑÎ:3/3 and the Poincaré theorem and the fact that the indices ÐK À K3 Ñ are pairwise relatively prime imply that for any 5 of the groups K3 , ÐK À K3" ∩ â ∩ K35 Ñ œ $4 :34 4 /3

and so kK3" ∩ â ∩ K35 k œ

9ÐKÑ / #4 :3 34 4

Also, for any 3, ¹K3 ì ,4Á3 K4 ¹ œ kK3 k¹,4Á3 K4 ¹ œ and so

9ÐKÑ 9ÐKÑ œ 9ÐKÑ :3/3 #4Á3 :4/4

K3 ì ,4Á3 K4 œ K

If L œ K$ ∩ â ∩ K8 , then 9ÐLÑ œ :"/" :#/# , which is solvable by the Burnside :; theorem. If L is simple, then L is abelian and so L is cyclic of prime order, which is false. Hence, let R be a minimal normal subgroup of L . Then R is elementary abelian of exponent, say :" and so is contained in any Sylow :" subgroup of L . But K# ∩ L has order :"/" and so R ü K# ∩ L . Now,


313

R K" ∩L Ÿ R and so R K Ÿ R K# ÐK" ∩LÑ Ÿ R K# Ÿ K# and so the normal closure R K is a proper nontrivial normal subgroup of K, a contradiction.

Exercises 1. 2.

Show that the classes of cyclic groups, abelian groups and nilpotent groups do not have the extension property. Show that if E – F is abelian or cyclic, then any refinement EüLüF is also abelian or cyclic.

Nilpotent Groups 3. 4. 5. 6. 7. 8. 9.

Can a nontrivial centerless group be nilpotent? a) Prove that any finite nilpotent group is supersolvable. b) Find an example to show that not every finite supersolvable group is nilpotent. Let K be a finite group. Prove that K is nilpotent if and only if every nontrivial quotient group of K has a nontrivial center. Let K be nilpotent but not abelian. Let E Ÿ K be maximal with respect to being normal in K and abelian. Prove that E œ GK ÐEÑ. Hint: Show that GÎE ü KÎE and that GK ÐEÑÎE ∩ ^ÐKÎEÑ Á Ö"×. For any even positive integer 8, prove that every group of order 8 is nilpotent if and only if 8 is a power of #. Prove that a nilpotent group is supersolvable if and only if it satisfies the ascending chain condition on subgroups. If L is nilpotent of class - and O is nilpotent of class . , prove that L } O is nilpotent of class maxÐ-ß .Ñ.

Solvable Groups 10. Prove that W% is solvable but not supersolvable. 11. Prove that W8 is solvable for 8 Ÿ %. 12. Assuming that E& is the smallest nonabelian simple group (which it is), prove that every group of order less than '! is solvable. 13. Let R be a nontrivial proper normal subgroup of a group K. Let : be a prime and : ± 9ÐR Ñ. Let L be a Hall :w -subgroup of K. Prove that R ∩ L and R LÎR are Hall :w -subgroups of R and KÎR , respectively. 14. Prove that the following are equivalent for a finite group K: a) K is solvable. b) Every nontrivial normal subgroup of K has a nontrivial abelian quotient group. c) Every nontrivial quotient group of K has a nontrivial abelian normal subgroup.

314


/7 15. Let K be a finite group of order 8 œ :"/" â:7 . Prove that K is solvable if and only if the composition length of K is - œ /3 . 16. Prove that a solvable group with a composition series must be finite. 17. Let K be a nontrivial finite solvable group. a) Prove that b: ÐKÑ is nontrivial for some prime :, that is, K has a nontrivial normal :-subgroup. b) Prove that b; ÐKÑ is nontrivial for some prime ;, that is, K has a normal subgroup L for which KÎL is a nontrivial ; -group. 18. Let K be a finite group. Prove directly that an abelian series can always be refined into a cyclic series with prime order factor groups. 19. Prove that the following are equivalent: a) Any finite group of odd order is solvable. b) Any finite nonabelian simple group has even order. 20. a) Prove that a finite group K is solvable if and only if W w Á W for all subgroups W Á Ö"× of K. b) Prove that if K contains a nonabelian simple subgroup W , then K is not solvable. c) Show that W w Á W for all subgroups of the dihedral group H#8 , showing that H#8 is solvable. Hint: Find an abelian subgroup R of index #. How do subgroups interact with R ? 21. A subgroup L of a group K is abnormal if

+ − ØLß L + Ù for all + − K. Prove that the normalizer of a Hall subgroup of a solvable group is abnormal.

