Ischia Group Theory 2006: Proceedings of a Conference in Honor of Akbar Rhemtulla

ISCHIA GROUP THEORY 2006 Proceedings of a Conference in Honour of

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ISCHIA GROUP THEORY 2006 Proceedings of a Conference in Honour of

Akbar Rhemtulla Ischia, Naples, Italy

29 March - 1 April 2006

edited by

Trevor Hawkes University of Warwick, UK

Patrizia Longobardi University of Salerno, Italy

Mercede Maj University of Salerno, Italy

r pWorld Scientific N E W JERSEY

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LONDON

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SINGAPORE

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BElJlNG

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SHANGHAI

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HONG KONG

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TAIPEI

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CHENNAI

Published by World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224

USA ofice: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 U K ofice: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-PublicationData Ischia Group Theory 2006 (2006 : Naples, Italy) Ischia Group Theory 2006 : proceedings of a conference in honor of Akbar Rhemtulla, ISCHIA, Naples, Italy, 29 March-1 April 2006 I edited by Trevor Hawkes, Patrizia Longobardi, and Mercede Maj. p. cm. ' Includes bibliographlcal references. ISBN-13: 978-981-270-735-2 (pbk. : alk. paper) ISBN-10: 981-270-735-2 (pbk. : alk. paper) 1. Representations of groups--Congresses. 2. Nilpotent groups--Congresses. 3. Sylow subgroups--Congresses. 4. Group theory--Congresses. I. Rhemtulla, Akbar. 11. Hawkes, Trevor O., 1936- 111. Longobardi, Patrizia. IV. Maj, Mercede. V. Title. QA176.183 2006 5 12'.22--d~22 2007035469 Photo on the cover by Enzo Rando (www.enzorando.it)

British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.

Copyright 0 2007 by World Scientific Publishing Co. Re. Ltd

All rights reserved. This book, or parts thereoi may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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Contents

Preface

vii

Sponsors

viii

Akbar Rhemtulla

ix

Bibliography

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Conference Program

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List of Participants

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A Combinatorial Property of Burnside Varieties of Groups Alireza ABDOLLAHI

1

A Note on Ends of Operator Algebras Tullio G. CECCHERINI-SILBERSTEIN and Aryeh Y.SAMET- VAILLANT

7

Minimal Quasinormal Subgroups of Groups

13

John COSSEY, Stewart STONEHE W E R and Giovanni ZACHER On Certain Saturated Formations of Finite Groups

22

Alma D'ANIELLO, Clorinda DE VIVO and Gabriele GIORDANO Groups with Few Non-normal Subgroups

33

Maria DE FALCO, Francesco DE GIOVANNI and C a m e l a MUSELLA Groups with Conditions on Infinite Subsets

46

Costantino DELIZIA and Chiara NICOTERA Some Generalizations of the Probabilistic Zeta Function

56

Eloisa DETOMI and Andrea LUCCHINI Groups with Proper Subgroups of Certain Types

73

Martyn R. DIXON, Martin J. EVANS and Howard SMITH Counting Conjugacy Classes of Subgroups in Finite p-groups, I

Gustavo A . FERNANDEZ-ALCOBER and Leire LEGARRETA V

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On Certain Classes of Generalized Periodic Groups Gerard ENDIMIONI Nielsen Equivalence Classes and Stability Graphs of Finitely Generated Groups Martin J. EVANS

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103

Counting Conjugacy Classes of Subgroups in Finite pgroups, I1 Manuel EGIZII DI MARCO, Gustavo A. FERNANDEZ- A LCOBER and Leire L E GAR R E T A

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Positive Laws on Large Sets of Generators and on Word Values Gustavo A . FERNANDEZ-ALCOBER and Pave1 SHUMYATSK Y

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Two Applications of the Hughes Subgroup of Finite pgroups Norbert0 GAVIOLI, Avinoam M A N N and Carlo M. SCOPPOLA

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Free Products and Higman-Neumann-Neumann of Lattice-Ordered Groups Andrew M. W . GLASS

Type Extensions 147

Attaching to a Profinite Space Wolfang HERFORT and Wolfram HOJKA

166

On Groups with Two Infinite Conjugacy Classes Marcel HERZOG, Patrizia LONGOBARDI and Mercede M A J

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Centralizers of Nonabelian Irreducible Equivalent G-groups Paz JIMENEZ-SERA L

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Some New Methods for Almost Regular Automorphisms Evgeny I. KHUKHRO

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Non-Properness of Amenable Actions on Graphs with Infinitely Many Ends Soyoung MOON and Alain V A L E T T E Groups with Finitely Many Maximal Normalizers Carmela SICA and Maria T O T A Locally Graded Groups with Few Non- (Torsion-by-Nilpotent) Subgroups Nadir TRABELSI

227 234

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Preface

The papers of this volume represent the proceedings of the Conference entitled Ischia Group Theory 2006 which took place a t the Jolly Hotel in Ischia (Naples, Italy) from 29th March to 1st April 2006 in honour of Akbar Rhemtulla. The organizing committee consisted of Trevor Hawkes, Patrizia Longobardi and Mercede Maj. The local committee was formed by Costantino Delizia, Chiara Nicotera, Carmela Sica and Maria Tota, all from the Universith di Salerno. The articles that follow are contributions by the Conference speakers and participants and are mostly in areas close to Akbar Rhemtulla’ s interests. In particular, the following subjects are represented: combinatorial group theory, varieties of groups, orderable groups, profinite groups, probabilistic methods in group theory, graphs connected with groups, conjugacy and subgroup structure, saturated formations. A poster session on various topics connected with themes of the Conference augmented the scientific programme. It is our pleasure to thank: 0 0 0

0

all the speakers and the partecipants, the authors for their contributions, the referees, who gave generously of their time t o review the papers and offer valuable feedback to the authors, the staff of the Jolly Hotel for being so accommodating and supportive, the friendly and patient staff a t World Scientific Publishing for their help and advice, and for producing these proceedings in a professional and timely manner, all the sponsors, the local commettee for its precious help and, in particular, Costantino Delizia and Maria Tota for their tecnical support in producing these proceedings, and, of course, Akbar Rhemtulla for allowing us t o celebrate with him. Trevor Hawkes, Patrizia Longobardi and Mercede Maj July 2007

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SPONSORS

P.R.I.N. Programmi di Ricerca di Interesse Nazionale Progetto dal titolo: “Teoria dei Gruppi e Applicazioni”

G.N.S.A.G.A. Gruppo Nazionale per le Strutture Algebriche, Geometriche e lor0 Applicazioni (1.N.d.A.M. - Istituto Nazionale di Alta Matematica “F. Severi”)

D.M.I. Dipartimento di Matematica e Informatica dell’Universit8. di Salerno

D.F. Dipartimento di Fisica “E. R. Caianiello”dell’Universit8. di Salerno

Universith degli Studi di Salerno

BANCA CARIME Gruppo BPU banca - Filiale di Caste1 San Giorgio (SA)

Regione Campania

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Akbar Rhemtulla

The Ischia Group Theory Conference (29th March-1st April 2006) was held in honour of Akbar Rhemtulla following his retirement in July 2004. A great many of the nearly 100 participants have known Akbar professionally and personally during his 40-year association with the Mathematics Department at the University of Alberta. Akbar was born in 1939 in Zanzibar, where he attended the Aga Khan primary school and the Government secondary school up t o Ordinary-Level examinations (roughly age 17). He completed his last two years of secondary education in Uganda, taking GCE Advanced-Levels at Makerere College in Kampala. When I first met Akbar, he was already a second-year undergraduate a t Makerere, which was then an external college of London University and awarded London degrees. Akbar gained a first-class honours BSc degree in Mathematics in 1963. The early 60s were an exciting and hopeful years in African history and by the time Akbar graduated, Uganda had become an independent nation and Makerere had broken its links with London to become one of the 3 colleges in the federal University of East Africa. In his final undergraduate year Akbar had chosen Q u a n t u m Mechanics and Group Theory as “special topics”, and he subsequently won a research scholarship from the Leverhulme Foundation to study under Philip Hall at Kings College, Cambridge. For the next three years Akbar and I attended Halls Part I11 lectures together and shared in the excitement and intellectual stimulus Cambridge offered research students. Philip Hall examined both of our PhD dissertations and held our viva voce examinations on the same day in June 1967. The title of Akbars dissertation was Problems of bounded expressibility in free products and a minimality property of polycyclic groups. Akbar had intended to return to East Africa at the end of his studies, but political turmoil in the region made that impossible. After submitting his PhD, he stayed on in Cambridge as a UNESCO Fellow, working on a project devised by Philip Hall. He then accepted a post-doctoral fellowship at the University of Alberta in Edmonton, where the rigours of the climate did not deter him from taking a tenure-track position the following year. The course of his academic career was set, and indeed his personal life too, for Antoinette (Toine), whom he had met in Cambridge, was already his wife when they reached Canada in August 1967. During that month, they hade married in Utrecht (Toines home town), honeymooned in Paris, returned to

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Utrecht and crossed t o Liverpool; from there they sailed to Montreal and travelled for 3 days on the Canadian Pacific railway to reach Edmonton in time for Akbar t o be in post on September 1st. They soon put down roots, and in the fullness of time bought a house and started a family in this oilrich flourishing province; here their three children, Jeanine, Anil and Mijke were born and raised, and eventually fled the nest. Akbar wrote on many aspects of group theory but several threads trace back to the influence of Philip Hall, in particular, his work on classes of infinite soluble groups and on orderable groups. He always had a good eye for apposite counter examples that set limits on what theorems are possible. With Roberta Botto-Mura he published a survey volume on orderable groups in the Lecture Notes in Pure and Applied Mathematics series. Although I cannot do justice, in this short appreciation, t o the full range

of Akbars published work, I would like give a flavour with a few selected theorems: (1967) In his first published paper, Akbar characterized polycyclic groups among finitely-generated soluble groups by the property that the intersection of any number of conjugate subgroups is already the intersection of finitely many of them. (1973) Any torsion-free group that is a residually pgroup for infinitely many primes p is a totally-orderable group.

(1987, 1988) A subgroup H is said to be elliptically embedded in a group G if for each subgroup K of G there exists an integer n = n ( K ) such that the subgroup ( H , K )generated by H and K is equal t o ( H I Y ) ~ . In two joint papers, Akbar and John Wilson make links between elliptical embedding and subnormality. They show, for instance, that a cyclic elliptically-embedded subgroup of a torsion-free soluble group is subnormal, and that a subgroup of a polycyclic-by-finite is elliptically embedded if and only if it is subnormal in a subgroup of finite index.

(1995) An open question of Zelmanov’s is whether every torsion-free nEngel group is locally nilpotent. (George Havas and Michael Vaughan-Lee have provided an affirmative answer when n = 4.) With Y.K. Kim, Akbar gives a positive answer for general n within the universe of totally-orderable groups. (2003) Let G be an R*-group (defined by the property that g = 1 is the only element for which a product of conjugates of g equals 1). It is known that every partial order can be extended t o a total order if G is

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metabelian, but the question is open if G is merely abelian-by-nilpotent. Akbar, Patrizia Longobardi and Mercede Maj have proved this extendibility theorem for R*-groups that are both abelian-by-nilpotent and nilpotent-byabelian. (This result has an interesting consequence: a rational polynomial of positive degree with no positive real roots always divides some rational polynomial with positive coefficients.) Akbar has been a wonderful collaborator throughout his career. Nearly 70 of more than 80 items on his list of publications to date are joint work his Erdos number is 2 - and he took full advantage of the generous travel and research grants available to Canadian academics. Many of us a t this conference, as guests of the Mathematics Department at the University of Alberta, have shared Akbars enthusiasm for group theory and have enjoyed the warm hospitality he and Toine always offered - their visitors never lacked for a convivial social life during their stay. He is a respected teacher, a loyal and generous colleague, and conscientious member of both his Department and the wider academic community in Canada. He has supervised the research of a dozen successful PhD students and has sat on or chaired numerous committees; he did a stint as Head of Department and was on the Board of Directors on the Canadian Mathematical Society, as well as a local director for the Pacific Institute for the Mathematical Sciences.

I know I speak for the conference members in wishing Akbar a long, happy and fruitful retirement. Trevor Hawkes, July 2007

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Bibliography 1. A. H. Rhemtulla, A minimality property of polycyclic groups, J. London Math. SOC.42 (1967), 456-462. 2. A. H. Rhemtulla, A problem of bounded expressibility in free products, Proc. Cambridge Philos. SOC.64 (1968), 573-584. 3. R. H. Lagrange and A. H. Rhemtulla, A remark o n the group rings of order preserving permutation groups, Canad. Math. Bull. 11 (1968), 679-680. 4. A. H. Rhemtulla, O n a problem of L. Fuchs, Studia Sci. Math. Hungar. 4 (lS69), 195-200. 5. A. H. Rhemtulla, C o m m u t a t o r s of certain finitely generated soluble groups, Canad. J. Math. 21 (1969), 1160-1164. 6. J. D. Dixon, J. Poland and A. H. Rhemtulla, A generalization of Hamiltonian and nilpotent groups, Math. Z. 112 (1969), 335-339. 7. A. H. Rhemtulla, A property of groups with n o central factors, Canad. Math. Bull. 12 (1969), 467-470. 8. N. D. Gupta and A. H. Rhemtulla, A note o n centre-by-finite-exponent varieties of groups, J. Austral. Math. SOC.11 (1970), 33-36. 9. R. Dark and A. H. Rhemtulla, O n Ro-closed classes and finitely generated groups, Canad. J. Math. 22 (1970), 176-184. 10. R. J. Hursey and A. H. Rhemtulla, Ordered groups satisfying the m a x i m a l condition locally, Canad. J. Math. 22 (1970), 753-758. 11. A. H. Rhemtulla and A. P. Street, Maximal sum-free sets in finite abelian groups, Bull. Austral. Math. SOC.2 (1970), 289-297. 12. C. K. Gupta, N. D. Gupta and A. H. Rhemtulla, Dichotomies in certain finitely generated soluble groups, J. London Math. SOC.3 (1971), 517-525. 13. A. H. Rhemtulla and A. P. Street, Maximal sum-free sets in elementary abelian p-groups, Canad. Math. Bull. 14 (1971), 73-80. 14. N. D. Gupta and A. H. Rhemtulla, O n ordered groups, Algebra Universalis 1 (1971/72), 129-132. 15. A. H. Rhemtulla, Right-ordered groups, Canad. J. Math. 24 (1972), 891-895. 16. A. H. Rhemtulla, Residually Fp-groups, f o r m a n y p r i m e s p , are orderable, Proc. Amer. Math. SOC.41 (1973), 31-33. 17. R. Botto-Mura and A. H. Rhemtulla, Solvable groups in which every m a x i m a l partial order is isolated, Pacific. J. Math. 51 (1974), 509-514. 18. N. D. Gupta, F. Levin and A. H. Rhemtulla, Chains of varieties, Canad. J. Math. 26 (1974), 190-206. 19. R. T. Botto-Mura and A. H. Rhemtulla, Ordered solvable groups satisfying the maximal condition o n isolated subgroups and groups with finitely m a n y relatively convex subgroups, J. Algebra 36 (1975), 38-45. 20. R. Botto-Mura and A. H. Rhemtulla, Solvable R*-groups, Math. Z. 142 (1975), 293-298. 21. R. Botto-Mura and A. H. Rhemtulla, Orderable groups, Lecture Notes in Pure and Applied Mathematics, 27, Marcel Dekker Inc., New York-Basel, 1977. 22. R. Botto-Mura and A. H. Rhemtulla, A class of right-orderable groups, Canad. J. Math. 29 (1977), 648-654.

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23. R. Botto-Mura and A. H. Rhemtulla, Extensions of orderable groups, Canad. Math. Bull. 20 (1977), 393-395. 24. R. Botto-Mura and A. H. Rhemtulla, Subdirect products of 0*-groups, Algebra Universalis 8 (1978), 23-27. 25. G. H. Cliff and A. H. Rhemtulla, Permuting the elements of a finite solvable group, Canad. Math. Bull. 22 (1979), 327-330. 26. A. C. Kim, B. H. Neumann and A. H. Rhemtulla, More Fibonacci varieties, Bull. Austral. Math. SOC.22 (1980), 385-395. 27. A. H. Rhemtulla, Groups of finite weight, Proc. Amer. Math. SOC.81 (1981), 191-192. 28. A. H. Rhemtulla, Finitely generated non-Hopfian groups, Proc. Amer. Math. SOC.81 (1981), 382-384. 29. A. H. Rhemtulla, Polycyclic right-ordered groups, Algebra, Carbondale 1980 (Proc. Conf., Southern Illinois Univ., Carbondale, Ill., 1980), 230-234, Lecture Notes in Math., 848, Springer, Berlin, 1981. 30. B. Hartley, J. C. Lennox and A. H. Rhemtulla, Cyclically separated groups, Bull. Austral. Math. SOC.26 (1982), 355-384. 31. A. H. Rhemtulla and B. A. F. Wehrfritz, Isolators in soluble groups of finite rank, Rocky Mountain J. Math. 14 (1984), 415-421. 32. A. H. Rhemtulla, A. Weiss and M. Yousif, Solvable groups with n-isolators, Proc. Amer. Math. SOC.90 (1984), 173-177. 33. M. S. Klamkin and A. H. Rhemtulla, T h e ballot problem, Math. Modelling 5 (1984), 1-6. 34. D. Meier and A. H. Rhemtulla, O n torsion-free groups of finite rank, Canad. J. Math. 36 (1984), 1067-1080. 35. A. H. Rhemtulla and H. Smith, A finite index property of certain solvable groups, Canad. Math. Bull. 27 (1984), 485-489. 36. D. Meier and A. H. Rhemtulla, Rank restricting properties of finitely generated soluble groups, Arch. Math. (Basel) 44 (1985), 216-224. 37. A. H. Rhemtuila, Characteristic properties of soluble groups of finite rank, J. Korean Math. SOC.22 (1985), 135-142. 38. A. H. Rhemtulla and J. S. Wilson, On elliptically embedded subgroups of soluble groups, Canad. J. Math. 39 (1987), 956-968. 39. A. H. Rhemtulla and J. S. Wilson, Elliptically embedded subgroups of polycyclic groups, Proc. Amer. Math. SOC.102 (1988), 230-234. 40. A. H. Rhemtulla and S. Sidki, Factorizable infinite solvable groups, J. Algebra 122 (1989), 397-409. 41. A. H. Rhemtulla and A. R. Weiss, Groups with permutable subgroup products, Group theory (Singapore, 1987), 485-495, de Gruyter, Berlin, 1989. 42. M. R. Darnel, A. M. W.Glass and A. H. Rhemtulla, Groups in which every right order is two-sided, Arch. Math. (Basel) 53 (1989), 538-542. 43. R. D. Blyth and A. H. Rhemtulla, Rewritable products in FC-by-finite groups, Canad. J. Math. 41 (1989), 369-384. 44. P. S. Kim and A. H. Rhemtulla, Permutable word products in groups, Bull. Austral. Math. Soc. 40 (1989), 243-254. 45. P. Longobardi, M. Maj and A. H. Rhemtulla, Periodic groups with permutable