Polycyclic Groups 22. a)

Let K be a polycyclic group with a cyclic series of length 8. Prove that K is 8-generated. b) Let E be an 8-generated abelian group for 8 ". Prove that E is polycyclic. 23. Prove that the following are equivalent: a) K is polycyclic. b) Every subgroup of K is finitely generated and solvable. c) Every normal subgroup of K is finitely generated and solvable. 24. Prove that a group K is polycyclic if and only if it is solvable and satisfies the maximal condition on subgroups, that is, if and only if every nonempty collection of subgroups of K as a maximal member. 25. a) Let E  F have an infinite cyclic factor group. Let E œ L!  L"  â  L 7 œ F be a proper refinement of E  F . Describe the factor groups of this refinement.


315

b) Let K be polycyclic. Prove that the number of steps whose factor group is infinite is the same for all cyclic series for K. Hint: Any two cyclic series have isomorphic refinements.

Supersolvable Groups 26. Prove that any supersolvable group is countable. 27. Prove that if KÎR is supersolvable and R is cyclic, then K is supersolvable. 28. Prove that a group is supersolvable if and only if it has a series in which each factor group is cyclic of prime order or cyclic of infinite order. Hint: Recall that E « F and F ü K implies E ü K . 29. Let K be supersolvable. Prove that if L is a maximal subgroup of K , then ÐK À LÑ is prime. Hint: First assume that L – K and look at KÎL . Then assume that L is not normal in K and factor by the normal interior L ‰ . Conclude that it is sufficient to consider L ‰ œ Ö"×. Consider the subgroup E in the first step Ö"×  E in a normal cyclic series and how it interacts with L . 30. Prove that if K is supersolvable, then Kw is nilpotent. Hint: Let Ö"× œ K!  â  K8 œ K be a normal cyclic series for K and consider the series Ö"× œ K! ∩ Kw  â  K8 ∩ Kw œ Kw which is normal and cyclic as well. Let F œ K5" and E œ K5 . To show that the series is central, it is sufficient to show that \ EÐF ∩ Kw Ñ EKw ³ Ÿ ^Œ E E E But \ÎE is cyclic and normal in KÎE. What can be said about ÐKÎEÑw ?

Radicals and Residues Definition Let ^ be a class of groups. Let K be a group. 1) If the partially ordered set ^ÐKÑ œ ÖL ü K ± L − ^× has a top element, it is called the ^-radical for K and is denoted by b^ ÐKÑ. 2) If the partially ordered set KÎ^ œ ÖL ü K ± KÎL − ^× has a bottom element, it is called the ^-residue for K and is denoted by b^ ÐKÑ.

316


31. Show that the ^-radical b^ ÐKÑ and the ^-residue b ^ ÐKÑ are characteristic subgroups of K (if they exist). 32. Let ^ be a class of groups closed under subgroup, quotient and join if at least one factor is normal. Let K be a group and let L ü K . Assume that all mentioned radicals and residues exist. Prove the following: a) If L Ÿ b^ ÐKÑ, then b^ ÐKÑ K Ÿ b^ Œ L L and the inclusion may be proper. b) If L Ÿ b ^ ÐKÑ, then b^ Œ

K L

œ

b ^ ÐKÑ L

33. Let ^ be a class with the extension property: R − ^ß KÎR − O implies K − ^. Prove that the following hold for any group K: a) The ^-radical of KÎb^ ÐKÑ is trivial, that is, b^ Œ