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subgroup products, Math. Proc. Cambridge Philos. SOC.106 (1989), 431-437. 46. A. H. Rhemtulla, Periodic extensions of ordered groups, Ordered algebraic structures (Curasao, 1988), 65-69, Math. Appl., 55, Kluwer Acad. Publ., Dordrecht, 1989. 47. A. H. Rhemtulla, Groups with m a n y elliptic subgroups, Groups-Korea 1988 (Pusan, 1988), 156-162, Lecture Notes in Math., 1398, Springer, Berlin, 1989. 48. M. Curzio, J. Lennox, A. H. Rhemtulla and J. Weigold, Groups with m a n y permutable subgroups, J. Austral. Math. SOC.(Ser. A) 48 (1990), 397-401. 49. S. H. Nazzal and A. H. Rhemtulla, Centrality in abelian-by-polycyclic groups, Arch. Math. (Basel) 56 (1991), 333-342. 50. P. S. Kim, A. H. Rhemtulla and H. Smith, A characterization of infinite metabelian groups, Houston J. Math. 17 (1991), 429-437. 51. P. Longobardi, M. Maj, A. H. Rhemtulla and H. Smith, Periodic groups with m a n y permutable subgroups, J. Austral. Math. SOC.(Ser. A) 53 (1992), 116-119. 52. M. Curzio, P. Longobardi, M. Maj and A. H. Rhemtulla, Groups with m a n y rewritable products, Proc. Amer. Math. SOC.115 (1992), 931-934. 53. P. Longobardi, M. Maj and A. H. Rhemtulla, Infinite groups in a given variety and Ramsey's theorem, Comm. Algebra 20 (1992), 127-139. 54. P. Longobardi, M. Maj and A. H. Rhemtulla, Coverang a group with asolators of finitely m a n y subgroups, Glasgow Math. J. 35 (1993), 253-259. 55. P. Longobardi, M. Maj and A. H. Rhemtulla, Residually solvable PSP-groups, Boll. Un. Mat. Ital. B 7 (1993), 253-261. 56. Y . K. Kim and A. H. Rhemtulla, Orderable groups satisfying a n Engel condition, Ordered algebraic structures (Gainesville, FL, 1991), 73-79, Kluwer Acad. Publ., Dordrecht, 1993. 57. A. H. Rhemtulla and H. Smith, O n infinite solvable groups, Infinite groups and group rings (Tuscaloosa, AL, 1992), 111-121, Ser. Algebra, 1,World Sci. Publ., River Edge, NJ, 1993. 58. A. H. Rhemtulla and H. Smith, O n infinite locally finite groups, Canad. Math. Bull. 37 (1994), 537-544. 59. Y . K. Kim and A. H. Rhemtulla, W e a k maximality condition and polycyclic groups, Proc. Amer. Math. SOC.123 (1995), 711-714. 60. P. Longobardi, M. Maj and A. H. Rhemtulla, Groups with n o free subsemigroups, Trans. Amer. Math. SOC.347 (1995), 1419-1427. 61. Y . K. Kim and A. H. Rhemtulla, O n locally graded groups, Groups - Korea '94 ( Pusan) de Gruyter, Berlin (1995), 189-197. 62. Y . K. Kim and A. H. Rhemtulla, Groups with ordered structures, GroupsKorea '94 (Pusan) de Gruyter, Berlin (1995), 199-210. 63. P. Longobardi, M. Maj, A. Mann and A. H. Rhemtulla, Groups with m a n y nilpotent subgroups, Rend. Sem. Mat. Univ. Padova 95 (1996), 143-152. 64. J. Poland and A. H. Rhemtulla, T h e number of conjugacy classes of nonnormal subgroups in nilpotent groups, Comm. Algebra 24 (1996), 3237-3245. 65. P. Longobardi, M. Maj and A. H. Rhemtulla, Subclasses of locally m i n i m a x groups closed u n d e r normal j o i n s , J. London Math. SOC.55 (1997), 341-347.

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66. M. Akhavan-Malayeri and A. H. Rhemtulla, C o m m u t a t o r length of abelianby-nilpotent groups, Glasgow Math. J. 40 (1998), 117-121. 67. A. Mohammadi Hassanabadi and A. H. Rhemtulla, Criteria f o r commutativi t y in large groups, Canad. Math. Bull. 41 (1998), 65-70. 68. R. La Haye and A. H. Rhemtulla, Groups with a bounded number of conjugacy classes of non-normal subgroups, J. Algebra 214 (1999), 41-63. 69. P. Longobardi, M. Maj and A. H. Rhemtulla, W h e n i s a right orderable group locally indicable?, Proc. Amer. Math. SOC.128 (2000), 637-641. 70. C. Delizia, A. H. Rhemtulla and H. Smith, Locally graded groups with a nilpotency condition o n infinite subsets, J. Austral. Math. SOC.Ser. A 69 (2000), 415-420. 71. A. H. Rhemtulla and D. Rolfsen, Local indicability in ordered groups: braids and elementary amenable groups, Proc. Amer. Math. SOC.130 (2002), 25692577. 72. A. H. Rhemtulla, Orderable groups, Proceedings of the International Conference on Algebra and its Application (ICAA 2002) (Bangkok), 47-55, Chulalongkorn Univ., Bangkok, 2002. 73. M. Akhavan-Malayeri and A. H. Rhemtulla, Products of commutators in free groups, Internat. J. Algebra Comput. 13 (2003), 231-240. 74. V. V. Bludov, A. M. W. Glass and A. H. Rhemtulla, Ordered groups in which all convex j u m p s are central, J. Korean Math. SOC.40 (2003), 225-239. 75. P. Longobardi, M. Maj and A. H. Rhemtulla, O n solvable R*-groups, J. Group Theory 6 (2003), 499-503. 76. A. H. Rhemtulla and M. Shirvani T h e residual finiteness of ascending HNNextensions of certain soluble groups, Illinois J. Math. 47 (2003), 477-484. 77. A. H. Rhemtulla and H. Smith, O n solvable R*- groups o f f i n i t e rank, Comm. Algebra 31 (2003), 3287-3293. 78. C. Delizia, C. Nicotera and A. Rhemtulla, Torsion-free groups with rank restricting properties, Comm. Algebra 33 (2005), 2765-2770. 79. V. V. Bludov, A. M. W. Glass and A. H. Rhemtulla, O n centrally orderable groups, J. Algebra 291 (2005), 129-143. 80. Y. K. Kim and A. H. Rhemtulla, O n orderable poly Engel groups, Comm. Algebra 34 (2006), 3023-3027. 81. C. Delizia, M. R. Moghaddam and A. H. Rhemtulla, T h e structure of Bell groups, J. Group Theory 9 (2006), 117-125. 82. Peter A. Linnell, Akbar H. Rhemtulla and Dale P.O. Rolfsen, Invariant group orderings and Galois conjugates J. Algebra, to appear.

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WEDNESDAY, MARCH 29 09.20 Welcome Greetings Chairman : Wolfgang Her fort 09.30 A. MANN Breadth of elements in polycyclic groups

10.20 C.K. GUPTA Around test elements 11.10 Coffee Break 11.30 A. LUCCHINI Finite groups with pmultiplicative probabilistic zeta function 12.20 V. BLUDOV Ordered solvable groups 13.30 Lunch Break

Chairman: Peter Neumann 15.30 S. STONEHEWER Minimal quasinormal subgroups of groups 16.20 A. GLASS The lattice-ordered group analogue of Higman’s Embedding Theorem 17.10 Coffee Break Chairman: Carlo Maria Scoppola 17.30 R. BRANDL

Arithmetic properties related t o commutators 18.00 N. GAVIOLI Irreducible nilpotent modular Lie algebras and wreath products 18.20 M. MASSA The probabilistic zeta function of the alternating group Alt(p 18.40 A. MONTINARO Large 2-transitive arcs 21.30 Recital of classical Neapolitan songs sponsored by Banca di Salerno - Credit0 Cooperativo

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+ 1)

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THURSDAY, MARCH 30 09.30 Social trip: island tour by boat with a stop in S. Angelo 13.30 Lunch Break Chairman: Andrea Lucchini 15.30 A. OLSHANSKIY On the behavior of isoperimetric functions for finitely presented groups 16.20 F. MENEGAZZO Complements of minimum normal subgroups 17.10 M. PELLEGRINI A generalized Cameron-Kantor Theorem 17.30 Coffee Break Chairman: M. Dolores PBrez-Ramos 17.50 F . LEINEN Positive definite functions of stable isometry groups 18.30 E. PACIFIC1 Character degree graphs that are complete graphs 18.50 A. PAVAN Computing the Frattini subgroup of a polycyclic group

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FRIDAY, MARCH 31 Chairman: Julio P. Lafuente 09.30 G. ENDIMIONI Polynomial functions of groups 10.20 D.J.S. ROBINSON On inert subgroups and inertial groups 11.10 Coffee Break 11.30 M. HERZOG On a generalization of FC-groups 12.20 M. DIXON Groups with proper subgroups of certain types 13.30 Lunch Break Chairman: Lino Di Martino 15.30 K . GRUENBERG Relation gaps and a theorem of Rhemtulla 16.20 P. KROPHOLLER Modular invariants of finite orthogonal groups 17.00 H. HEINEKEN Groups all of whose elements have prime power order 17.30 Coffee Break 18.00 Concert

20.30 Social Dinner

xx SATURDAY, APRIL 1 Chairman: Akbar Rhemtulla 09.30 M. EVANS Nielsen equivalence classes and stability graphs of finitely generated groups 10.20 E. KHUKHRO Finite groups with automorphisms whose centralizers have small rank 11.10 Coffee Break 11.30 C. CASOLO Groups with all subgroups subnormal 12.20 B. HUPPERT How to shuffle cards

13.30 Lunch Break Chairman: Trevor Hawkes 15.00 G. FERNANDEZ-ALCOBER Positive laws on word values 15.50 P. MORAVEC Schur multipliers and power endomorphisms of groups 16.20 0. KEGEL Ideals in group algebras

17.10 Coffee Break

REGISTERED PARTICIPANTS Alireza Abdollahi, University of Isfahan, Iran Maria J. Asiain, Universidad P2iblica de Navarra, Spain Marina Avitabile, Universitci dell 'Aquila, Italy M. Gokhan Benli, Middle East Technical University, Turkey Cansu Betin, Middle East Technical University, Turkey Mariagrazia Bianchi, Universitci d i Milano, Italy Vasily Bludov, Irkutsk State University, Russia Victor Bovdi, University of Debrecen, Hungary Rolf Brandl, Universitat Wiirzburg, Germany Brunella Bruno, Universitci di Padova, Italy Daniela Bubboloni, Universitci di Firenze, Italy Carlo Casolo, Universita d i Firenze, Italy Tullio G. Ceccherini-Silberstein, Universitci del Sannio, Italy Maria Rosaria Celentani, Universitci d i Napoli Federico 11, Italy Gabriella Corsi, Universitci di Firenze, Italy Mauro Costantini, Universita d i Padova, Italy Eleonora Crestani, Universita di Padova, Italy Giovanni Cutolo, Universitci di Napoli Federico 11, Italy F'rancesca Dalla Volta, Universitci d i Milano - Bicocca, Italy Erika Damian, Universitci d i Brescia, Italy Alma D'Aniello, Universitci di Napoli Federico 11, Italy Paola D'Aquino, Seconda Universitci di Napoli, Italy Ulderico Dardano, Universitci d i Napoli Federico II, Italy F'rancesco de Giovanni, Universitci d i Napoli Federico 11, Italy Willem de Graaf, Universitci di Trento, Italy Rosa De Nicola, Universitci di Salerno, Italy Costantino Delizia, Universith di Salerno, Italy Eloisa Detomi, Universitci di Padova, Italy Lino Di Martino, Universitci d i Milano - Bicocca, Italy Martyn Dixon, University of Alabama, U.S.A. Silvio Dolfi, Universita di Firenze, Italy

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GBrard Endimioni, Universite' de Provence, France Kivanc Ersoy, Middle East Technical University, Turkey Martin Evans, University of Alabama, U.S.A . Maria-Josk Felipe, Universidad Polite'cnica de Valencia, Spain Gustavo Fernandez-Alcober, Universidad del Pais Vasco, Spain Carla Fiori, Universitci di Modena e Reggio Emilia, Italy Norbert0 Gavioli, Universitci dell 'Aquila, Italy Anna Gillio, Universitci di Milano, Italy Andrew Glass, University of Cambridge, U.K. Karl Gruenberg, Queen Mary University London, U.K. Chander Kanta Gupta, University of Manitoba, Canada Trevor Hawkes, University of Warwick, U.K. Hermann Heineken, Universitat Wurzburg, Germany Wolfgang Herfort, University of Tecnology, Vienna, Austria Marcel Herzog, Tel Aviv University, Israel Bertram Huppert, Universitat Mainz, Germany Diana Imperatore, Universitci d i Salerno, Italy Paz Jimenez Seral, Universidad de Zaragoza, Spain Z. Yalcin Karatas, Middle East Technical University, Turkey Otto Kegel, Universitat Freiburg, Germany Evgenii Khukhro, Cardiff University, U.K. Christina Krause, Universitat Oldenburg, Germany Peter Kropholler, University of Glasgow, U.K. Julio P. Lafuente, Universidad Pliblica d e Navarra, Spain Leire Legarreta, Universidad del Pais Vasco, Spain Felix Leinen, Universitat Mainz, Germany Antonella Leone, Universitci di Napoli Federico 11, Italy Patrizia Longobardi, Universitci di Salerno, Italy Andrea Lucchini, Universitci d i Brescia, Italy Annamaria Lucibello, Universitci d i Salerno, Italy Maria Silvia Lucido, Universitci d i Udine, Italy Antonio Machi, Universith di Roma - La Sapienza, Italy Mario Mainardis, Universith d i Udine, Italy Mercede Maj, Universitci di Salerno, Italy Avinoam Mann, Hebrew University of Jerusalem, Israel Marilena Massa, Universitci di Milano - Bicocca, Italy Katayoon Mehrabadi, Tarbiat Modarres University, Tehran, Iran Federico Menegazzo, Universith d i Padova, Italy Alessandro Montinaro, Universitci d i Lecce, Italy

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Primoi Moravec, University of Ljubljana, Slovenia Marta Morigi, Universith di Bologna, Italy Jose M. Muiioz-Escolano, Universidad de Zaragoza, Spain Peter Neumann, The Queen's College, Oxford, U.K. Chiara Nicotera, Universita di Salerno, Italy Alexander Olshanskiy, Moscow State University, Russia Emanuele Pacifici, Universith d i Milano, Italy Laura Paladino, Universitci di Pisa, Italy Andrea Pavan, Universitci di Padova, Italy Marco Pellegrini, Universita d i Milano - Bicocca, Italy M. Dolores Perez-Ramos, Universidad de Valencia, Spain Andrea Previtali, Universith Insubria-Como, Italy Akbar Rhemtulla, University of Alberta, Canada Derek J. S . Robinson, University of Illinois, U.S.A. Alessio RUSSO,Seconda Universith di Napoli, Italy Valentina RUSSO,Universitci dell 'Aquila, Italy Carlo Maria Scoppola, Universitci dell 'Aquila, Italy Luigi Serena, Universith di Firenze, Italy Carmela Sica, Universitd d i Salerno, Italy Salvatore Siciliano, Universita di Lecce, Italy Ernest0 Spinelli, Universitd d i Lecce, Italy Stewart Stonehewer, University of Warwick, U.K. Maria Clara Tamburini, Universith Cattolicu di Bresciu, Italy Antonio Tortora, Universita di Salerno, Italy Maria Tota, Universita di Salerno, Italy Nadir Trabelsi, University of Setif, Algeria Erkan Murat Turkan, Middle East Technical University, Turkey Pinar Ugurlu, Middle East Technical University, Turkey Angela Valenti, Universita di Palerrno, Italy Giovanni Vincenzi, Universith di Salerno, Italy Thomas Weigel, Universitb d i Milano - Bicocca, Italy Giovanni Zacher, Universitci di Padova, Italy Claudio Paolo Zuccari, Universith di Firenze, Italy

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A C O M B I N A T O R I A L PROPERTY OF BURNSIDE VARIETIES OF GROUPS ALIREZA ABDOLLj\III' Departm.etit of Math,emati.c.s, IUn1:i:ersity of ~.sfuhari, Isfahan, 81746-73441: Iran and Institute for Studies in Theoretical Physics and Mathematics (IPM) * E - m a i l : a . a b d o l l a h i ~ ~ m a f h .ac.ir ui. ht t p ://s ci. ui.a c .ir/math/iVe w/ahd o 1la hi. h t m Let G be an infinite group and n E {2;3}. It is proved that zn = 1 for all z E G if and only if 1~ E X n for all infinite subsets X of G, where

x n := {z*zg'.'In,12, E X}. Keyurords: Comhinatorial conditions; Burnside variety of groups

1. I n t r o d u c t i o n and results

Let w - xFf'l .-.x:+* be a (reduced) word in t.he free group of rank n E N , on the letters 51:. . . ,x,, where €1,. . . c t E {-1, l}. Suppose that G is a group and X I .,. . X, are n non-empty subsets of G. Define ~

. w ( X 1 , .. . , X n ) := {ail . . .a;t

For example, if w1 = X

~

~

~ l ( X 1 , X z=) {

~

I aj

E Xij, 1 5 j

- 1 -2

~ w2 ~ =Z2,~

x2X z1xz Z , and

5 t}. wg = zy,

then

~ ~ - ~I Wb, C~: E-Xi, ~ ~b ?dd E X z } ,

,ti)z(Xl,Xz) = (a-lh,'h,'~dldz I a , c € Xi,

bl,bz,$l,d2 E

Xz},

and w 3 ( X 1 )= (a.bc I a , b , c E XI}. Let W and V be t.wo non-empty subsets of the free group of rank n E N. Let P ( V , W ) be the class of all groups G such that for every n-tuple (91,. . . ,g,) of elements of G there exist v E V and w E W such that v # w and v(g1,. . . , g n ) = w(g1,. . . , g n ) (cf. [9]). We denote by P*(V> IV) (respectively P#(V, W ) )the class of all groups G satisfying thc following condition: G t P*(v, W ) (respectively G t P#(v. w))if arid only if for it11 ii1fiuit.e

1

2 subsets X I , . . . , X, of G there exist 'u E V and w E W (respectively, also there exist elements a1 E X i , . . . , a, E X n ) such that 'u # w and 1~ E vw-l(X1,. . . , X,) (respectively w(a1,.. . , a n ) = w(a1,. . . ,a,)). Clearly, we have .F U P( V,W )C_ P* (V, W )and

FUP(V,W) c P # ( V , W )c P*(V,W), where F is the class of finite groups. The following questions arise naturally.

Question 1.1. For which non-empty subsets V and W of a free group of finite rank, the equality

P ( v,W ) u .F = P# (V,W ), holds?

Question 1.2. For which non-empty subsets finite rank, the equality

V and W of a free group of

P#(V,W ) = P*(V,W ) , holds?

Question 1.3. For which non-empty subsets finite rank, the equality

P(

V and W of a free group of

v,W )u F = P* (V,W ) ,

(*I

holds? Question 1.1 considered by many people, where the pair (V, W )is of the form ({I}, {w}) with w is a non-trivial word in a free group (see e.g., [1,8,10]).Note that P({'u},{w}) = P({vw-'}, (1)) and P # ( { v } ,{w}) = P#({vw-'}, (1)). Question 1.3 first appeared in [3], where this question has been answered positively for V = (~1x2)and W = {xp1}, and in this case P(V, W ) is the class of abelian groups. If n > 1 is an integer and

V = {x1 . . .x,} then

P(V, W ) is exactly the

and W = (

~ ~ ( .1. ).xu(,)

I CT E S,}

class of n-permutable groups (see [7]). Also if

v = W = {xu(l). . . xu(n)1

0

E

S,}

then P(V,W ) is precisely the class of n-rewritable groups (see [6]). Therefore the main result of [4]says that Question 1.3 has positive answer for

V

= W = {%(1)

.

' '

Xu(,)

I

E

Sn},

3

and the result of [5] says that the equality (*) in Question 1.3 is true, whenever

V

= {XI

. . x,} and W = ( x , ( ~ ). . . x ~ (I ~ E) Sn}.

Let n > 0 be an integer. It seems that the 'simplest' case of Question 1.3 is when V = {P} and W = (1). Note that in this case P ( V , W ) is t,he variety of groups satisfying the law x R = 1, i.e. the Burnside variety of finite exponent n of groups. For this latter choice for (V, W ) Question 1.1 has positive answer (see [lo]).Here we positively answer Question 1.3 when V = {x'} or V = {x3} and W = (1). Theorem 1.1. If n E {2,3}, then P ( { x " } ,(1)) U .F = P * ( { x n } (1)). ,

2. Proofs

We begin with a result which shows that the equality (*) of Question 1.3 is true on the class of residually finite groups, whenever V and W are finite. Proposition 2.1. Let V and W be any non-empty finite subsets of the free group of rank n E N. Then every infinite residually finite P*(V,W)-group belongs to the class P(V,W ) . Proof. Let 91, . . . , gn be arbitrary elements of G and

S = (v(g1,.. . , g R ) w - ' ( g l , .. . , g n ) I v # w , u E V and w

E

W}.

Suppose, for a contradiction, that 1 $ S . Since G is residually finite and S is finite, there exists a normal subgroup N of finite index in G such that S n N = 0 . Now by considering infinite subsets N g l , . . . , Ng,, there exist z1 E V and w E W such that v # w and 1~ E vw-l(Ngl,. . . , Ng,) and so v(g1,.. . , g n ) w - l ( g l , . . . , g n ) E N , which is a contradiction. 0 For proving the main result, we first show some general results on infinite subsets of an arbitrary group similar to the main result of [2]. Theorem 2.1. Let w(zl,.. . , x n ) be a word in the free group of rank n > 1 such that all the letters X I , . . . , x , occur in 20. Suppose further that w = vlxzv2 where E is a non-zero integer, v l , v 2 are two (possibly empty) words such that only the letters X I , . . . , xi-1,x,+1 . . . , x , occur an them. Then every infinite subset Y of a group with the property that

a,b E

Y, a'

= b'

aa

=b

4

contains un infinite subset X such that for every n distinct elements E X ,we have ~ ( y i ,. . ,yn) # 1.

~ i ,. ..ryn

Proof. List the elements of Y under some well order I as y1, y2,. . . . Let s E Y ( n )the set of all n-element subsets of Y, and write the elements of s in the ascending order yj,, . . . , yj,, jl < . . . < j,. Create n! 1 sets as follows: One Y, for any a E S, and 2.Put s E Y, if w ( ~ j , ( l~. .) . ,y J O ( %= ) )1 and put s E 2 if s # Y, for all D E S,. Then by Ramsey's Theorem [ll], there exists an infinite subset X of Y such that X ( " ) C Y, for some a or X ( " ) C 2.If X(")C Y, for some a then for any sequence jl < j 2 < . . < j , (after restricting the order 5 to X)we have w(yj,(,, . . . , yjO(,)) = 1. Then

+

where v1 and 112 are evaluated on (Y.7m(1) . ,Yjm(t-I), Y.7m(t+l),' . . , Y J m ( % , )and theirs values do not depend on yJOcl,. Let a ( i ) = k . Now since X is infinite, there exist two sequences as follows: 1 . .

jl
aX, where [ a i , x ]= ai+l, 1 a of order 4:

< i < 3, [a4,x] = 1. Then H

has an automorphism

16

Also a2 is conjugation by a3. So there is an extension G = H ( a ) , where a2 = u3, [ui,a]= 1,1 i 4, and [ x , u ]= u1u2x4. Since B a G and G / B has a modular subgroup lattice, A = B ( a ) is quasinormal in G. But

<
aE,

(5)

which is of course the wreath product C, ? E , where C, denotes the cyclic group of order p . Set Y = (y) and A = B >a Y . Then

A is quasinormal in G, since Y is quasinormal in E . As Y-module, B is the sum of p 2 indecomposables of rank p . Therefore the Frattini subgroup

@(A= ) A’

=

[B,Y]

17

has rank p 2 ( p - 1) and B/A’ has rank p2. Set X = (x).Then @ ( A ) Xis n o t a subgroup. For, if it is, then

[ @ ( A )XI , 6 B n @ ( A ) X= @ ( A ) ,


O

and this contradiction proves the lemma.

0

Lemma 3.6. Let G be a soluble-by-finite m i n i m a x group with finitely m a n y subgroups which are neither normal n o r polycyclic. If X i s a non-polycyclic subgroup of G, then X G / X G i s finite. Proof. Clearly, the set of all subgroups Y of G such that X 5 Y 5 X G is finite. As maximal subgroups of soluble-by-finite minimax groups have finite index, it follows that the index IXG : XI is finite. Moreover X has finitely many conjugates in G by Lemma 3.3 and hence the group X G / X G is finite. 0

Corollary 3.7. Let G be a soluble-by-finite m i n i m a x group with finitely m a n y subgroups which are neither normal n o r polycyclic. If GIG' i s finite, t h e n every subgroup of infinite index of G is finitely generated. Proof. Let X be any subgroup of G which is not finitely generated. It follows from Lemma 3.6 that also the core X G of X is not finitely generated, and so the factor group G / X G has finitely many non-normal subgroups.

40

Then G / x G either is finite or nilpotent (see [5]). Since all nilpotent homomorphic images of G are finite, the subgroup X has finite index in G. The lemma is proved. 0 We can now prove the following result.

Lemma 3.8. Let G be a soluble-by-finite group with finitely m a n y subgroups which are neither normal n o r polycyclic. T h e n either G is polycyclicby-finite o r its commutator subgroup G' i s periodic. Proof. Assume for a contradiction that the statement is false, and choose a counterexample G for which the set { X I , . . . , Xk} of all subgroups which are neither normal nor polycyclic has smallest order k ; of course k > 0 , since the result holds in the case of groups with polycyclic non-normal subgroups (see [2]).It follows from Lemma 3.4 that all periodic subgroups of G are finite. In particular, the largest periodic normal subgroup T of G is finite and G / T is likewise a minimal counterexample, so that without loss of generality we may suppose that G contains no periodic non-trivial normal subgroups. Moreover, G is a minimax group by Lemma 3.5 and hence it is also residually finite. Thus the Fitting subgroup A of G is a torsion-free nilpotent group and G / A is finitely generated and abelian-byfinite. Clearly, A is not finitely generated and hence also its centre Z ( A ) is not finitely generated (see [ 6 ] , Lemma 2.6). Let (Ki)iElbe a collection of G-invariant subgroups of finite index of Z ( A ) such that

n

Kz = (1).

iEI

Thus every factor group G / K i has finitely many non-normal subgroups. As G' is not periodic, it follows that there exists i for which G / K i is not a Dedekind group, so that G/Ki is periodic (see [5]) and G / A is finite. On the other hand, A / Z ( A )is torsion-free, so that A = Z ( A ) is abelian and G is a finite extension of a torsion-free abelian subgroup. For every i = 1,.. . , k, the subgroup N G ( X ~has ) fewer than k subgroups ) finite index which are neither normal nor polycyclic; moreover, N G ( X ~has in G , so that it is not polycyclic-by-finite. Thus the minimal choice of G yields that NG(X,)' is periodic and hence even finite. In particular, each X i is finite and hence the normal closure

N=

(Xi, ..., X;)G

is likewise finite by Dietzmann's Lemma. Thus N = (1) and each subgroup X i is abelian. Clearly, G is not nilpotent and so G' cannot be finitely generated (see [l],Lemma 3.19). Moreover, there exists a G-invariant subgroup L of G' such that G'IL is a finite group whose order is divisible

41

by at least three different prime numbers (see [9], Part 2, Theorem 10.34); then L is not finitely generated and so G I L has finitely many non-normal subgroups. It follows that GIG' is finite (see [5]), so that the index IG : Xi1 is finite by Corollary 3.7 and hence each element of X i has finitely many conjugates in G. Since by Lemma 3.6 the group X G / X G is finite for each non-polycyclic subgroup X of G, there exists an element x of G such that ( x ) is~not polycyclic (see [3], Theorem 5.5). In particular, z has infinitely many conjugates, so that it does not belong to any X i and hence every proper subgroup of ( x ) containing ~ (z)is polycyclic. Obviously, the normal subgroup B = An(x)Gis not polycyclic and so ( x ) = ~ B(x);moreover, B I B n (z) does not contain proper subgroups of finite index and hence it is periodic. As the index IG : is finite by Corollary 3.7, it follows that A has rank 1 and contains a G-invariant subgroup Q of finite index which is isomorphic t o Q, for some prime number p . Then G/CG(Q) is isomorphic to a finite non-trivial group of automorphisms of Q, and so there exists g E G such that as = for all a E Q. The element g has finite order and Q n ( 9 ) = (1). If T is the set of all odd primes q # p , we have that

is an infinite set of non-normal subgroups of G which of course are not polycyclic. This contradiction completes the proof of the lemma. 0 Lemma 3.9. Let G be a n infinite polycyclic-by-finite group with finitely m a n y subgroups which are neither normal n o r polycyclic. T h e n all nonnormal subgroups of G are polycyclic.

Proof. Assume that the statement is false, and choose a counterexample G for which the set { X I ,. . . , X k } of all subgroups which are neither normal nor polycyclic has smallest order k. Then each X i has fewer than k subgroups which are neither normal nor polycyclic, so that all non-normal subgroups of X i are polycyclic and hence X,l is finite (see 121, Theorem 2.13). It follows that also the normal closure K = ( X i , . . . , X i ) G is finite. If T is the largest finite normal subgroup of G, we obtain that all non-normal subgroups of G/T are polycyclic, so that G / T must be polycyclic (see [2], Theorem 2.13) and hence T is not soluble. Suppose that G contains a finite subgroup E which is neither normal nor soluble, and let A be a torsion-free abelian non-trivial normal subgroup of G. By hypothesis the set T of all prime numbers p such that EAp is a normal subgroup of G is infinite, so that

E= n E A p P€T

42

is likewise normal in G, and this contradiction shows that all finite nonnormal subgroups of G are soluble. In particular, all non-normal subgroups of T are soluble and so T contains a minimal non-soluble normal subgroup N such that T I N is a Dedekind group. Then N is normal in G and GIN has finitely many non-normal subgroups, so that GIN is abelian and all non-normal subgroups of G are polycyclic (see [2], Theorem 2.13). This contradiction completes the proof. 0

Lemma 3.10. Lei G be a non-periodic group such that the set of all subgroups which are neither normal n o r polycyclic is finite and non-empty. If G / Z ( G ) is finite, then G is m i n i m a x and all its periodic subgroups are normal.

Proof. Let X and Y be non-polycyclic subgroups of the centre Z(G)such that XnY = { 1);then both factor groups G / X and G/Yhave finitely many non-normal subgroups and hence at least one of them must be periodic (see [ 5 ] ) .It follows that the subgroup T consisting of all elements of finite order of Z ( G ) is either finite or a finite extension of a Priifer subgroup and Z ( G ) = T x A , where A is a torsion-free abelian group with finite Prufer rank. Let B be a finitely generated subgroup of A such that A I B is periodic. Assume for a contradiction that G is not minimax, so that for each integer k 2 3 the group A / B 2 k has infinitely many non-trivial primary components and hence it contains a subgroup of the form

where each factor is infinite and has no elements of order 2. Then G/Lk,l and G/Lk,2 are Dedekind groups with elements of order 8 (see [5]), so that they are abelian and

It follows that

G' 5

n

B2' = (1)

k23

and this contradiction shows that G is a minimax group. Suppose now that A is finitely generated, so that T is infinite and the finite residual J of G is a group of type po3 for some prime p . In particular G is min-by-max. Clearly, the non-periodic group GIJ has finitely many non-normal subgroups, so that it is abelian and G' is contained in J . Thus all non-normal subgroups of G are polycyclic (see [2], Theorem 2.13), a contradiction. Therefore the subgroup A is not finitely generated. Finally, let H be any periodic subgroup of G. Clearly, A contains infinitely many

43

subgroups which are not finitely generated and so there exists a subgroup V of A such that H V is normal in G. As H is characteristic in H V , it is a normal subgroup of G. 0 Let A be a torsion-free abelian group of finite rank n. Recall that A is an C-group if it is not finitely generated but all its subgroups of rank less than n are finitely generated. Obviously, every torsion-free abelian group of rank 1 which is not finitely generated has the property 2; examples of C-groups of rank greater than 1 have been produced by L. Fuchs [4]. We shall say that an C-group A is an C; -group if it contains a finitely generated subgroup B such that AIB is a group of type pa for some prime number p .

Lemma 3.11. Let G be torsion-free abelian group containing a n C l subgroup of finite index. T h e n there exists a p r i m e number p such that the additive group Q, of rational numbers whose denominators are powers of p i s a homomorphic image of G . Proof. Let A be an 21-subgroup of finite index of G, and let B be a finitely generated subgroup of A such that AIB is a group of type pa ( p prime). Consider a subgroup C of B such that BIC is infinite cyclic, and let EIC be the subgroup consisting of all elements of finite order of AIC. Clearly E has rank less than A , so that it is finitely generated and hence EIC is finite. It follows that the torsion-free group AIE is isomorphic to Q,. On the other hand, the subgroup T I E of all elements of finite order of G I E is finite, so that GIE splits over T I E and Q, is a homomorphic image of G. 0

We are now in a position to prove the main result of the paper. It completes the classification of locally graded groups with finitely many subgroups which are neither normal nor polycyclic.

Theorem 3.12. Let G be a locally graded non-periodic group with finitely m a n y subgroups which are neither normal n o r polycyclic. T h e n all nonnormal subgroups of G are polycyclic. Proof. The group G is soluble-by-finite by Lemma 3.2. Assume that the statement is false, and suppose first that the commutator subgroup G’ of G is finite. Then every finitely generated subgroup of G has finitely many conjugates and it follows from Lemma 3.3 that G has finite conjugacy classes of subgroups. Thus the factor group G / Z ( G )is finite (see [ 7 ] ) ;in particular, all subgroups of G’ are normal in G by Lemma 3.10 and so G is soluble. Let X be a non-normal subgroup of G which is not polycyclic. Clearly,

44

G I X ' is likewise a counterexample and hence replacing G by G / X ' it can be assumed without loss of generality that X is abelian. If X contains a Prufer subgroup P , the factor group G / P is not abelian and contains only finitely many non-normal subgroups, a contradiction since G I P is not periodic (see [5]). As G is a minimax group by Lemma 3.10, it follows that the subgroup Y consisting of all elements of finite order of X is finite and hence X = Y x A , where A is a torsion-free subgroup. The intersection C = A n Z(G) contains a finitely generated subgroup B such that C / B is a non-trivial direct product of Prufer subgroups. Let D / B be a Prufer subgroup of C I B and assume that D # C ; then G / D is a non-Dedekind group having only finitely many non-normal subgroups and so G / C is a Dedekind group, a contradiction. Therefore C / B is a Prufer group. Let r be the rank of C , and let U be any subgroup of C with rank smaller than r ; as G/U is neither periodic nor abelian, it has infinitely many non-normal subgroups and hence U must be finitely generated. Thus C is an &-group and it follows from Lemma 3.11 that Q, is a homomorphic image of A for some prime number p . It follows that { A Q 1 q # p } is an infinite set of subgroups of A which are not finitely generated, and hence there exist distinct prime numbers q1 and q 2 such that both AQ1and A42 are normal subgroups of G. Therefore A = ( A Q 1 , A Q 2and ) , so also X = Y A , is normal in G. This contradiction shows that G' is infinite. Moreover, we have by Lemma 3.8 and Lemma 3.9 that G' is periodic. Let, H be m y proper G-invariant subgroup of G'. Then G / H is neither periodic nor abelian, so that it has infinitely many non-normal subgroups and hence H must be finite. Therefore G' has no infinite proper G-invariant subgroups. As G' is periodic, it has only finitely many infinite subgroups; thus G' is a Cernikov group, and so by hypothesis even a group of type poo for some prime p . If G' 5 Z(G), then G is a minimax group with finite residual G' and every abelian subgroup of G is min-by-max (see [l], Theorem 2.13), so that all non-normal subgroups of G are polycyclic (see [a],Theorem 2.13), a contradiction. Assume now that G' is not contained in Z ( G ) , so that G = R E , where R is a divisible abelian normal psubgroup of G and E is finitely generated (see [l],Theorem 2.11). Clearly, G' is contained in R and hence there exists a subgroup V such that R = G' x V . If V # {l},we have that G' contains a proper subgroup K such that K V is normal in GI so that V itself is a normal subgroup of G and G / V has finitely many non-normal subgroups, contradicting the fact that G/V is neither periodic nor abelian. Therefore R = G' and hence G is a min-bymax group with finite residual G'. Another application of Theorem 2.13 of [2] yields that all non-normal subgroups of G are polycyclic. This last 0 contradiction completes the proof of the theorem.

45

References 1. S. FRANCIOSI - F. DE GIOVANNI - L.A. KURDACHENKO: ‘‘On groups with many almost normal subgroups”, Ann. Mat. Pura Appl. 169 (1995), 35-65. - F. DE GIOVANNI - M.L. NEWELL:“Groups with polycyclic 2. S. FRANCIOSI non-normal subgroups”, Algebra Colloq. 7 (2000), 33-42. - F. DE GIOVANNI - M.J. TOMKINSON: “Groups with 3. S. FRANCIOSI polycyclic-by-finite conjugacy classes”, Boll. Un. Mat. Ital. 4B (1990), 35-55. 4. L. FUCHS:“Infinite Abeiian Groups”, Academic Press, New York (1970). 5. N.S. HEKSTER- H.W. LENSTRA:“Groups with finitely many non-normal subgroups”, Arch. Math. (Basel) 54 (1990), 225-231. 6. L.A. KURDACHENKO - A.V. TUSHEV - D.I. ZAICEV:“Modules over nilpotent groups of finite rank”, Algebra and Logic 24 (1985), 412-436. 7. B.H. NEUMANN: “Groups with finite classes of conjugate subgroups”, Math. 2. 63 (1955), 76-96. - J.S. WILSON:“On certain minimal conditions for infinite 8. R.E. PHILLIPS groups”, J . Algebra 51 (1978), 41-68. 9. D. J.S. ROBINSON: “Finiteness Conditions and Generalized Soluble Groups”, Springer, Berlin (1972). 10. G.M. ROMALIS - N.F. SESEKIN:“Metahamiltonian groups”, Ural. Gos. Univ. Mat. Zap. 5 (1966), 101-106. 11. G.M. ROMALIS- N.F. SESEKIN:“Melahamiltonian groups 11”, Ural. Gos. Univ. Mat. Zap. 6 (1968), 52-58. 12. G.M. ROMALIS - N.F. SESEKIN: “Metahamiltonian groups 111”, Ural. Gos. Univ. Mat. Zap. 7 (1969/70), 195-199.