K b^ ÐKÑ

œ Öb^ ÐKÑ×

b) The ^-residue of b ^ ÐKÑ is b ^ ÐKÑ, that is, b^ Ðb^ ÐKÑÑ œ b^ ÐKÑ 34. Let ^ be the class of finite groups. a) Show that there are groups with no ^-radical. b) Show that there are groups with no ^-residue. 35. Let K be a nontrivial finite group. Prove that the following are equivalent: a) K is solvable. b) For every proper normal subgroup O – K, the factor group KÎO has a nontrivial :-radical b: ÐKÎOÑ for some prime :, that is, KÎO has a nontrivial normal :-subgroup. c) For every nontrivial characteristic subgroup O of K, the ; -residue b ; ÐOÑ of O is proper in O for some prime ; , that is, there is a proper normal subgroup E of O such that OÎE is a nontrivial a ; -group.

Additional Problems 36. Let K be a group. Let > œ >K . 5 a) Prove that KÐ5Ñ Ÿ >K ÐKÑ. b) Prove that derlenÐKÑ Ÿ nilclassÐKÑ. 37. Let K be a group. Let > œ >K . Prove the following: a) Ò>5 ÐKÑß >4 ÐKÑÓ Ÿ >54" ÐKÑ Hint: Use induction. Use the three subgroups lemma on Ò>5" ÐKÑß >4" ÐKÑÓ œ ÒÒ>5 ÐKÑß KÓß >4" ÐKÑÓ. b) >5>4 ÐKÑ ÐKÑ Ÿ >54" ÐKÑ for 5ß 4 ".


317

c) Ò>5 ÐKÑß ' 4 ÐKÑÓ Ÿ ' 45" ÐKÑ for 4 5  ". d) KÐ5Ñ Ÿ >j5 ÐKÑ, where j5 œ #5  ". Hence, derlenÐKÑ is less than or equal to the smallest integer 5 for which j5 - ³ nilclassÐKÑ and so derlenÐKÑ Ÿ ilog# Ð-  "Ñj

Chapter 12

Free Groups and Presentations

Throughout this chapter, \ denotes a nonempty set of formal symbols and \ " denotes the set of formal symbols ÖB" ± B − \×. Further, we assume that \ and \ " are disjoint and write \ w œ \ “ \ " .

Free Groups The idea of a free group J\ on a nonempty set \ is that J\ should be the “most general” possible group containing \ , that is, the elements of \ should generate J\ but have no relationships within J\ . In this case, \ is referred to as a set of free generators or a basis for the free group J\ . To draw an analogy, if Z is a vector space, then a subset U of Z is a basis for Z if and only if for any vector space [ and any assignment of vectors in [ to the vectors in U , there is a unique linear transformation from Z to [ that extends this assignment. This property and the analogous property that defines free groups are best described using univerality. Definition Let \ be a nonempty set. A pair ÐJ ß ,À \ Ä J Ñ where J is a group has the universal mapping property for \ (or is universal for \ ) if, as pictured in Figure 12.1, for any function 0 À \ Ä K from \ to a group K, there is a unique group homomorphism 70 À J Ä K for which 70 ‰ , œ 0 The map 70 is called the mediating morphism for 0 . In this case, we say that 0 can be factored through , or that 0 can be lifted to J . The group J is called a free group on \ and \ is called a set of free generators or a basis for J . The map , is called the universal map for the pair ÐJ ß ,Ñ. We use the notation J\ to denote a free group on \ .


319

320


κ

X

F

∃!τf

f

G Figure 12.1 It is clear that the universal map , is injective. Moreover, ,\ generates J\ , for if Ø,\Ù  J\ , then Theorem 4.17 implies that there are distinct group homomorphisms 5ß 7 À J\ Ä K into some group K that agree on Ø,\Ù. Therefore, if 0 œ 5l\ œ 7 l\ , then the uniqueness condition of mediating morphisms is violated. Hence, ,\ generates J . For these reasons, it is common to suppress the map , and think of \ as a subset of J\ . The following theorem says that the universal property characterizes groups up to isomorphism. Theorem 12.1 Let \ be a nonempty set. If ÐJ ß ,Ñ and ÐKß -Ñ are universal for \ , then there is an isomorphism 5 À J ¸ K connecting the universal maps, that is, for which 5‰, œProof. There are unique mediating morphisms 7- À J Ä K and 7, À K Ä J for which 7- ‰ , œ -