GROUPS WITH CONDITIONS ON INFINITE SUBSETS COSTANTINO DELIZIA and CHIARA NICOTERA Dipartimento di Matematica e Informatica, Universita di Salerno, via Ponte don Melillo, Fisciano (SA), 84084, Italy E-mail: [email protected]. [email protected] The purpose of this paper is to present a comprehensive overview of known results on certain combinatorial problems which are related to infinite subsets of a group. Moreover, in the last section of this paper we prove that if G is a finitely generated locally graded group and every infinite subset of G contains different elements x and y such that [z, y, y] = 1, then G/Zz(G) is finite. Keywords: Engel conditions, locally graded groups, infinite subsets, twogenerator subgroups.

1. Introduction During a conference of the Australian Mathematical Society in 1975, Paul Erdos posed the following question:

Suppose that every infinite set of elements of a group G contains a pair which commute. Does exist an upper bound for the order of (finite) subsets of G consisting of pairwise non-commuting elements?

B.H. Neumann [31] proved that for a group G the previous property is equivalent to being centre-by-finite] and therefore IG : Z(G)I is the required upper bound. Neumann’s proof uses Ramsey’s Theorem [7] stating that every infinite graph contains either an infinite complete subgraph or an infinite totally disconnected subgraph. In fact, groups arising in Erdos’ question belong to the class of groups whose commutativity graph has no infinite totally disconnected subgraphs. Since Neumann’s result many other authors have dealt with similar problems where the commutativity is replaced by a different group theoretical property. Let X be a class of groups. Given a group GI let rx.(G) be the simple graph whose vertices &re the elements of GI and different vertices z and y are connected by an edge if the subgroup ( z l y ) belongs t o the class K . ]

46

47

The group G is said t o be an X"-group if the graph (G) has no infinite totally disconnected subgraphs. X"-groups are discussed in Section 2. Suppose now X is a variety defined by the two-variable law w(z, y ) = 1. Given a group G, let rX*(G) be the simple graph whose vertices are all elements of G, and different vertices x and y are connected by an edge if w ( z , y ) = 1. The group G is said to be an X*-group if the graph r x * ( G ) has no infinite totally disconnected subgraphs. Of course, every X"-group is an X*-group. X*-groups are discussed in Section 3. Although no examples of varieties X with X * X o are known, it is a very hard problem to prove results concerning with X*-groups, since the related condition is very weak. Variations of these questions involving several infinite subsets of a group are discussed in Section 4. Finally, in Section 5 we prove our result stating that if G is a finitely generated locally graded group and every infinite subset of G contains different elements x and y such that [x,y, y ] = 1, then G / Z z ( G )is finite. 2. XO-groups

If A denotes the variety of all abelian groups, defined by the law [x,y] = 1, then A" = A*. With this notation we have:

Theorem 2.1 (B.H. Neumann [31]).G is a n d o - g r o u p if and only if G / Z ( G ) is finite. J.C. Lennox and J . Wiegold studied NO-groups, where class of all nilpotent groups.

N

denotes the

Theorem 2.2 (J.C. Lennox and J. Wiegold [25]). Let G be afinitely generated soluble group. T h e n G i s a n NO-group if and only if it is finiteby-nilpotent. Such a characterization does not hold in general. Indeed, for each prime p 2 5, M.R. Vaughan-Lee and J . Wiegold [41] constructed a countable locally finite group of exponent p which is perfect, and such that each of its 2-generator subgroups is nilpotent of bounded class. A group G has finite depth if the lower central series of G stabilizes after a finite number of steps. That is, if there exists a positive integer k such that 7k(G) = ~ k + l ( G )The . smallest positive integer k with this property is called the depth of G. Let 0 denote the class of all groups with finite depth, and let 0 k denote the class of all groups having depth at most k . Clearly, every NO-group is an 0"-group. So the following result generalizes Theorem 2.2.

48

Theorem 2.3. Let G be a finitely generated soluble group. T h e following conditions are equivalent: (1) G i s finite-by-nilpotent; (2) G i s a n 0"-group (A. Boukaroura [9]); (3) G i s a n X"-group where X i s the class of all Cernikov-by-nilpotent groups (N. Lemnouar [24]).

A. Boukaroura [9] also proved that a finitely generated soluble group is an Rk-group if and only if it is a finite extension of a group in which every 2-generator subgroup is nilpotent of class a t most k . Theorem 2.4. Let G be a finitely generated soluble group, and let X be one of the following classes: (1) of all polycyclic groups (J.C. Lennox and J . Wiegold [25])] or (2) of all supersoluble groups (J.R.J. Groves [17]), or

(3) of all nilpotent-by-finite groups (N.Trabelsi [39]), or (4) of all finite-by-nilpotent groups (N.Trabelsi [39]), or (5) of all torsion-by-nilpotent groups (N.Trabelsi [40]).

T h e n G i s a n Xo-group i f and only if it belongs t o the class X . If k > 1 is any integer, let Nk denote the class of all nilpotent groups with nilpotency class at most k . There are several results concerning with finitely generated soluble NkO-groups.

Theorem 2.5 (C. Delizia [lo]). Let G be a finitely generated soluble group. T h e n G i s a n NZo-group if and only if G/Zz(G) i s finite. For each prime p , L.-C. Kappe [18] constructed a group G, which is nilpotent of class 3 with exp(G,) = p 2 for p = 2,3, exp(G,) = p for p 2 5. Such groups have finite coverings by normal N2-subgroups, and therefore they are N2"-groups. But G,/Z2(G,) is infinite for all primes p . So one cannot remove the assumption that G is finitely generated in Theorem 2.5, even if the solubility is replaced by the nilpotency.

Theorem 2.6 (A. Abdollahi and B. Taeri [6], C. Delizia [13]). A finitely generated soluble NkO-group i s a finite extension of a nilpotent torsion-free group in which the nilpotency class of 2-generator subgroups does not exceed k . If G i s metabelian, t h e n G/Zk(G) i s finite. M.F. Newman [32] showed that for k 2 3 there exist finitely generated soluble NkO-groups G with derived length 3 such that G/Zk(G) is infinite.

49

So in the last sentence of Theorem 2.6 the assumption that G is metabelian cannot be removed. Theorem 2.7 (C. Delizia [13]). Let G be a finitely generated soluble NkO-group of derived length d . T h e n GlZkd-1 (G) is finite.

C. Deli& [13] also showed that if G is a finitely generated soluble N3O-group then G/Z4(G) is finite. Finitely generated residually finite N k o groups have also been investigated. Theorem 2.8 (C. Delizia [12]). Let G be a finitely generated residually finite group. T h e n G i s a n N2O-group if and only if G/Zz(G) i s finite.

A group G is said to be locally graded if every non-trivial finitely generated subgroup of G has a non-trivial finite quotient. This class of groups frequently appears in literature, mainly in the study of groups that do not have infinite finitely generated simple groups as subgroups. Locally soluble groups and locally residually finite groups are locally graded. The class of locally graded groups is subgroup closed, but it is not closed under taking homomorphic images, since free groups are locally graded. Sometimes, results that are known to hold true for soluble or residually finite groups can be extended to the class of locally graded groups [22]. The following theorem is essentially a generalization of most of the previous results concerning Nko-groups. Theorem 2.9 (C. Delizia, A. Rhemtulla and H. Smith [15]). Let G be a finitely generated locally graded N;co-group. T h e n there is a positive integer c depending only o n k such that G/Z,(G) i s finite.

A group G is an (Nk)-group if each infinite subset of G contains k elements which generate a nilpotent subgroup of class at most k.

+1

Theorem 2.10 (P. Longobardi and others [30]). A finitely generated group G as a n (Nk)-grOup if and only if G/Zk(G) i s finite.

The authors pointed out that (Nk)-groups have non-trivial FC-centre, so they are hyperabelian-by-finite. Therefore no solubililty assumptions are required in Theorem 2.10. It is also proved that every torsion-free ( N k ) group is nilpotent of class at most k. Given a positive integer k , let 8 k be the class of all k-Engel groups, that is, groups satisfying the identity [z, ky] = 1. Theorem 2.11 (P. Longobardi [26]). Let G be a finitely generated locally graded &ko-group. T h e n G is finite-by-(k-Engel).

50

3. &*-groups Recall from the previous section that the class &I, of all k-Engel groups is a variety defined by the two-variable law [ x , k y ] = 1. It is really hard to classify &k*-groups, since the related condition is very weak. The following results are known.

Theorem 3.1 (P. Longobardi and M.Maj [27]). Let G be a finitely generated soluble &k*-grOUP. T h e n G i s finite-by-nilpotent. More precisely:

Theorem 3.2 (A. Abdollahi [3]). Let G be a finitely generated soluble &k*-group. T h e n there i s a positive integer c depending only o n k such that G/Z,(G) i s finite. I f G i s metabelian, t h e n G/Zk(G) i s finite. For k

=2

we have the following results.

Theorem 3.3. Let G be a finitely generated &z*-group. If G is (1) soluble (A. Abdollahi [2]), or

(2) residually finite (C. Delizia and C. Nicotera [14]), then G/ZZ(G) is finite. In Section 5 we will prove that Theorem 3.3 can be generalized to the class of locally graded groups.

Theorem 3.4. Let G be a finitely generated locally graded &Z*-group. T h e n G/Zz(G) i s finite. For k = 3 we mention the following result:

Theorem 3.5 (A. Abdollahi [3]). Let G be a finitely generated soluble &S*-group. T h e n G i s a finite extension of a nilpotent group in which the nilpotency class of 2-generator subgroups does n o t exceed 5’. 4. Conditions involving several infinite subsets Let V be a variety of groups defined by the law w(y1, . . . , y,) = 1, and let Vg denote the class of all groups G in which, for any infinite subsets XI,. . . , X,,there exist q E XI, . . . , 2 , E X, such that w(z1,. . . ,x,) = 1. Obviously V U 3 & Vfl,where 3 is the class of all finite groups. It is known that for many varieties V and for many words w the equality V U 3 = Vfl holds.

51

Theorem 4.1. T h e equality V U F . = V t holds for the variety V defined by the law w = 1, when w is one of the following words: ( 1 ) [ y , z ] (B.H. Neumann [31]); ( 2 ) [y, zI2 (P. Longobardi and M. Maj [28]); (3) ~ [ Y ~ ~ Y ~ I [ P.S. Y ~ , Y~~ Ii I ~ m9 1 ) ; (4) [ y l ,. . . ,yk+l] (P. Longobardi, M. Maj and A.H. Rhemtulla ~91); (5) [y,z,z] (L.S. Spiezia 1351, [36]); (6) [ y ,z , z , z ] (L.S. Spiezia [37]); (7) ( y ~ ) ~ ( y ~ z(A. ~ ) Abdollahi -l [l]); (8) [ y ' l L r z ]where , m E { 3 , 6 } U { 2 k l k 2 0) (A. Abdollahi and B. Taeri [51); (9) [ y m , z " ] , where m E {2klk 2 0) (A. Boukaroura [S]); (10) [y", z ] [ y z"]-', l where n E { f 2 , 3 } (B. Taeri [38]); (11) ( y ~ ) ~ ( z y(A. ) - ~Abdollahi and B. Taeri [5]); (12) 91... y n (G. Endimioni [16]); (13) ( y y y ? . . . x F ) ~ where , m E { 2 k \ k 2 0) (A. Boukaroura [S]).

Theorem 4.2. Let V be a variety of groups defined by the law w a n infinite Vtf-group G is a V-group in the following cases:

= 1.

Then

( 1 ) w = [ y l ,. . . , y k , y l ] and G i s hyperabelian or locally soluble or locally finite (C. Delizia [ l l ] ) ; ( 2 ) w = [y, k z ] and G i s locally graded (A. Abdollahi [3]); (3) w = [ y n , z ] [ y , z n ] - ' ,where n E { - 3 , 4 } and G has finitely m a n y elem e n t s of order 2 or 3 (B. Taeri [38]). Statement ( 2 ) in Theorem 4.2 had been previously proved by L.S. Spiezia [35] for finitely generated soluble groups, and by 0. Puglisi and L.S. Spiezia [34]for locally soluble and locally finite groups. It is still an open question whether the equality V U F = Vtf holds for any variety V and for any word w (see Problem 15.1 in The Kourovka notebook [ 2 3 ] ) But . it is known that there exist general situations in which the previous equality holds.

Theorem 4.3 (G. Endimioni [IS]). Let V be a variety of groups defined by the law w = 1. T h e n a n infinite Vu-groupG is a V-group in the following cases: (1) G is locally nilpotent;

52

(2) G i s finitely generated and soluble, and every finitely generated soluble V-group i s polycyclic; (3) G i s locally soluble o r locally finite, and every finitely generated soluble V-group is nilpotent.

A. Abdollahi [4] also investigated varieties V and certain classes X of groups for which the equality (V U F)n X = Vfl n X holds. Several other variations of the discussed problems appear in the literature. For instance, P.S. Kim, A. Rhemtulla and H. Smith [20] proved that if G is an infinite group with the property that for every four infinite subsets Y1, Y2, Y,, Y4 of G there exist elements yyi E Y, (i = 1 , 2 , 3 , 4 ) such that the subgroup (yl,yz, y3,y4) is metabelian, then G itself is metabelian. Let m be a positive integer and n = 2m. As a generalization of the previous result, A. Rhemtulla and H. Smith [33] proved that if G is an infinite group with the property that for every n infinite subsets Y1 , . . . , Y, of G there exist elements yi E Y i ( i = 1,.. . ,n ) such that the subgroup (yl, . . . ,y,) is soluble with derived length at most m, then G itself is soluble with derived length at most m. 5. Proof of Theorem 3.4 We will prove that every finitely generated locally graded &2*-groupis residually finite. Then the result will follow from (2) of Theorem 3.3. SteD 1. If G is a n group t h e n ( x ) ( Y )is finitely generated f o r all x and y in G , that i s G i s restrained (211. Of course, we may assume that y has infinite order. Thus the set

{zyi :i

> 1)

is infinite. Since G is an &2*-group, there exist different integers i , j such that

Clearly

>

1

53 Then it is straightforward t o see that [ x y i ,x y j , x y j ]

= (xY~-~x-1x-YixY'),Y3.

Thus from [ x y i ,x y j , x y j ] = 1 it easily follows that x Y ' ~ Y ' - ~ x - ~ x - = ~ ~ 1. If j > i , write xY' = xYixx-Yz-'; if i > j , write xYi = zY'zY'-~~-~. In both cases we conclude that (xy" : n

2 0) I (ZY"

: 172.1

Now starting from the infinite set { z y i : i argument, we can prove that (xy" : n

for suitable integers h, k

5 0) 5

(xy" : lnl


0 for some t E N, then, by 0 < Prob:(t) 5 Prob&(t) and, by Proposition 3.1, YG(S)is finite for every simple group S. This also implies that rn,O(G) is finite for every integer n, since there are a t most two non-isomorphic simple groups of order n, say S 1 and Sz,and m;(G) = V L ~ ( S ~x ~S;G(S2)) ( ~ ' )= y~(S1)+ Y G ( S 1 such that Pn # 0 and let H a N with IN : HI = n; then N / H is a simple group of order at least ISI.Actually, n = IS1 and since is not a prime-power, if rs = n where r and s are coprime, then ,BT = ps = 0 and Pn # PTPs = 0 in contradiction with the 0 fact that the series D $ ( s ) is multiplicative.

1st

71

Proposition 5.3. The group G is pro-nilpotent if and only i f D;(s) = PG(S)

Proof. If G is pro-nilpotent, then every maximal subgroup is normal, hence p ( H , G ) # 0 only if H a G and p a ( H , G) = p ( H ,G ) . Therefore, D;(s) = pG(S).

To prove the converse, consider the intersection T of the normal subgroups M of G with the property that G / M is cyclic of prime order. Then S non abelian

S abelian

We have YC(S)

D z ( s ) = D z / T ( ~ ) D ; ( ~with )

Dg(s)= S non abelian

and PG(s)= PG/T(s)PG,T(s) where (see for example [5])

On the other hand, since G / T is abelian, the two Dirichelt series P G / T ( s ) and D Z I T ( s )coincides and consequently

PG,T(s)= D;(s). Assume by contradiction that G is not pro-nilpotent: it is not difficult to see that there exists at least a maximal subgroup M of G which is not normal and such that IG : MI is either a prime power or an odd integer. Let M be the set of proper subgroups H of G such that H T = G and IG : Hi is either odd or a prime power and let m = minHEM IG : HI. If H E M and IG : HI = m, then H is a maximal subgroup of G and p ( H , G ) = -1; hence, by looking at the definition of PG,T(s),we deduce that b, # 0. On the other hand, since PG,T(s) = D;(s), if b, # 0, then m must be divisible by I S1 for some non abelian simple group S, so rn is even and divisible by at least two different primes, a contradiction. 0 References 1. Nigel Boston, A probabilistic generalization of the R i e m a n n zeta function, Analytic number theory, Vol. 1 (Allerton Park, IL, 1995) 138 (1996), 155162. 2. Kenneth S. Brown, The coset poset and probabilistic zeta f u n c t i o n of a finite group, J. Algebra 225 (2000), no. 2, 989-1012.

72

3. Eloisa Detomi and Andrea Lucchini, Crowns and factorization of the probabilistic zeta f u n c t i o n of a finite group, J. Algebra 265 (2003), no. 2, 651-668. 4. Eloisa Detomi and Andrea Lucchini, Profinite groups with multiplicative probabilistic zeta f u n c t i o n , J. London Math. SOC.(2) 70 (2004), no. 1, 165-181. 5. Eloisa Detomi and Andrea Lucchini, Crowns in profinite groups and applications, Noncommutative algebra and geometry, Lect. Notes Pure Appl. Math., vol. 243, Chapman & Hall/CRC, Boca Raton, FL, 2006, 47-62. 6. Philip Hall, T h e eulerzan functzons o f a group, Quart. J. Math. (1936), no. 7, 134-151. 7. Alexander Lubotzky and Dan Segal, Subgroup growth, Progress in Mathematics, vol. 212, Birkhauser Verlag, Basel, 2003. 8. Andrea Lucchini, T h e X-Dirichlet polynomial of a finite group, J. Group Theory 8 (2005), no. 2, 171-188. 9. Avinoam Mann, Positively finitely generated groups, Forum Math. 8 (1996), no. 4, 429-459. 10. Avinoam Mann, A probabilistic zeta f u n c t i o n for arithmetic groups, Internat. J. Algebra Comput. 15 (2005), no. 5-6, 1053-1059. 11. Avinoam Mann and Aner Shalev, Sample groups, maximal subgroups, and probabilistic aspects of profinite groups, Israel J. Math. 96 (1996), part B, 449-468. 12. Richard P. Stanley, Enumerative combinatorics. Vol. 1, Cambridge Studics in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 1997. 13. John S. Wilson, Profinite groups, London Mathematical Society Monographs. New Series, vol. 19, The Clarendon Press Oxford University Press, New York, 1998.

GROUPS WITH PROPER SUBGROUPS OF CERTAIN TYPES Martyn R. Dixon* and Martin J. Evans**

Department of Mathematics University of Alabama Tuscaloosa, A L 35487-0350, U.S.A. * email:[email protected]. edu ** email: [email protected] Howard Smith

Department of Mathematics Bucknell University Lewisburg, PA. 17837, U . S . A . email:[email protected] This paper represents the content of a talk given by the first author at the Ischia Group Theory Conference, 2006.

Keywords: locally (soluble-by-finite), finite rank, soluble

1. Introduction Let P be a group theoretical property or class of groups. As usual, we shall say that G is a P-group, or that G E P , if G has the property P or belongs to the class P . We let 7 denote the class of groups G in which every proper subgroup of G is a P-group. If P is a subgroup closed property then of course P 5 P and, for the most part, we will assume that this is the case. We are interested in two questions: (1) For which properties P do we have 7 = P? (2) If P # 7, what can be said concerning the structure of the minimal non-P groups in 7 ,those groups in 7 \ P? The types of classes P that we are interested in here are:

5 the class of finite groups 'u the class of abelian groups

73

74 %(%,)

the class of nilpotent groups (of class at most c)

G(6d)the class of soluble groups (of derived length at most d ) % the class of groups of finite rank.

Recall that a group G is of finite rank r if every finitely generated subgroup of G is at most r-generator and r is the least integer with this property. We shall also consider product classes such as P = 6%,the class of d 1 groups G that are soluble-by-finite rank. Various authors have considered groups in which all subgroups are normal, or all subgroups are subnormal, or all subgroups are inert etc. but we shall not discuss such work here. Our notation when not explained is that used in [l]. 2. A test case and a brief history It is well-known that U %. On the other hand, since LU = U, it is clear that non-finitely generated groups in are abelian. What can be said concerning finitely generated %-groups? Finite groups with all proper subgroups abelian (or merely nilpotent) were classified in [2-51. The classification of infinite finitely generated %groups appears to be intractable because of such exotic examples as the Tarski monsters, those infinite simple 2-generator groups in which every proper subgroup is cyclic of prime order, first constructed in [6]. To avoid such problems it is standard to restrict attention t o the class of locally graded groups, where a group G is locally graded if every nontrivial finitely generated subgroup has a nontrivial finite image, and this we do from now on. If G is an infinite finitely generated locally graded %-group then of course there is a free abelian normal subgroup N such that GIN is finite. If q is a prime not dividing the order of GIN then the Schur-Zassenhaus theorem ensures that GINQsplits over NINq and it is now quite easy to see that G is metabelian. However, even more is true: infinite finitely generated locally graded %-groups are themselves abelian due to the following result of Zaicev [7] which, in our opinion, deserves to be better known.

a

Theorem 2.1 (Zaicev). Every infinite soluble %-group as a n 6;d-g'roup. This result has itself been generalized to the class of locally graded groups in Lemma 2.1 of [8]. The extension from the soluble case to the locally graded case needs a very similar argument to the easy one given just before Theorem 2.1.

75

Theorem 2.2 (Dixon-Evans). Let G be a n infinite locally graded group. T h e n G i s a n 6d-group.

6-

It follows that infinite locally graded a-groups are abelian and, more generally, each locally graded %-group is finite or in Gd. A much more difficult problem arises when we remove the bound on the derived lengths of the proper subgroups and attempt to prove a corresponding result for in Section 3. The and return t o G-groups. We first discuss the class class 3 was considered in the paper of Newman and Wiegold [9], which appeared several years before the construction of the Heineken-Mohamed groups in [lo].The Heineken-Mohamed groups are infinite soluble groups in % \ !Yl with the additional property that every subgroup is also subnormal. Further examples of Heineken-Mohamed type groups have since appeared in [ll]and [12], but it is well-known that if G is a non-nilpotent soluble group with all proper subgroups subnormal and nilpotent then GIG' Z Cpm, for some prime p . (The original Heineken-Mohamed groups are subgroups of Cp) Cpm.)The following important result of Asar [13] shows that at least locally graded %groups are not too complicated. Theorem 2.3 (Asar). Let G be a locally graded 8 - g r o u p . T h e n G i s soluble.

A less well-known property of the original Heineken-Mohamed groups is that they are examples of @-groups that are not; %&groups. This latter fact follows from Lemma 1 of [lo] since if X 5 G then XG' 5 G. Hence XG'IG' is finite and X is abelian-by-finite. We mention here that groups (and m-groups) do not differ too much from those of HeinekenMohamed type. First we have a theorem of Napolitani and Pegoraro [14].

m-

Theorem 2.4 (Napolitani-Pegoraro). Let G be a locally graded group.

( i ) If G E (ii) If G E

then either G E t h e n either G E

%zor G i s periodic. %5 or G i s periodic.

One way of viewing this is that in the universe of non-periodic groups (for example) m - g r o u p s are themselves nilpotent-by-finite. This result built on earlier work of Bruno [15], who discussed periodic locally graded %&groups G and proved, in particular, that if G $ 2lUg then 0 0

G is metabelian and GIG' E C p m for , some prime p When G is not a pgroup then G' is a p'-group and G = G' M A, where A S Cpm

76 0

If G is a pgroup then all proper subgroups of G are ascendant and hypercentral, and for all proper subgroups H of G , HG' $; G

Now, Asar's theorem (Theorem 2.3) is actually much more general than discussed above. Otal and Peiia [16] initiated the study of groups in which all proper subgroups are nilpotent-by-Cernikov and, by combining the work of Asar [13] with further results of Napolitani and Pegoraro [14] it is in fact possible to prove

Theorem 2.5 (Asar). If G is a locally graded group a n d all proper subgroups of G are nilpotent-by-cernikov t h e n G i s itself nilpotent-by-Cernikov. In particular G i s soluble. Thus the periodic m - g r o u p s are necessarily nilpotent-by-Cernikov. We remark also that Bruno and Napolitani [17] have recently shown, using Theorem 2.5, that a locally graded group in which every proper subgroup is nilpotent of class at most c-by-Cernikov is itself nilpotent of class at most c-by-Cernikov. 3. The class

and groups of related type

---

In this section we are interested in the classes G , 6 3 ,G% and so on. What is the structure of groups in these more general classes? The first extra complication that arises here is the presence of simple groups, such as As, in the class 6?. On the other hand, if G is an infinite locally graded group and G E g , and if G is not finitely generated then it is clear that G is locally soluble. Furthermore,

Lemma 3.1. If G i s a n infinite finitely generated locally graded c - g r o u p t h e n G is soluble. Proof. Since G is locally graded there is a normal subgroup N of G such that GIN is a nontrivial finite group. Since G E it is clear that N is soluble, of derived length d say. Let \GIN1 = k . If X G then X N / N Z X / ( X n N ) is soluble of derived length at most k and hence X is soluble of derived length at most d k . Thus G E G;d+k and it follows from Theorem 2.2 that G is actually soluble of derived length at most d k . 0

+

The following problem is as yet unsolved: 0

Is every locally soluble - g r o u p soluble?

+

77

This is probably the most important open problem in this area. In Theorem 1 of [18] it is shown that an infinite periodic locally soluble - g r o u p that is not soluble must be a perfect locally nilpotent pgroup for some prime p. We turn now to the more general class of locally graded --groups but we are immediately confronted with the following open problem. 0

What can be said concerning locally graded groups of finite rank?

At this stage it is reasonable, then, to restrict attention to a smaller class of groups, where something is known concerning this latter question, and a natural candidate is the class of locally (soluble-by-finite) groups. We note however that Cernikov [19] defined a rather broad generalization of the class of locally (soluble-by-finite) groups which we now describe. Let L, R, P, P be the usual closure operations as defined in [l]. Thus if ?;, is a class of groups then 0 0

0 0

G

E L2J if every finite subset of G is a subset of a 3)-group G E R’Z) if for each 1 # x E G there is a normal subgroup N , of G such that x $! N, and GIN, E 9 G E PZJ if G has an ascending series each of whose factors is a !2J-group G E P?2Jif G has a descending series each of whose factors is a !&group

We let A denote the set of closure operations (L, R, P,P} and define the class X to be the A-closure of the class of periodic locally graded groups; the class X is Cernikov’s class. Certainly X is an immense class of groups and it is quite easy t o prove the following result.

Lemma 3.2. Every X-group is locally graded. On the other hand it seems to be unknown whether X exhausts the class of all locally graded groups. 0

Is there a locally graded group that is not an X-group?

The importance of Cernikov’s paper for us lies in the following result which is proved in [19]. Theorem 3.1 (N. S. Cernikov). An X-group of finite rank is almost locally soluble. This result directly generalizes major theorems of Sunkov [20] and Lubotzky and Mann [21], who obtained results of the same type for locally finite and residually finite groups respectively.

78

In [22] we proved the following variant of Cernikov’s theorem

Theorem 3.2. Let G be a locally (soluble-by-finite) group. Suppose that all locally soluble subgroups of G have finite rank. Then G i s almost locally soluble. In view of Theorem 3.1 a number of results concerning X-groups satisfying certain rank conditions can be obtained and we now discuss the class Xn In 1231 we proved the following result.

m.

Theorem 3.3. Let G be a X-group with all proper subgroups soluble-byfinite rank. T h e n either (a) G i s (ii) G i s (iii) G i s (iv) G i s

locally soluble, or soluble-by-finite rank and almost locally soluble, or soluble-by-PSL(2, F ) , or soluble-by-Sz(F),

where F is a n infinite locally finite field with n o infinite proper subfields.

As we indicated earlier, i t is unknown whether case (i) can actually arise without the group also being of the type in case (ii). We note also that P S L ( 2 , F ) and Sz(F) are =-groups which are not 6%-groups. We do not know what sorts of soluble groups can occur in (iii) and (iv) of Theorem 3.3. In particular one problem that often crops up in this type of work, and which requires further study, in our view, is the following question. a

What is known concerning extensions of elementary abelian pgroups by P S L ( 2 , F ) (or Sz(F)), where p is the characteristic of F?

We first obtained Theorem 3.3 in the locally (soluble-by-finite) case; a somewhat involved transfinite induction then allowed us to deduce the result for X-groups. The reader is referred t o [23] for details. When we replace “soluble-by-finite rank” by “soluble-by-finite” our results can be strengthened so that they apply t o the class of all locally graded groups, as follows (see Theorem C of [23]).

Theorem 3.4. Let G be a locally graded group with all proper subgroups soluble-by-finite. T h e n either

(i) G is locally soluble, or (ii) G is soluble-by-finite, or (iii) G is soluble-by-PSL(2, F ) , or

79

(iv) G i s soluble-by-Sz(F), where F i s a n infinite locally finite field with n o infinite proper subfields. 4. Methods

There is no doubt that a major difficulty one encounters in proving theorems of the kind given above is the possible presence of infinite simple groups. In [24]we obtained the following result about simple locally (soluble-by-finite) groups which is easy t o prove and yet has important consequences.

Proposition 4.1. Let G be a countably infinite simple locally (soluble-byfinite) group. T h e n G contains a locally soluble, residually finite, proper subgroup R, and for each integer i 2 1 subgroups X i , Ri such that:

(a) &+I 5 Ri 5 R 5 X i 5 Xi+l, (ii) G = Ui>lXi, (iii) Ri Q X i , and (iv) X i I R i i s finite.

It follows that G i s locally residually finite and R has the property that for every g E G there exists a positive integer n = n ( g ) such that gn E R. A key ingredient in [24] is the notion of an inert subgroup. A subgroup H of a group G is inert if IH : H n H9l is finite for each g E G. (The study of inert subgroups was initiated by Belyaev in [25] and [26].) The idea here clearly is that H is not moved very far under the action of conjugation. Certainly normal subgroups and finite subgroups of a group are inert and it is not difficult t o show that the subgroups X i and R occurring in Proposition 4.1 are also inert. Because of the additional properties that R enjoys, we have termed the subgroup R of Proposition 4.1 super-inert. The key to proving results about simple groups is now the following. Theorem 4.1. Let G be a countable simple locally (soluble-by-finite) group and suppose that R i s a super-inert subgroup of G with nontrivial HirschPlotkin radical. T h e n G i s locally finite. The idea of the proof is to show first that each of the inert subgroups X i of Proposition 4.1 has nontrivial Hirsch-Plotkin radical H P ( X B ) . It is then possible, using a result of Belyaev (see Theorem 1.4 of [25]), to deduce that X i / H P ( X i ) is an FC-group. By a well-known fact it follows that

80

X i H P ( X i ) / H P ( X i ) is locally finite, and then one can deduce that G is locally (nilpotent-by-finite) . The techniques used by Napolitani and Pegoraro in [14] then enable us t o show that G is locally finite. Lemma 4.1. Let G be a n infinite simple locally (soluble-by-finite) group with all proper subgroups soluble-by-finite rank. T h e n G PSL(2, F ) or G E S z ( F ) f o r some infinite locally finite field F , all of whose p'roper subfields are finite.

Proof. If G is countable then a corresponding super-inert subgroup R (as in Proposition 4.1) has nontrivial Hirsch-Plotkin radical by Lemma 10.39 of [l]so G is locally finite, by Theorem 4:l. If G is uncountable then by Theorem 4.4 of [27] G has a local system consisting of countably infinite simple subgroups, each of which is locally finite by the above argument; hence G is locally finite in any case. Now if H is a proper subgroup of G then there exists N a H such that N is soluble and H / N has finite rank. By a theorem of Sunkov [20], H / N is almost locally soluble and hence H itself is almost locally soluble. The result now follows by a theorem of Kleidman and Wilson [28]. 0 The proof of Theorem 3.3 hinges in part on the following result. Lemma 4.2. Let G be a locally (soluble-by-finite) group with all proper subgroups soluble-by-finite rank. T h e n either

( i ) G is almost locally soluble, or (ii) G i s (locally soluble)-by-simple. Proof. By Lemma 1 of [23], G has a locally soluble radical S and G / S has trivial locally soluble radical. Suppose that G/S is not simple and let N / S be a nontrivial proper normal subgroup of G / S . Then N / S is solubleby-finite rank and it is easily seen that N / S therefore has finite rank. By Cernikov's theorem [19] it follows that N / S is almost locally soluble and hence finite. Thus G/CG(N/S) is finite. If G = CG(N/S) then N / S is abelian and hence trivial, contrary t o the choice of N . Hence CG(N/S) is a proper subgroup of G and is therefore soluble-by-finite rank. By Cernikov's theorem CG(N/S) is almost locally soluble, whence G too is almost locally soluble. The result now follows. 0 The lemma can be improved substantially by replacing (ii) above with: (ii)* G is soluble-by-simple. This follows quite easily using the next result which requires use of the Burnside Basis Theorem (see Lemma 6 of [23]).

81

Lemma 4.3. Let X be a group, Y a locally nilpotent normal subgroup of X and U a soluble normal subgroup of Y of derived length k such that Y / U has rank r . T h e n Ux i s soluble of derived length at m o s t k(r 1).

+

Finally we remark that some of the techniques used in t h e proofs of the results presented here can be used in attempting t o classify @- and =-groups, and we hope t h a t the details of this classification will appear in a forthcoming paper.

Acknowledgments This pa.per is dedicated to Akbar Rhemtulla, on the occasion of his retire-

ment. T h e authors would also like t o thank t h e conference sponsors for financial support. References 1. D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups vols. 1 and 2, Ergebnisse der Mathematik und ihrer Grenzgebiete (SpringerVerlag, Berlin, Heidelberg, New York, 1972). Band 62 and 63. 2. G. A. Miller and H. Moreno, Non-abelian groups in which every subgroup is abelian, R u n s . Amer. Math. SOC.4, 398 (1903). 3. K. Iwasawa, Ueber die Struktur der endlichen Gruppen, deren echte Untergruppen samtlich nilpotent sind, Proc. Phys.-Math. SOC.Japan (3) 23, 1 (1941). 4. L. Rkdei, Die endlichen einstufig nichtnilpotenten Gruppen, Publ. Math. Debrecen 4 , 303 (1956). 5 . 0. J. Schmidt, Groups all of whose subgroups are nilpotent, Mat. Sb. 31, 366 (1924). 6. A. Y . Ol’shanskii, Groups of bounded exponent with subgroups of prime order, Algebra i Logika 21, 553 (1982), English transl. in Algebra and Logic, 21 (1982), 369-418. 7. D. I. Zaicev, Stably solvable groups, Izv. Akad. Nauk SSSR Ser. Mat. 33, 765 (1969), English transl. in Math. USSR-Izv.,~(1969), 723-736. 8. M. R. Dixon and M. J. Evans, Groups with the minimum condition on insoluble subgroups, Arch. Math. 72, 241 (1999). 9. M. F. Newman and J. Wiegold, Groups with many nilpotent subgroups, Arch. Math. 15, 241 (1964). 10. H. Heineken and I. J. Mohamed, A group with trivial centre satisfying the normalizer condition, J . Algebra 10, 368 (1968). 11. B. Bruno and R. E. Phillips, On multipliers of Heineken-Mohamed type groups, Rend. Sem. Mat. Univ. Padova 85, 133 (1991). 12. F. Menegazzo, Groups of Heineken-Mohamed, J . Algebra 171, 807 (1995). 13. A. 0. Asar, Locally nilpotent pgroups whose proper subgroups are hypercentral or nilpotent-by-Chernikov, J. London Math. SOC.61, 412 (2000).

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14. F. Napolitani and E. Pegoraro, On groups with nilpotent by Cernikov proper subgroups, Arch. Math. 69,89 (1997). 15. B. Bruno, On groups with “abelian by finite” proper subgroups, Boll. Un. Mat. Ital. B (6) 3,797 (1984). 16. J. Otal and J. M. Peiia, Groups in which every proper subgroup is Cernikovby-nilpotent or nilpotent-by-Cernikov, Arch. Math. 51,193 (1988). 17. B. Bruno and F. Napolitani, A not,e on nilpotsnt-by-Chernikov groups, Glasg. Math. J . 46,211 (2004). 18. M. R. Dixon, M. J. Evans and H. Smith, Groups with various minimal conditions on subgroups, Ukrainian Math. J. 54,957 (2002). 19. N. S. Cernikov, A theorem on groups of finite special rank, Ukrain. Mat. Zh. 42,962 (1990), English transl. in Ukrainian Math. J. 42,(1990), 855-861. 20. V. P. Sunkov, On locally finite groups of finite rank, Algebra i Logika 10,199 (1971), English transl. in Algebra and Logic,lO (1971),127-142. 21. A. Lubotzky and A. Mann, Residually finite groups of finite rank, Math. Proc. Camb. Phil. SOC.106,385 (1989). 22. M. R. Dixon, M. J. Evans and H. Smith, Locally (soluble-by-finite) groups of finite rank, J . Algebra 182,756 (1996). 23. M. R. Dixon, M. J. Evans and H. Smith, Groups with all proper subgroups soluble-by-finite rank, J . Algebra 289,135 (2005). 24. M. R. Dixon, M. J. Evans and H. Smith, Embedding groups in locally (soluble-by-finite) simple groups, J . Group Theory 9,383 (2006). 25. V. V. Belyaev, Inert subgroups in infinite simple groups, Sibirskia Matematischeskil Zhurnal 34, 17 (1993), English transl. in Siberian Mathematics Journal, 34 (1993) 606-611. 26. V. V. Belyaev, Locally finite groups containing a finite inseparable subgroup, Sibirskia Matematischeski; Zhurnal34, 23 (1993), English transl. in Siberian Mathematics Journal, 34 (1993) 218-232. 27. 0. H. Kegel and B. A. F. Wehrfritz, Locally Finite Groups, North-Holland Mathematical Library (North-Holland, Amsterdam, London, 1973). Volume 3. 28. P. B. Kleidman and R. A. Wilson, A characterization of some locally finite simple groups of lie type, Arch. Math. 48,10 (1987).