and

7, ‰ - œ ,

7- ‰ 7, ‰ - œ -

and

7, ‰ 7- ‰ , œ ,

and so

But the identity maps +K and +J are the unique mediating morphisms for which +K ‰ - œ -

and

+J ‰ , œ ,

and so 7- ‰ 7, œ + K

and 7, ‰ 7- œ +J

which shows that the maps 7- and 7, are inverse isomorphisms. Definition Let \ be a nonempty set. A word AÐB" ß á ß B8 Ñ over \ w (or the equation AÐB" ß á ß B8 Ñ œ ") is a law of groups if AÐ+" ß á ß +8 Ñ œ " for all groups K and all +3 − K.


321

The following theorem says that what's true in a free group is true in all groups. Theorem 12.2 Let \ be a nonempty set and let AÐB" ß á ß B8 Ñ be a word over \ w . If ÐO\ ß ,Ñ is universal on \ , then the following are equivalent: 1) AÐ,B" ß á ß ,B8 Ñ œ " in J\ 2) AÐB" ß á ß B8 Ñ is a law of groups. Proof. Let K be a group and let +3 − K for 3 œ "ß á ß 8. The function sending B3 to +3 can be lifted to a unique homomorphism 5À J\ Ä K for which 5,B3 œ +3 and so AÐ+" ß á ß +8 Ñ œ 5AÐ,B" ß á ß ,B8 Ñ œ 5" œ "

Cauchy's theorem says that any group is isomorphic to a subgroup of a symmetric group. There is an analog for quotients of free groups, but first we require a definition. Definition Let Y œ ÖK3 ± 3 − M× be a family of groups. A subgroup O of the direct product } Y is called a subdirect product of the family Y if the restricted projection maps 33 lO À O Ä K3 are surjective for all 3 − M . If K is a group, then the identity map +À K Ä K can be lifted to an epimorphism 5À JK q» K and so K is isomorphic to a quotient of the free group JK . More generally, if \ is a set for which cardÐ\Ñ cardÐKÑ, then any surjection 0 À \ q» K can be lifted to an epimorphism 5À J\ q» K and the induced map 5À J\ ÎO ¸ K shows that K is isomorphic to a quotient group of the free group J\ . Moreover, we have freedom to choose the values of 7 ÐBOÑ for B − \ arbitrarily, but O depends on that choice. More generally, if Y œ ÖK3 ± 3 − M× is a nonempty family of groups and if \ is a set for which cardÐ\Ñ cardÐK3 Ñ for all 3 − M , then there are isomorphisms 73 À J\ ÎO3 ¸ K3 for all 3 − M , where 73 ÐBO3 Ñ can be specified arbitrarily, but O3 depends on that choice. Now, the “Chinese” map 5À J\ Ä } J\ ÎO3 3−M

defined by 5ÐAÑÐ3Ñ œ AO3 for A − J\ has kernel M œ +O3 . Hence, if 5 is the induced embedding, then the composition }73 ‰ 5À J\ ÎM ä } 3−M K3 shows that J\ ÎM is isomorphic to a subdirect product of the family Y . Moreover, we can specify that the element BM be sent to any element of } K3 , for all B − \ (again at the expense of O3 ). Theorem 12.3 Let K be a group, let Y œ ÖK3 ± 3 − M× be a nonempty family of groups and let \ be a set.

322


1) If cardÐ\Ñ cardÐKÑ, then there is an isomorphism 7 À J\ ÎR ¸ K where we can choose the elements 7B for B − \ arbitrarily, but R depends on that choice. 2) Suppose that cardÐ\Ñ cardÐK3 Ñ and that we have specified isomorphisms 73 À K3 ¸ J\ ÎO3 for all 3 − M . Let M œ +O3 . Then J\ ÎM is isomorphic to a subdirect product of the family Y and the isomorphisms 73 can be used to specify the elements BM for B − \ arbitrarily in } K3 , but M depends on that choice.