COUNTING CONJUGACY CLASSES OF SUBGROUPS IN FINITE p-GROUPS, I GUSTAVO A. FERNANDEZ-ALCOBER Matematika Saila, Euskal Herriko Unibertsitatea, 48080 Bilbao, Spain E-mail: [email protected]

LEIRE LEGARRETA Matematika Saila, Euskal Herriko Unibertsitatea, 48080 Bilbao, Spain E-mail: 1eire.legarretaQehu.es In a series of two papers, we consider the problem of counting conjugacy classes of two families of subgroups in a finite pgroup G: the non-normal subgroups and the normalizer subgroups. We report on lower bounds for the number of such conjugacy classes, and propose some open problems. In this first paper, we consider the number of classes of non-normal subgroups.

Keywords: Finite p-groups; Conjugacy classes.

1. Introduction

A group G acts by conjugation on the set of its subgroups, and also on some classes of subgroups which are invariant under conjugation, such as the normal subgroups (the fixed points of the action), the non-normal subgroups, the cyclic non-normal subgroups and the normalizer subgroups. If the group G is finite, these types of subgroups are gathered in a finite number of conjugacy classes, say r , and we are curious about the properties of this number T . More precisely, we are interested in the case when G is a finite pgroup. How does T relate to other invariants of the group? In the remainder, let us write p(G), v ( G ) , v*(G) and w ( G ) to denote the number of (conjugacy classes of) normal subgroups, the number of classes of non-normal subgroups, of cyclic non-normal subgroups and of normalizer subgroups of G, respectively. If the order of G is p n , it is a basic fact that G has at least n 1 normal subgroups and, if G is not

+

83

84

+ +

cyclic, at least n p 1. Thus p(G) grows with n and, if G is not cyclic, also with p . Obviously, this is not the case for v ( G ) , v*(G) and w(G), as Dedekind groups show. (Recall that a Dedekind group is a group in which every subgroup is normal, and that finite Dedekind p-groups fall into two classes: abelian groups and Hamiltonian 2-groups, i.e. direct products of the quaternion group Q8 and an elementary abelian 2-group.) Even if we avoid Dedekind groups, we cannot say that these numbers of classes grow with n or p . For example, consider the group

Mpn = ( a ,b I apn-'

= b p-l,

n-2

[a,b]=ap

),

where n 2 3 if p > 2, and n 2 4 if p = 2. Then v ( M p n )= v*(Mpn) = 1 and w ( M p n )= 2 for all p and n. We can argue, however, that this group is almost abelian, since the derived subgroup has order p and the nilpotency class is 2. Hopefully, the further G is from being abelian, the larger v ( G ) , v*(G)and w(G) should become. As a natural measure of non-commutativity in a finite pgroup, we can take the nilpotency class c of G. Is it true that v ( G ) ,v*(G)and p ( G ) grow with c? Brand1 [4] suggested a more precise version of this question: if G is not Hamiltonian, does the inequality v(G) 2 c - 1 hold? Poland and Rhemtulla [15] answered this question in the affirmative. Later, Li Shirong [14] proved that v*(G)2. c-1 for odd p . In this case, it is not true any more that the bound holds for all 2-groups with the only exception of Hamiltonian groups: it is not difficult t o see that v* (G) = 2 for every 2-group of maximal class G. As for w(G), Gavioli, Legarreta, Sica and Tota [lo] have obtained the bound w ( G ) 2 c for odd p , the case p = 2 being still unsettled. Hence we can say that in most cases all three functions v(G),v*(G)and w(G) grow with c. Do they also grow with the prime p if the class is greater than 2? It is the purpose of this paper t o describe some positive answers to this question in the cases of v(G) and w ( G ) . In [8] we obtain the bound

v(G) 2 p ( c - 2)

+ 1,

(1)

with the only exception of Hamiltonian groups and generalized quaternion groups, in which case v(G) 2 2(c - 2) (note that p = 2 in these exceptional cases). On the other hand, Egizii di Marco [5] has obtained that

for every odd prime p. In fact, in the case of v(G) we can provide better bounds by replacing the nilpotency class c with other invariants which are usually greater. More

85

precisely, let k = k ( G ) and IG : Z(G)I = pe. Then

C = C(G) be defined by means of IG’l

+

v(G) 2 p ( k - 1) 1

=pk

and

(3)

for every p-group which is not Hamiltonian or generalized quaternion, and

v(G) 2 p(C- 3)

+2

(4)

for all odd primes p . These bounds are obtained in [8] and [9],respectively. Since k 2 c - 1, it is clear that (3) is an improvement of (1).On the other hand, we have C 2 c unless G is cyclic. Furthermore, if G / Z ( G ) is not a group of maximal class (which will most commonly happen) then l 2 c 1 and (4) is sharper than (1). This serves as a common introduction to the two papers on the subject in this volume, this one and [6]. In this first part we focus on the bounds for v(G). The proofs of (3) and (4) are lengthy, and we refer the reader t o [8] and [9] for full details. However, by skipping the more technical arguments, it is possible t o convey the main ingredients of these proofs without being long. This has been our goal in this paper, as well as proposing some open problems on this matter. The relation between w ( G ) , c and p will be considered in the second paper [6] of this series, written jointly by the two authors of this part and M. Egizii di Marco.

+

2. Conjugacy classes of non-normal subgroups and the

order of the derived subgroup In this section we outline the proof of the following result. Full details can be found in [8] by the authors.

Theorem 2.1. Let G be a f i n i t e p-group and suppose that IG’l = p k . T h e n v(G) 2 p ( k - 1) + 1, unless G i s a Hamiltonian group or a generalized quaternion group. For these exceptional 2-groups, v ( G ) = 2 ( k - 1). In order to simplify the exposition, we assume in the remainder that p is an odd prime, since the case p = 2 is more technical. The reason for this is that the intersection of all non-normal subgroups in a finite p-group G is trivial for p > 2 but can be non-trivial for p = 2. This result of Blackburn can be found in Theorem 1 of [2].Thus, if N # 1 is a normal subgroup of G and p > 2, then there is at least one non-normal subgroup of G not containing N , and consequently v(G) 1 v ( G / N ) 1.

+

86

As an appetizer, let us prove the weaker result v ( G ) 2 k . This is immediate by induction on k : if N is a normal subgroup of G of order p then v ( G / N ) 2 k-1 by the induction hypothesis, and then v ( G ) 2 v ( G / N ) + l 2 k , as desired. This straightforward proof suggests a strategy to obtain the bound v ( G ) 2 p ( k - 1) 1: prove that v ( G ) 2 v ( G / N )+ p for every normal subgroup N of order p , provided that k 2 2. (In the case k = 1, which is the base for the induction, we do not need anything since the bound holds trivially.) Unfortunately, the following example shows that the difference v(G)- v ( G / N ) can take any value, independently of p .

+

Example 2.1. Let p be a fixed odd prime. For n > m group given by the following presentation:

2

1, consider the

G = ( a , b I apn = bPm = 1, [a,b] = apn-m). Then the subgroup N = ( u p " - ' ) is the only central subgroup of G of order p . It is not difficult to see that any (non-normal) subgroup of G that does not contain N is conjugate to one of the subgroups ( b ) , ( b p ) ,. . . , (bpm-l). As a consequence, v ( G ) = v ( G / N ) m.

+

Despite this, in some cases we get a positive result.

Theorem 2.2. Let G be a finite p-group with non-cyclic centre, and let N be a n y normal subgroup of G of order p . If k 2 2 , then v ( G ) 2 v ( G / N )+ p .

Proof. Let T be a central elementary abelian subgroup of order p 2 containing N , and let T I , .. . , Tp+l be all the subgroups of T cf order p . Of course, we may assume Tl = N . Since any subgroup which contains Ti and Tj for different i and j must also contain T , we get

+ + . . . + U(G/Tp+l)- p v ( G / T ) = v ( G / N )+ C ( v ( G / T i ) v ( G / T ) ) .

v ( G ) 2 v ( G / T l ) v(G/Tz) P+l

-

i=2

Now G/Ti is not abelian for any i , since the derived subgroup G' has order at least p 2 . It follows that v(G/Ti) 2 v ( G / T )+ l for all i , and consequently

v ( G )2 4 G I N ) + P .

0

Thus, in a general induction framework, we may assume that G is a finite pgroup with cyclic centre. Let N be the only central subgroup of G of order p . Now we split the analysis into the following three cases: (i) G is a regular pgroup.

87

(ii) G is a pgroup of maximal class. (iii) G is not a regular group nor a group of maximal class. If G is a group of maximal class, then the structure of G is well understood (see for example [7]) and one can find directly, without using induction, p ( k - 1) 1 classes of non-normal subgroups. If G is not regular nor of maximal class, then another result of Blackburn (Theorem 6 in [3]) states that the number s of subgroups of order p satisfies the congruence

+ s

G

+ + . . + pp-’

1 p

(mod p”).

(5)

Now we can use the following result. Theorem 2.3. Let G be a finite p-group and suppose S i s a set of subgroups of G closed under conjugation, of cardinality s. If s = SO s l p . . Snpn is the p-adic decomposition of s, t h e n the number of conjugacy classes of subgroups building u p S i s at least SO s1 ’ . . Sn.

+ +. +

+ + +

Proof. Since the length of every conjugacy class of subgroups of G is a ppower, we get a decomposition of s in the form s = t o t l p . . tmp7”, where ti 2 0 is the number of classes of length pi. Thus it suffices to prove the following number theoretical result: if s is a positive integer and s = SO s l p . . . s,pn = t o t l p + . . . + t,pm with 0 5 si 5 p - 1 and ti 2 0 for all i, then t o + tl + . . . t , 2 so + s1 + . . . + .s, This can be proved by induction on s. Let i be the first index for which si > 0 and let similarly j be the first index for which t j > 0. Then j 5 i and the result follows by applying the induction hypothesis t o s - p’ .

+ +. +

+

+ +

+

+

By applying this theorem to congruence ( 5 ) , we obtain that the subgroups of G of order p gather in at least p conjugacy classes. One of these classes corresponds to the normal subgroup N , and consequently we get v ( G ) 2 v ( G / N ) p - 1. If G has an elementary abelian subgroup of order p 3 , then we can choose a subgroup H of order p 2 such that H n N = 1. Hence v(G) 2 v ( G / N ) + p and we are done. Now the structure of the groups without elementary abelian subgroups of order p3 has been described by Blackburn in Theorem 4.1 of [l],and one can check that the result holds in every case. Thus it only remains to consider the case when G is a regular pgroup. In this setting, a fundamental role is played by the following concept.

+

Definition 2.1. Let G be a finite pgroup. A subset ( 9 1 , . . . , g T } of G is called a basis of G if every element g E G can be uniquely written in the form g = 9;”’ . . . g:T with 0 5 n, < o ( g z ) .

88

A fundamental result of P. Hall [ l l ]assures that regular pgroups have bases, that all bases have the same number r of elements (which is characterized by the condition IG : GPI = IO1(G)l = p'), and that the orders of the elements in any two bases are the same. If ( 9 1 , . . , , g T } is a basis of G , then the subgroup N is contained in at most one of the subgroups ( g i ) . Thus the subgroups of the type ( g ) , where g is an element of some basis of G , are a rich source of non-normal subgroups not containing N . Now the problem is to decide how many non-conjugate subgroups of this type there exist. If IG : @(G)I2 p 3 , then it is possible t o find at least p non-conjugate subgroups, and consequently v ( G ) 2 v ( G / N ) p . Thus we are only left with the case that IG : @(G)I= p 2 , i.e. that G is 2-generated. This is the hardest part of the proof, since we cannot rely on induction to prove the result, as shown by Example 2.1. We have to find directly the required number of non-conjugate non-normal subgroups, and again the use of bases is a key ingredient. The details are too technical, and we refer the reader to [8]. Poland and Rhemtulla [15] have classified the groups for which the bound v ( G ) 2 c - 1 becomes an equality. It follows from this classification that the equality can only hold for c 5 4, and for odd p in fact only for the groups Mpn of class 2. This indicates that the bound v ( G ) 2 c-1 can be improved, as we have done in Theorem 2.1 with the bound v ( G ) 2 p ( k - l ) + l . How sharp is this new bound? In this case, we should try to find examples satisfying the cquality for every p and every k . For p = 2 it suffices to consider the semidihedral groups SDzn for n 2 4, which satisfy k = n - 2 and v(SD2n) = 2n - 5. However, we do not know of any examples for odd primes, apart from the groups M p , for which k is only 1.

+

Problem 2.1. Find a sharp lower bound for the number of conjugacy classes of non-normal subgroups in a finite p-group G (p odd), an t e r m s of p and k . 3. Conjugacy classes of non-normal subgroups and the

order of central quotients Let now C be defined by the condition IG : Z(G)I = p e . It is well-known that k is bounded in terms of C, more precisely k 5 l ( e - 1 ) / 2 (see Theorem 9.12 of [ 1 2 ] ) .Thus a bound for v(G) in terms of implies a bound in terms of k . Is it possible to bound v ( G ) from below with a function of p and We give a positive answer t o this question for odd primes, which we describe in this section.

e

e?

89

In fact, we bound v(G) in terms of an invariant X which is usually greater than l. If z is an element of Z(G) of maximum possible order, we define X by means of the equality IG/(z)I = p x . Thus X 2 e, and the equality holds if and only Z(G) is cyclic. We write X(G) instead of X if we need t o emphasize the dependence of X on G. If m = pa is a power of p , let us put l(m)= a. With this notation, X(G) = 1(IGI) - l(expZ(G)). Our motivation for considering this invariant comes from the bound v(G) 2 X - 1, proved by La Haye and Rhemtulla in Theorem 1 of [13]. Is it possible t o introduce the prime p in this bound? Guided by the result obtained in terms of k and p in the last section, we could expect the inequality v(G) p(X - 2) 1 to hold. The next example shows this is not true.

+

>

Example 3.1. Let p be an odd prime and let n 2 3. Then the group G = ( a ,6 I upn = b P 2 = 1, [a,61 = upn-') satisfies that X = 4 and v(G) = p 2.

+

However, we can obtain a bound which is only slightly worse. Theorem 3.1. Let G be a non-abelian finite p-group, where p i s a n odd prime. T h e n v(G) 2 p(X - 3) 2.

+

Note that we have to leave apart abelian pgroups in the preceding theorem, since v(G) = 0 in that case, but X can be arbitrarily large. The proof of this theorem uses induction on the order of the group, and as in Theorem 2.1, we deal separately with the cases when Z(G) is cyclic or is not. In the former case, it is necessary t o consider the same three subcases as before: G regular, G of maximal class, and the rest of groups. Again, most of the effort has to be devoted t o regular pgroups. There is, however, an important difference which makes the proof of the bound in terms of X harder than in the case of k . If N is a normal subgroup of G of order p , then k(G) 5 k(G/N) 1, and due t o this fact, it is enough t o prove that v ( G ) 2 v ( G / N )+ p for the induction to work. On the other hand, all we can say about X is the following.

+

Theorem 3.2. Let G be a finite p-group and let N be a normal subgroup of G of order p . T h e n X(G) 5 X(G/N) 2.

+

Proof. Since X(G) - X(G/N) = 1(IGI) - Z(expZ(G)) - 1(IG/NI) =

2(exp Z ( G / N ) )- I(exp Z(G))

+ Z(expZ(G/N))

+ 1,

(6)

90

it suffices to see that expZ(G/N) 5 p expZ(G). For this purpose, choose an element g N of maximum order in Z(G/N). Since [g,G] 5 N and N is a central subgroup of order p , it follows that [gp,GI = [ g ,G]P = 1 and g p E Z(G). Consequently exp Z(G/N)

=o(gN)I o(g) I P exp Z(G),

as desired.

(7) 0

This makes it necessary to find 2 p classes, and not only p , of non-normal subgroups not containing N . However, if X(G) = X(G/N) 2 then the next fact gives a hint as to where to look for non-normal subgroups.

+

Theorem 3.3. Let G be a finite p-group and let N be a normal subgroup of G of order p such that X(G) = X(G/N) + 2. If g N i s a n element of m a x i m u m order in Z(G/N), t h e n ( g ) i s a non-normal subgroup which does n o t contain N . Proof. By (6), we have expZ(G/N) = p expZ(G). Then ( 7 ) implies that o ( g N ) = o ( g ) , and consequently ( 9 ) n N = 1. Since g N E Z(G/N), if the subgroup ( 9 ) is normal in G, then [g,G] 5 (g) n N = 1. Hence g E Z(G), and o ( g N ) 5 o(g) 5 exp Z(G) < exp Z(G/N), which is a contradiction. 0

In some cases, we choose an alternative approach to overcome this difficulty: instead of taking the quotient by a normal subgroup of order p , we factor out the subgroup T of all central elements of order p , i.e. T = Rl(Z(G)). Suppose IT1 = p 3 . Since expT = p , the proof of Theorem 3.2 applies almost verbatim, with the only exception that 1(G)- l(G/T) = s and not l. Consequently we have X(G) 5 X(G/T) s 1. However, our next result shows that the situation is better than expected.

+ +

Theorem 3.4. Let G be a finite p-group, and suppose that IR1(Z(G))l = p s . T h e n X(G) 5 X(G/Rl(Z(G))) s.

+

Proof. Put T = Rl(Z(G)), and let g T be an element of maximum order in Z(G/T). As in the proof of Theorem 3.2, we have gp E Z(G). Suppose by way of contradiction that X(G) = X(G/Rl(Z(G))) s 1. Following again the proof of Theorem 3 . 2 , it follows that o ( g T ) = o ( g ) and o ( g ) = pexpZ(G) > p . But the first equality implies that ( 9 ) n T = 1, and the second that 1 # ( g p ) I Z(G). This is clearly a contradiction. 0

+ +

This theorem has the following version for regular pgroups.

91

Theorem 3.5. Let G be a regularp-group, and suppose that IRl(G)I T h e n X(G) 5 X(G/RI(G)) s.

+

=ps.

If we apply the induction hypothesis to G/RI(G), we need to find s p classes of non-normal subgroups which do not contain R1(G). Now, since IRl(G)I 2 p 2 unless G is cyclic (recall that p is odd), all non-normal cyclic subgroups of G satisfy this property. Hence it suffices to see that u*(G) 2 s p in this case. These are the main ingredients to be used in the proof of Theorem 3.1. As already happened in Section 2, most of the work has t o be done in the regular pgroup case, and bases play a key role in our arguments. We finish this section with some open problems.

Problem 3.1. Does a bound similar to the one in Theorem 3.1 also hold for finite 2-groups? The next question tries to sharp Theorem 2.1 in the vein of what we have done in this section: instead of working with e, we have used the usually bigger value A. By symmetry, we consider the following dual version for k . Let G be a finite p-group, and define n = r;(G) by means of the following relation: K = l(G) - I(exp GIG'). In other words, if K is a normal subgroup of G such that the quotient G / K is cyclic of maximum possible order, then IKI = p". Note that n 2 k 1 unless G is cyclic.

+

Problem 3.2. Does the bound u(G) 2 p ( -~ 3 ) not abelian?

+ 2 hold i f p > 2 and G i s

Finally, we formulate a question of a different type about the behaviour of u(G).

Problem 3.3. L a Haye and Rhemtulla have proved that u(G) 5 1 or u(G) 2 p for a finite p-group ( L e m m a 3 of [15']). A r e there a n y other gaps in the possible values that u(G) can take? Computational evidence hints that the answer might be negative. Acknowledgments The authors are supported by the Spanish Ministry of Science and Education, grant MTM2004-04665, partly with FEDER funds, and by the Univcrsity of the Basque Country, grant UPVO5/99.

92

References 1. N. Blackburn, Generalizations of certain elementary theorems on pgroups, Proc. London Math. SOC.(3) 11,1-22 (1961). 2. N. Blackburn, Finite groups in which the nonnormal subgroups have nontrivial intersection, J . Algebra 3,30-37 (1966). 3. N. Blackburn, Note on a paper of Berkovich, J . Algebra 24,323-334 (1973). 4. R. Brandl, Groups with few non-normal subgroups, Cornm. Algebra 23,no. 6, 2091-2098 (1995). 5. M. Egizii di Marco, Norm and conjugacy classes of normalizers in finite pgroups, PhD thesis, Universith dell’Aquila, (L’Aquila, Italy, 2005), pp. viiS57. 6. M. Egizii di Marco, G.A. FernBndez-Alcober, L. Legarreta, Counting conjugacy classes of subgroups in finite pgroups, 11, this volume. 7. G.A. Fernhdez-Alcober, An introduction to finite pgroups: regular groups and groups of maximal class, Mat. Contemp. 20, 155-226 (2001). 8. G.A. Ferntindez-Alcober, L. Legarreta, Conjugacy classes of non-normal subgroups in finite nilpotent groups, preprint. 9. G.A. Fernhdez-Alcober, L. Legarreta, Conjugacy classes of non-normal subgroups and the order of central quotients in a finite pgroup, preprint. 10. N. Gavoli, L. Legarreta, C. Sica and M. Tota, On the number of conjugacy classes of normalisers in a finite pgroup, Bull. Austral. Math. SOC.73, 219230 (2006). 11. P. Hall, A contribution to the theory of groups of prime-power order, Proc. London Math. SOC.(2) 36,29-95 (1933). 12. B. Huppert, N. Blackburn, Finite Groups, 11 (Springer-Verlag, Berlin, 1982). 13. R. La Haye, A. Rhemtulla, Groups with a bounded number of conjugacy classes of non-normal subgroups, J. Algebra 214,41-63 (1999). 14. Li Shirong, The number of conjugacy classes of non-normal cyclic subgroups in nilpotent groups of odd order, J . Group Theory 1, 165-171 (1998). 15. J. Poland, A. Rhemtulla, The number of conjugacy classes of non-normal subgroups in nilpotent groups, Cornm. Algebra 24,no. 10, 3237-3245 (1996).

ON CERTAIN CLASSES OF GENERALIZED PERIODIC GROUPS G. ENDIMIONI C.M . I- CJniversitC de Provence 99, rue F. Joliot-Curie, F-13453 Marseille Cedex 13 *E-mail:[email protected] Let m be a positive integer. We say t h a t a group G belongs t o the class Ern if G contains m elements u1,. . . ,urn such t h a t ( U ; ~ X U I ) . . . (uG1xurn)= 1 for all x E G. Among other things, we show that any group in & is of finite exponent d F d i n g 9 and is nilpotent of class 5 3. Also we show that for any group G E B4,the subgroups Gs and G4 are respectively abelian and nilpotent of class 5 2.

Keywords: generalized periodic group, generalized periodic element.

1. Introduction

This paper is a developed version of the last part of my talk given at the Conference and devoted t o polynomial functions of groups. By definition, a polynomial function of a group G is a function of the form x H wOx'lw1.. . wn-lxEnwn, with W O ,. . . ,w, in G and where €1, . . . ,en are integers. Very naturally, this notion of polynomial function has led to a generalization of the notion of periodicity: according t o Fuchs [4],an element x of a group G is said t o be a generalized periodic element if there exist a positive integer m and elements u1,. . . ,u, E G such that (UT1XU1).

.

*

(.,'xum)

=

1.

Such elements are also called G-periodic elements by certain authors. Notice that if G is locally nilpotent, one sees without difficulty [8] that generalized periodicity is equivalent to periodicity. When all elements of G are generalized periodic elements, we say that G is a generalized periodic group. In the opposite direction, when the unity is the unique generalized periodic element, G is called an R*-group [5]. The notion of generalized periodicity occurs in particular in the theory

93

94

of orderable groups (namely groups admitting a total order which is left and right invariant). Indeed, it is easy to see that any orderable group is an P - g r o u p . Whether the converse is true remained an open question for several years (see for instance Problem 2 in [5] or Problem 1.47, p. 109 in [9]),but answered by Bludov [2] in the negative. Notice however that the converse holds in the class of groups which are both abelian-by-nilpotent and nilpotent-by-abelian [3,11]. Another question relating to generalized periodic groups is due t o GorEakov [6], who gave an example of a (nontrivial) generalized periodic group which is torsion-free. This group is not finitely generated, so GorEakov posed the following question (see Problem 3.1, p. 121 in [9]):does there exist a (nontrivial) generalized periodic group which is torsion-free and finitely generated? GoryuSkin [7] gave an example, proving that the answer is positive. We present here another example, due to Curkin (see p. 121 in [9]): one can show that the group G = ( a ,b I a-'b2a

= b-2,

b-'a2 b = C 2 )

is torsion-free, and it is generalized periodic since we have

z"a-1z2a)(b-1z2b)(b-la-122ab)

=1

for all z E G. Note that in Curkin's group, the relation above is independant of z. That leads us to distinguish certain groups among the generalized periodic groups. If m is a positive integer, we denote by t3, the class of groups G satisfying the following property: there exist m fixed elements u1, ... , u , E G s u c h t h a t h

( u ~ ~ z u .1. (u;'zu,) ).

= 1 (for all z E

G).

h

In particular, Curkin's group belongs to Bs. If we denote by t3, the class of groups of exponent dividing m, we have t3, 5 B,, and trivially, B1 = B1. Also it is very easy to prove that ,132 = &. On the other hand, we shall see 5 Bg. In the class in Section 3 that t33 # & but we shall prove that B4,a group can contain elements of infinite order. For example, the infinite since az2az2 = 1 for dihedral group D , = ( a ,b I a2 = b2 = 1) belongs to all z E D,. Nevertheless, in contrast t o &, we shall see in Section 4 that & does not contain a non trivial torsion-free group. Also we shall prove that for any group G E &, the subgroup G4 is nilpotent of class 5 2 and the subgroup Gs is abelian (where G" denotes the subgroup generated by all powers z", for all z E G). When m 2 5, it appears difficult to obtain significant information about an arbitrary group G E B, and it seems reasonable t o investigate such a A

A

&

A

A

A

95

question in a restricted class of groups. For instance, it is easy to prove that a finitely generated metabelian group in 8, is polycyclic. That leads to pose the question: is a finitely generated soluble group in 8, polycyclic ? In fact, during the Conference, C. Casolo has provided me an example showing the answer is negative: there exists in & a finitely generated soluble group of derived length 3 which is not polycyclic. With his kind agreement, I shall detail this example in the last section of this paper. A

A

2. Notation and preliminary lemmas

As usual, in a group G, the commutator of two elements x , y is defined by [X , Y 1 - 2-l y -l x y ; instead of [ [ x , y ] , z ] we , shall write [ x , y , z ] .If H is a subgroup of G, we denote by C G ( H )its centralizer in G. Recall that G" and that B, is the class of groups is the subgroup generated by satisfying the identity xm = 1. Before looking into the classes 83 and &, we establish three lemmas.

+

Lemma 2.1. In a group G , consider m 1 elements wo, . . . , wm such that W O . . . w , = 1. Then there exist m elements u1,. . . ,urn E G such that woxw1. . . wm-lxw, = ( u T 1 x u l ) . . (u;'xum) for all x E G. Proof. For each k E { I , .. . , m } ,put x E G, we have WOXWl

. . . W,-1XWm

Uk

= W k W k + l . . . w,.

Then, for any

= WO . . . w m ( u ; l x u l ) . . . (u;lxum)

and the result follows.

0

Lemma 2.2. Let m > 2 be an integer. Then, a group G belongs to grn if and only af there exist m - 2 elements a l , . . . ,am-2 E G such that ( a T 1 x u l ) . . (a&y2xam-2)x2= 1 for all LL: E G. Proof. It suffices to prove the part 'only if' since the converse is obvious. Thus suppose that G belongs to Em. Therefore, there exist elements u1,. . . , ~ ~ - u ,1 ,E G such that (UTIXU1).

. . (u,'lxum-l)(u,lxu,)

=1

for all 2 E G. It follows

(,m,,l,,l,kl).

. . ( ~ m ~ , 1 1 X ~ m - 1-1u )mX

= 1.

By replacing x by u1u;'x in this equality, we obtain a relation of the form x u l x u ~ x ...xu,-1x = 1, and so

.

2 1 1 X U ~ X . . XVrn-1X2

=

1.

96

By Lemma 2.1, since ~1x2122.. . m,-1 a l l .. . , am-2 E G such that W12U22..

. ZUm-1

=

= 1 when x = 1, there exist elements

(aylxa1).. . (a,',xa,-2)

for all x E G and the result follows.

0

Lemma 2.3. Let G be a group. Suppose that there exist integers r , s and an element a E G such that U - ~ X ' U = x-' for all x E G. Then the quotient group G/CG(G') belongs to B,+t,.

Proof. Let y be an element of G. Replacing x by y-lxy in the relation U - ~ Z ' U = x-', we obtain ~-~y-~x'ya= Y - ~ x - ' ~But . y-lx-'y = y-la-'xray, thus ~ - ~ y - ~ z=~y-la-lxTay ya and it follows [y-l, a-l]-lx'[y-l,

a-11

= 2'.

In other words, the commutator [y/-l,a-l] belongs to CG(G'). Hence, in the quotient G/CG(G'), the image of a lies in the centre of G/CG(G'). Therefore, using the relation U - ~ Z ' U = x-', we obtain

"'x

G

1 modulo CG(G'),

as desired. 3. The class

17

&

We summarize in the next theorem the main properties of groups belonging t o &. Recall that a group G is said t o be n-abelian if (xy)" = xnyn for all elements x and y in G. A characterization of these groups is due t o Alperin

PI. Theorem 3.1. Let G be a group in &. Then:

(a) (ia) (iii) (iv)

G is 3-abelian.

The subgroup G3 is included an the centre of G. G is of finite exponent dividing 9. G is nilpotent of class 5 3.

Proof. (i) By Lemma 2.2, there exists an element a E G such that a-1xax2 = 1 for all x E G. It follows that the map x H x-2 = uP1xu is an automorphism of G and so ( ~ - ' y - ' ) - ~ = ( ~ - ' ) - ~ ( y - ' ) - ~for all x,y E G. In other terms, we have the relation (yx)' = x2y2,which is equivalent to the required relation, namely ( ~ y =) x3y3. ~

97

(ii) It suffices to apply Lemma 2.3, with r = 1 and s = 2. (iii) Clearly, the centre of G (and so G3 too) satisfies the identity x3 = 1. Since G3 and GIG3 are both of exponent dividing 3, the result follows. (iv) By a result of Levi [lo], every 3-abelian group is nilpotent of class a t most 3 (in a more general context, that is also a consequence of Alperin’s results [l]). Since G is 3-abelian1 we obtain the desired result. 0

A finitely generated periodic nilpotent group being finite, we obtain immediately:

..

Corollary 3.1. A finitely generated group in ,133 i s a f i n i t e 3-group of exp o n e n t dividing 9. This result is sharp since there exist in f33 finite groups of exponent 9 exactly. For example, denote by Cg = . . . ,8} the additive group of integers modulo 9 and by cp the automorphism of Cg defined by cp(x) = 42. Clearly, the semidirect product G = Cg x (cp) is of order 27 and of exponent 9. Furthcrmore, it is easy to verify that a-1zax2 = 1 for all z E G, where a = (o,cp). Thus G belongs t o &.

{o,

4. The class

g4

As we said in Section 1, the fact that the infinite dihedral group belongs to 6 4 shows that a group in & can contain elements of infinite order. Also note that this group is not nilpotent. However, the next theorem shows that a group in & is an extension of a nilpotent group by a group of finite exponent.

Theorem 4.1. L e t G be a group in &. T h e n : (i) T h e subgroup G4 i s nilpotent of class 5 2; (ii) T h e subgroup Gs i s abelian. By a well-know result of Sanov [12], a finitely generated group of exponent dividing 4 is finite, and so nilpotent. Therefore, the first part of Theorem 4.1 shows that any finitely generated group G E is soluble, and even polycyclic. It follows that G/G8 is finite. Hence, thanks to the second part of Theorem 4.1, we obtain:

&

Corollary 4.1. A finitely generated group in @4 i s abelian-by-finite. The proof of Theorem 4.1 will be the direct consequence of a series of technical lemmas. In the following, we consider a group G in f34. Therefore,

98

by Lemma 2.2, there exist elements a, b E G such that

(a-'za)(b-'zb)z2 = 1 for all z E G. One can already note the relations

b-'ab

=

and -

u ' b a = bK3 obtained by replacing z by a and by b in (1). Lemma 4.1. For a n y x E G , iue have

'

'

(a - 1 z a )(b- zb) = (b- zb)( a - l z a ) . Proof. It follows from (1) the relation (a-'za)(b-'zb)

= z-'.

Substituting 2-l for z gives

(a-lz-'a)(b-'z-'b)

= x2

By multiplying (4) and ( 5 ) , we obtain the desired result.

0

In particular, Lemma 4.1 shows that we may permute the roles of a and b in equation (1).

Lemma 4.2. I f ( ( G ) denotes the centre of G , we have: 6)[a,bI E ((GI; (ii) a2 E ((G) and b2 E ((G). Proof. (i) First we substitute u-'zu for z in (1).We obtain ( a - 2 ~ (b-la-lzab) ~2) (a-'z2a)

=

1.

(6)

Now we map each side of (1)with the inner automorphism associated to a. That gives

(a-2za2)(u-lb-lzba)( c ~ - ~ z ' a=) 1. Relations (6) and (7) imply b-'a-'zab baza-'b-', we obtain

(7)

= a-lb-lzba; replacing then z by

[a,b ] - ' ~ [ ab], = Z,

(8)

99

as required. (ii) Substituting za-l for z in ( l ) ,we have

a-lsb-lza-lbza-lsn-l= 1

(9)

z ( b - l z b ) [ b ,a](a-lza)(a-2za-2) = 1.

(10)

whence

But we have shown that the commutator [b,a] = [u,b]-' belongs t o the centre of G; furthermore, a-'za and b-lsb commute by Lemma 4.1. Thus relation (10) can be written in the form z ( a - l z a ) (b-lzb) ( [ b ,a ] ~ - ~ z a = - ~I.)

We have (a-'za)(b-'zb) write

= z-'

by (1). Furthermore, using

[b,a]a-2 = (b-la-1b)a-l

(11)

(a), we

can

= a2.

Hence relation (11) becomes z-1a2zu-2 = 1.

(12)

It follows that a2 belongs to C(G). Since we can permute the roles of a and b, also b2 E C(G). 0 We may then deduce from Lemma 4.2 that torsion-free group.

& does not contain a non trivial

Corollary 4.2. We have: (i) as = b8 = 1; (ii) If G E & is torsion-free, then G = (1). Proof. (i) We can write 1 = [U2 ,b] = [a,b][a, b, U ][a,b] = [a, b]2

since a2 and [u,b] belong to C(G) by Lemma 4.2. We deduce from (2) that a-' = u-lb-lab = [ a , b ] ,whence a-' = [a,bl2 = 1, and so as = 1. Permuting the roles of a and b, we obtain also bs = 1. (ii) If G is torsion-free, then a = b = 1, and relation (1) becomes z4 = 1. That implies z = 1 and the proof is complete. 0 Lemma 4.3. W e have ab-' E Cc(G2).

100

Proof. There follows from (1) the relation

(a-lza)(b-lzb) = x-2

(13)

for all z E G. By Lemma 4.1, this relation can also be written in the form

(b-lzb)(a-lza) = z-2

(14)

Mapping both sides of relation (13) (respectively, (14)) by the inner automorphism associated to a (respectively, b) and using the fact that a2 and b2 belong to ((G), we obtain

z(a-lb-lzba)

= a-1z-2a

(15)

z(b-la-lzab) = b-1z-2b.

(16)

and

But

b-la-lzab

=

[a,b]-la-lb-lzba[a, b] = a-lb-lzba

since [a,b] belongs t o the centre of G by Lemma 4.2. Thus the lefthand sides of relations (15) and (16) are equal. It follows that .-1

whence z2ab-'

= ab-'z2.

x - 2 a = b-1x-2b,

This completes the proof of Lemma 4.3.

(17) 0

Lemma 4.4. A group G in & satisfies the identities: (2) [ 2 4 , y 2 , 2 ] = 1; (ii) iz8, y41 = 1.

r

Proof. (i) Let denote the quotient group G/Cc(G2). We write Z for the image in r of an element 5 E G. In particular, we have in I? the relation

-

(z-lq(;-lZ;)

= Z-2

(18)

for all z E G. But Z = b by Lemma 4.3, and so the relation becomes

-_ 1-2a z a = r 2

(19)

Hence we can apply Lemma 2.3. It follows that r/G(r2)belongs t o ,134. That means that for all z, y E GIwe have [E4, p] = ?, or, in other words, [z4,y2]E CG(G~). Thus [z4,y2, z2] = 1 for all z, y, z E GI as required. (ii) Replacing z by x2 in (l),we obtain

(a-122a)(b-12%) = z-4

(20)

101

whence a-l

x 4u = x - 4

(21)

since u-lx2u = bP1x2bby (17). Applying once again Lemma 2.3, we have G/Cc(G4)E f?,, whence the identity [x8,y4] = 1. 0

&.