Construction of the Free Group The notion of a free group can be defined constructively, without appeal to the universal mapping property. When a constructive approach is taken, one usually hastens to verify the universal mapping property, since this is arguably the most useful property of free groups. On the other hand, since we have chosen to define free groups via universality, we should hasten to give a construction for free groups. Let [ œ Ð\ w Ñ‡ be the set of all words over the alphabet \ w . As a shorthand, we allow the use of exponents, writing Ú BâB î Ý Ý Ý 8 factors B8 œ Û ðóñóò B" âB" Ý Ý Ý 8 factors Ü%

if 8  ! if 8  ! if 8 œ !

where % is the empty word. It is important to keep in mind that this is only a shorthand notation. Thus, for example, B% B# and B' are both shorthand for BBBBBB and so B% B# œ B' . However, B% B# is shorthand for BBBBB" B" but B# is shorthand for BB and so B% B# Á B# . Since the operation of juxtaposition on [ is associative and since the empty word % is the identity, the set [ is a monoid under juxatpostion. In an effort to form a group, we also want to require that BB" œ % œ B" B for all B − \ . More specifically, consider the following rules that can be applied to members of [ : 1) Removal rules: For =ß . − [ and B − \ , =BB" . Ä =. =B" B. Ä =. BB" Ä % B" B Ä % where one of = or . may be missing.


323

2) Insertion rules: For =ß . − [ and B − \ , =. Ä =BB" . =. Ä =B" B. % Ä BB" % Ä B" B where one of = or . may be missing. Let us refer to a finite sequence =" ß á ß =5 of applications of these rules as a reduction of ? to @ of length 5 (even though @ may have greater length than ?). The trivial reduction is an application of no rules and so ? is obtained from itself by the trivial reduction. Since the removal and insertion rules come in inverse pairs, reduction defines an equivalence relation ´ on [ . Let [ Î ´ denote the set of equivalence classes of [ , with ÒAÓ denoting the equivalence class containing A. Since equivalent words must represent the same group element, it is really the equivalence classes that are the candidates for the elements of the free group J\ on \ . Moreover, since ? ´ 7 Ñ for >3 − W and /3 − ™, which is not possible. It follows that \Î#E\ is a basis for E\ Î#E\ and so dimÐE\ Î#E\ Ñ œ k\ k

and similarly for ] Þ Thus, since E\ ¸ E] implies E\ Î#E\ ¸ E] Î#E] , we have k\ k œ dimÐE\ Î#E\ Ñ œ dimÐE] Î#E] Ñ œ k] k

and if W generates E\ , then

kW k kWÎ#E\ k k\ k

This completes the proof of part 1). For part 2), if J\ ¸ J] , then E\ ¸ J\ ÎJ\w ¸ J] ÎJ]w ¸ E]

and so part 1) implies that k\ k œ k] k. Finally, if W generates J\ , then

334


WÎJ\w œ Ö=J\w ± = − W× generates E\ œ J\ ÎJ\w and so kW k kWÎJ\w k k\ÎJ\w k œ k\ k

Theorem 12.17 Let E\ be free abelian on \ . Then all independent sets have cardinality at most k\ k. Proof. It is sufficient to prove the result for E œ { B−\ ™B where ™B œ ™ for all B. The set Z œ { B−\ B is a vector space over the rational field  and it is easy to see that a subset U œ Ö@3 ± 3 − M× © E is dependent in E if and only if U is linearly dependent over . But in the vector space Z , all sets of cardinality greater than k\ k are linearly dependent over  and therefore also over ™. The Nielsen–Schreier Theorem says that every subgroup of a free group is free and so the subgroups of free groups are very restricted. (Nielsen proved this result for finitely-generated groups in 1921 and Schreier generalized it to all groups in 1927.) Theorem 12.18 1) Any subgroup of a free group is free. 2) Any subgroup W of a free abelian group E\ is free abelian and rkÐWÑ Ÿ rkÐEÑ. Proof. We omit the difficult proof of part 1) and refer the interested reader to Robinson [26]. For part 2), we may assume that E\ œ { B−\ ØBÙ and that \ is well ordered. Since the elements of W have finite support, for 0 − W , we can let 3Ð0 Ñ be the largest index B for which 0 ÐBÑ Á ". For each B − \ , consider the set MB œ Ö0 ÐBÑ ± 0 − Wß 3Ð0 Ñ Ÿ B× Then MB Ÿ ØBÙ and so MB œ Ø0B ÐBÑÙ for some 0B − WB . We show that W is free on the set U œ Ö0B ± B − \ß 0B ÐBÑ Á "× If U does not span W , among those elements of W not in the span of U , choose an element 1 for which C œ 3Ð1Ñ is the smallest possible. Since 1ÐCÑ − MC œ Ø0C ÐCÑÙ, it follows that 1ÐCÑ œ 0C5 ÐCÑ for some nonzero 5 − ™. Then Ð10C5 ÑÐBÑ œ 1ÐBÑ0C5 ÐBÑ œ " for all B C and so 3Ð10C5 Ñ  C, which implies that 10C5 is in the span of U . But then


335

1 œ Ð10C5 Ñ0C5 is also in the span of U , a contradiction. Thus, U spans W . Also, U is independent, since if 0B/"" â0B/88 œ " where B3  B4 for 3  4, then applying this to B8 gives 0B/88 ÐB8 Ñ œ " and so /8 œ !. Similarly, /3 œ ! for all 3 and so U is independent. Hence, Theorem 12.15 implies that W is free over U . Also, it is clear that kU k Ÿ k\ k and so rkÐWÑ Ÿ rkÐEÑ.

Applications of Free Groups Sometimes free groups can help produce complements. Theorem 12.19 If 5À K q» J\ is an epimorphism, where J\ is free on \ , then O œ kerÐ5Ñ is complemented in K. Proof. Define a function 7 À \ Ä K by letting 7 B be a fixed member of 5" ÐBÑ. Since J\ is free on \ , there is a unique homomorphism 7 À J\ Ä K that extends 7 on \ . Since for any B − \ , 5 ‰ 7 ÐBÑ œ B it follows that 5 ‰ 7 œ +. In other words, 7 is a right inverse of 5 and so Theorem 5.23 implies that kerÐ5Ñ is complemented in K . With the help of free groups, we can provide an example of a finitely-generated nonabelian group with a subgroup that is not finitely generated. We have already proved (Theorem 2.21) that if K is an 8-generated abelian group, then every subgroup of K can be generated by 8 or fewer elements. Theorem 12.20 Let \ œ ÖBß C× and let J\ be the #-generated free group on \ . Let K œ ØWÙ, where W œ ÖC5 BC5 ± 5  !× Then K is isomorphic to the free group J^ on a countably infinite set ^ œ ÖD" ß D# ß á × and so is not finitely generated. Proof. Consider the function 0 À ^ Ä K defined by 0 ÐD5 Ñ œ C5 BC5 . Then there is a unique mediating morphism 7 À J^ Ä K for which 7 ÐD5 Ñ œ C5 BC5 . It is clear that 7 is surjective.

336


In addition, if /

/

7 ÐD3"3" âD3737 Ñ œ % where 35 Á 35" , /35 Á ! and 7 ", then C3" B/3" C3" 3# B/3# C3# 3$ B/3$ âB/37" C37" 37 B/37 C37 œ % in K Ÿ J\ . Since the left-hand side can be reduced to % using only removal steps, it follows that 35 œ 35" for all 5 and so 7 œ " and C3" B/3" C3" œ % But the left-hand side of this equation reduces to % by removal steps if and only if /" œ !, which is false. Hence, 7 is injective and therefore an isomorphism. We can also provide an example of a group K with a subgroup L Ÿ K for which +L+"  L . Theorem 12.21 Let J\ be the free group on \ œ ÖBß C×. Then J\ has a subgroup L for which BLB"  L . Proof. Let L consist of the empty word % and the set of all words of the form A œ B8" C5" B8# C5# âB8< C5< B8

Fundamentals of Group Theory: An Advanced Approach

Fundamentals of Group Theory: An Advanced Approach

Fundamentals of Group Theory: An Advanced Approach