Proof of Theorem 4.1. (i) Let G be a group in It follows from Lemma 4.4 that G satisfies the identity [x4,g4, z4] = 1. Clearly, that implies that G4 is nilpotent of class a t most 2. (ii) The second identity of Lemma 4.4 implies the identity [x8, y 8 ] = 1, and so G8 is abelian. 0 5. Example Let G be a finitely generated soluble group in Em, of derived length T . Clearly, the quotient GIG‘ is finite, of exponent dividing m. Thus G is polycyclic when r 5 2. The aim of this last section is to show that G need not be polycyclic when T = 3 . More precisely, we present an example of a finitely generated soluble group of derived length 3 which is in &, and which is not polycyclic. This example is due to C. Casolo, and I wish to thank him for permitting to include it in this paper. First consider the integral group ring ZD, of the infinite dihedral group D , = ( a ,b I u2 = b2 = 1).Let A denote the direct product A = D , x C2, where C2 is the multiplicative group {-1,l). For any (u, E) in A, we define = EUW for all w in ZD,. Then a map v ( ~ , . ): ZD, + Z D , by (P(~,~)(w) (P(%,~)is an automorphism of the additive group of the ring ZD,. In order to avoid some ambiguity, we shall distinguish the ring ZD, and its additive ~ )(P(%,~)is group, which will be denoted by B. Clearly, the map : ( u ,H a homomorphism from A to Aut(B). Finally, we define G as the semidirect product G = B X $ A. In other words, any element of G can be written in the form ( w ,u,E ) (w E B , u E D,, e = f l ) ,with an operation defined by

(w,u,E)(W’,u’,E‘) = (w CeuW’,uu’,EE’). Put a = (0,a, l), 7 = (0,1, -l), and consider an arbitrary element x = (w, u,E) in G. An easy calculation shows that the element y = ax2az2is of the form (w’, 1,l).A similar calculation gives y ~ y q= 1. That implies the relation

102

Thus G belongs t o &. Moreover, it is casy t o sce t h a t G is finitely generated and soluble of derived length 3. Finally, G is not polycyclic since B is not finitely generated.

References 1. J. L. Alperin, A classification of n-abelian groups, Canad. J. Math. 21 (1969) 1238-1244. 2. V. V. Bludov, A n example of a n unorderable group with strictly isolated identity, Algebra and Logic 11 (1972) 341-349. 3. V. V. Bludov and E. S. Lapshina, O n ordering groups with a nilpotent commutant, Siberian Math. J. 44 (2003) 405-410. 4. L. F’uchs, Partially ordered algebraic system, (Pergamon Press, London, 1963). 5. L. Fuchs, O n orderable groups, in Proc. Internat. Conf. Theory of Groups, Austral. Nat. Univ. Camberra, 1965, pp. 89-98, Gordon and Breach Science Publishers (1967). 6. Yu. M. GorEakov, A n example of a G-periodic torsion-free group (Russian), Algebra i Logika 6 (1967) 5-7. 7. A. P. GoryuSkin, A n example of a finitely generated G-periodic torsion-free group, Siberian Math. J. 14 (1973) 146-148. 8. H. A. Hollister, O n a condition of Ohnishi, Proc. Amer. Math. SOC.19 (1968) 1337-1340. 9. E. I. Khukhro and V. D. Mazurov (eds), The Kourovka notebook: unsolved problems in group theory, 15th edition, Inst. of Math., Russian Academy of Sciences, Novosibirsk, (2002). 10. F. Levi, Notes on group theory. I, 11, J. Indian Math. SOC.8 (1944) 1-9. 11. P. Longobardi, M. Maj and A. Rhemtulla, O n solvable R*-groups, J. Group Theory 6 (2003) 499-503. 12. I.N. Sanov, Solution of Burnside’s problem for exponent 4 (Russian), Leningrad State Univ. Annals, Mat. Ser. 10 (1940) 166-170.

NIELSEN EQUIVALENCE CLASSES AND STABILITY GRAPHS OF FINITELY GENERATED GROUPS M. J. EVANS Department of Mathematics, The University of Alabama, Tuscaloosa, A L 35487-0350, U.S.A. E-mail: [email protected] Let G be a d-generator group. In general, for each integer n 2 d, there exist . . , gn) of elements of G such that (91,.. . , gn) = G. many ordered n-tuples (91,. Probably the most natural way to classify such n-tuples is by collecting them into the Nielsen equivalence classes of G on n generators. This note contains a brief and informal introduction to the theory of Nielsen equivalence classes and also introduces the stability graph r ( G ) of G, a graph that records relationships between Nielsen equivalence classes of G on n generators for varying n . It is a fairly faithful transcription of the author's talk at Ischia Group Theory Conference, 2006 although some additional information is included. Keywords: Nielsen equivalence, T-systems.

1. Introduction Let G be a finitely generated group and let d number of generators of G. For each n 2 d let

=

d ( G ) denote the minimal

V ( G , n )= { ( g l , g 2 , ' . . , g n ) l g i E G , ( g 1 1 g 2 , . . . , g n ) = G I , the set of generating n-vectors of G. Given v = ( g l , g 2 , . . . ,gn) E V ( G , n )we can obtain a (not necessarily different) vector in V ( G , n )by applying one of the following elementary

Nielsen transformations: (1) permuting the entries of v, -1 (2) inverting an entry of v, v -+ ( g l , . . . , g i , . . . , g n ) , (3) multiplying one entry of v on the right or left by another entry, v (91,. . . ,g i g j , . . . ,gn) where i # j .

103

+

104

We say that u, v E V ( G ,n ) are Nielsen equivalent if u can be changed into v by a finite sequence of these moves. If u, v are Nielsen equivalent we write u + v or u-v. (Clearly is an equivalence relation and so V ( G ,n) is the disjoint union of the Nielsen equivalence classes of G o n n generators.) The reader has probably noticed a similarity between the elementary Nielsen transformations and the standard set of generators of Aut(Fn), the automorphism group of the free group of rank n. This similarity is not a coincidence; indeed it is possible to view much of the material below as part of the study of a certain action of Aut(F,) on V ( G ,n). In this work we will not discuss this action, which is well-documented (see for instance [1,2]). As an illustration, we note that if (a,b,c) E V ( G , 3 ) ,for some 3generator group GI then ( a ,b, c)-(a, b, b-'a2c) since we may transform (a,b,c) -+ (a,b,ac) -+ (a,b,a2c) 4 (a,b-l,a2c) -+ (a,b-l,b-la2c) + ( a ,b, b-'a2c). Indeed, it is easy to see that if w E ( a ,b) we may transform ( a ,b, c) -+ ( a ,b, wc) or (a, b, c) -+ ( a ,b, c w ) or even ( a ,b, c) -+ ( a ,b, W - ~ C Z U ) . Of course similar remarks apply to elements of V ( G ,n ) for other n. There is another type of transformation we can apply to v = (91,. . . , g n ) E V ( G ,n ) that produces an element of V ( G ,n ) ,namely:

-

(4)

v

-+

( a ( g i ) ,. . . ,a ( g n ) ) for some a E Aut(G).

If u E V ( G ,n) can be transformed into v by a finite sequence of transformations of the types given in (1)-(4), we write u N T v. Clearly y-is again an equivalence relation; we call the equivalence classes of V ( G , n ) under N T the T,-systems of G. Note that each Tn-system of G is a union of Nielsen equivalence classes of G on n generators. The study of T-systems was initiated by B.H. Neumann and H. Neumann [3]. This note is a brief, informal introduction to Nielsen equivalence classes and related matters. We have made no attempt to be encyclopaedic. Our purpose is t o convey something of the flavour of the subject and t o this end we have taken a few liberties. Most notably, in the interest of clarity, we have not always given the strongest known result if its statement is distractingly technical. Accordingly, the interested reader is encouraged to consult the original sources. 2. The case n = d ( G ) Let G be a 2-generator group and suppose that ( a ,b ) , (c, d ) E V ( G ,2) are Nielsen equivalent. Since [b,a] = [a,b]-', [a, b-'1 = ([a, bib-')-', [a,ba] =

105

[a,b]” and [a,ab] = [a,b], it is easy to see that [a,b] is conjugate in G to [c,d] or [c,4 - l . If follows that if ( w , ~ )(y,z) , E V(G,2) and there is no a E Aut(G) with ~ ( [ w , z ] = ) [y,z]*’, then (w,z) and ( y , n ) do not belong to the same T2-system of G.

Example 2.1. Let G = As, the alternating group of degree 5, and consider V = ((123),(12345)),~= ( ( 1 2 3 ) , ( 1 3 5 2 4 ) ) E V(&,2). NOW [ ( 1 2 3 ) , ( 1 2 3 4 5 ) ]= (142) whereas [ ( 1 2 3 ) , ( 1 3 5 2 4 ) ]= ( 1 4 5 3 2 ) . Since these two commutators have different orders we deduce that A5 has at least two T2-systems. (In fact A5 has exactly two T2-systems [4].) Using the above method, which is usually attributed to G. Higman, many 2-generator groups can be shown t o have more than one T2-system. Let us record a little of what is known in this area. Let t2(G) denote the number of T2-systems of the group G and let k ( S m )denote the number of conjugacy classes of S,, the symmetric group of degree m. The following theorem is taken from [5].

Theorem 2.1.

pi)

For all integers m 2 2 we have: t2(S2mf3), t2(S2m+4), t2(A2m+5), t2(A2m+4) 2 k ( S m ) * (ii) Let p be a prime and m a positive integer such that pm # 2,3,4,5,7,9. Then tz(PSL(2,p”)) is at least (p” - 2)/m if this is an integer and at least ((p” - 2)/m] 1 otherwise. (iii) For all positive integers m the Suzuki group Sz(22mf1) is such that t 2 ( ~ 2 ( 2 ~ ~ +is’ )at ) least [22m/(2m I)] 1.

+

+ +

It is reasonable to conjecture that tz(G) > 1 for each finite simple group G. For infinite groups things are even worse: A.M. Brunner [6] has shown that ( a ,bl a-lb2a = b3) has infinitely many T2-systems. An interesting result of R. Guralnick and I. Pak [7] shows that, in a very strong sense, Higman’s method cannot be extended to groups G with d(G) > 2. To explain their result we first let w be a non-trivial word in the free group F, = (51,. . . , zn() and let qW : V(G, n) -+ G be the associated map $w(gl , . . . ,g,) = w(g1, . . . , gn). We say that w is invariant on Tsystems if the set of Aut(G)-conjugates of {+;l(g1,. . . ,gn)} is constant on all generating n-vectors in a T.,-system. Thus, Higman’s method depends

106

on the fact that for n = 2 the commutator [ X I , xz] is invariant on T-systems. The theorem of Guralnick and Pak is the following: Theorem 2.2. For eve? non-trivial word w E Fk where k 2 3, there exists a finite group G such that w is not invariant on T-systems. Nevertheless] we still have the following result which is due to M.J. Dunwoody [8]. Theorem 2.3. To each pair of integers n > 1 and N > 1 and every prime p , there exists a finite p-group G which is nilpotent of class 2 and has at

least N T,-systems. We shall see in Theorem 3.1 below that necessarily d ( G ) = n in this theorem. In general we expect a group G to have ‘many’ Nielsen equivalence classes on d(G) generators although there are notable exceptions, some of which are discussed below. 3. A first look at the case n

> d(G)

In the sequel we deal almost exclusively with results about V ( G ,n) where n > d(G).Many of our results are of the following form: if n - d(G) is ‘sufficently large’ then G has a unique Nielsen equivalence class on n generators. The meaning of ‘sufficiently large’ will depend on the nature of the group G under consideration. Such results are ‘stability theorems’ in an obvious sense. Our prototype is the following theorem of M.J. Dunwoody [9] which stands in sharp contast t o Theorem 2.3. Theorem 3.1. Let G be a finite soluble group and let n > d(G). Then G has a unique Nielsen equivalence class on n generators. Dunwoody stated his result for T,-systems but his argument establishes the stronger theorem we have given here. Similar remarks apply t o a number of the results given below. Theorem 3.2. Let G be a group, let n > d(G) and suppose that

(i) G = PSL(2,p) where p is an odd prime, (Gilman 1111, or (ii) G = PSL(2,am) where m 2 2, (Evans [lo]), or (iii) G is u Suzuki group S Z ( ~ ~ ” where + ~ ) m 2 1, (Evans [lo]), or (iv) G = 11111, the Muthieu group of order 11.10.9.8, (Nettles [ l l ] ) ,or

107

(v) G = P S L ( 2 , 3 P ) where p is a prime, (McCullough and Wanderley [12lj, O r (vi) G is finitely generated and nilpotent, (Evans [2]). Then all elements of V ( G ,n) are Nielsen equivalent. Parts (i)-(v) of this theorem are proved by brute force. The proofs all show how to transform u into v for arbitrary u, v E V ( G ,n ) and depend on very detailed knowledge of the subgroup lattice of the group G involved. It seems unlikely that such methods can be used t o tackle many more families of finite simple groups. In contrast, part (vi) depends only on the corresponding (easy) result for abelian groups and that G’ 5 F r a t ( G ) for nilpotent groups G. This is an appropriate point at which to mention a conjecture of J. Wiegold that has had an enormous influence on the author’s approach to our subject. Conjecture 3.1 (Wiegold’s conjecture). Let G be afinite simple group and let n > 2. Then G has a unique T,-system.

We will not discuss this conjecture in detail although some of the results below are obviously relevant. Instead we direct the reader to a paper of I. Pak [13] which records most of what was known a t the time of its publication. Pak’s paper contains many interesting results on a variety of topics and is warmly reconmiended to the reader. 4. Stability graphs, train tracks and unsupported nodes

Let G be a d-generator group, n 2 d , v = (gI,g2,.. . ,gn) E V ( G ,n ) and r E N. We define v * r = (gI,g2,. . . , g n , 1,.. . , 1 ) E V ( G , n r ) where, of course, we intend that there are r 1s at the end of this vector. We proceed to define a levelled graph r ( G ) that we call the stability graph of G. (The meaning of the word ‘levelled’ in this context will soon become clear.) Let [v]denote the Nielsen equivalence class containing v. The nodes (or vertices) of r ( G ) at level n are the Nielsen equivalence classes [v] of G on n generators. A node [v] at level n is joined by an edge to the node [v * 11 at level n 1. All edges of r ( G ) arise in this way. Consider the hypothetical stability graph of a group G given in Fig. 1. We intend that r ( G ) has a unique node at level n for each n 2 d 2 and say that r ( G ) is a bamboo stalk from level d 2. Let [a], [b],[c],[d]be the four nodes at level d , from left to right. Here, of couse, a, b, c, d E V ( G ,d ) are pairwise inequivalent. Now [a]and [b]are

+

+

+

+

108

level d

+2

level d

+1

level d

Fig. 1. A hypothetical stability graph r ( G )

+

-

joined t o the same node at level d 1 and so a * 1 b * 1. On the other hand c * 1 + d * 1 although c * 2 N d * 2. We say that r ( G ) has a train track of length one starting a t level d. More generally a stability graph r ( G ) has a train track of length r starting a t level n if there exist u, v E V ( G ,n) such that u * r * v * r . Let us show that the length of a train track in r ( G ) is less than d(G).

Theorem 4.1. Let u , v E V ( G , n ) where G is a finitely generated group and n 2 d = d(G). Then u*d and v * d are Nielsen equivalent. Consequently r ( G ) is a tree. Proof. Let G = ( X I ,..., ~ d ) u , = ( g l , . . . , g n ) and v = (hl, . . . , h,). Then u * d = (91, . . . ,g,,l, . . . , 1) 4 (91,...,gnrxl,..., z d ) + (1,.. . ,1,2 1 , . . . , Z d ) + ( h l , .. * , h,, 2 1 , . . . ,Zd) -+ ( h l ,. . . , h,, 1 , . . . ,1) = v * d as required. The final statement is now obvious. 0 Before investigating further properties of the graph in Fig. 1 we discuss a very interesting class of two-generator groups G such that r ( G ) has no train tracks. The following definition is due to J.L. Brenner and J. Wiegold ~41.

109

Definition 4.1. A group G has spread n if given non-trivial there exists z E G such that (gi,x) = G for i = 1,.. . , n.

91,

. . . ,gn

E

G

Theorem 4.2. Let G # 1 be a group of spread 2 and let u , v E V ( G , n ) for some n 2 d(G) = 2. T h e n u * 1 and v * 1 are Nielsen equivalent. Consequently r ( G ) has n o train tracks. Proof. Let u = ( g l , . . . , g n ) and v = ( h l ,. . . ,h,). On permuting the entries of u and v if necessary we may assume that that g1 # 1 # hl and so there exists z E G such that ( g 1 , z ) = (h1,z)= G. We may now transform u * 1 = (91,. . . ,gn, 1 ) --+ (91,. . . , g n , x ) -+ (g1,1,. . 1 , ~+) (g11h1, 1 , . . . 1 1 , x ) + ( h l ,1,.. .111z) ( h ~ , . .r .h n , ~ ) 4 ( h l , .. . , hn, 1) = v * 1. The proof is complete. 0 -+

A well-known consequence of the classification of finite simple groups is that such groups are 2-generator. Although it is unknown whether all finite simple groups have spread two, Guralnick and Shalev [15] have shown the following: Theorem 4.3.

(i) All finite simple groups have spread one. (ii) Almost all finite simple groups have spread two. Therefore, for almost all finite simple groups GIthe stability graph r ( G ) has no train tracks. Returning to Fig. 1 , let [v]be the node at level d+l that is not connected to a node a t level d. Thus v is not equivalent to a vector of the form u * 1 where u E V ( G ,d). We say that [v] is an unsupported node at level d 1. To explain the significance of such nodes we need a few definitions. Let F, denote the absolutely free group of rank n, freely generated by 2 1 , . . . , x,. We say that y E F, is a primitive element of F, if there exist yz, . . . , yn E F, such that (y, y2, . . . , y,) = F,. Similarly, if 1 < r < n, we say that z 1 , . . . , z, are associated primitive elements of F, if there exist z,+l,.. . , z, such that 21.. . , z , generate F,. Clearly each v = (91,. . . ,gn) E V ( G ,n) where n 2 d(G) determines an epimorphism 8, : F, -+ G such that &(xi) = gi for i = 1 , . . . ,n. It is not difficult to show (see, for instance, [2, Lemma 1.51) that if v E V ( G ,n) then v --+ u * r for some u E V ( G ,n - r ) if and only if Ice.(&) contains r associated primitive elements z 1 , . . . , z, € F,. Thus the existence of an unsupported node at level n > d(G)in the stability graph of a group G is equivalent to tlie existence of R a F , with F,IR G such that R contains no primitive element

+

=

110

of F,. These comments have two consequences that will be very useful in the next section. Proposition 4.1. Let F,/R = G and suppose that n > d ( G ) .

( i ) If R does not contain a primitive element of F, then r ( G ) has a n unsupported node a t level n. (ii) If R contains k associated primitive elements of F, but not k -k 1 associated primitive elements and k 1 5 n - d ( G ) , then F(G) has a train track of length k starting at level n - k .

+

According to [16, p. 911, F. Wauldhausen asked a question that is equivalent t o ‘If R a F, is such that d ( F , / R ) < n, does R necessarily contain a primitive element of F,?’ In the next section we discuss examples of G.A. Noskov [17] that show the answer to this question is ‘no’. Clearly Noskov’s examples provide us with stability graphs that contain unsupported nodes. To conclude this section we note that Dunwoody’s theorem shows that the graph in Fig. 2 is the stability graph r ( G ) of each finite soluble group; of course the number of nodes at level d varies from group to group. It is also the stability graph of each of the groups that appear in Theorem 3.2 and, conceivably, of each finite simple group or even each finite group.

Fig. 2.

r ( G ) for finite soluble groups G

111

5. The existence of unsupported nodes and train tracks In this section we sketch a proof of the following theorem which establishes the existence of unsupported nodes and train tracks. Part (i) follows from a result of G.A. Noskov [17] (Theorem 5.2 below) and part (ii) from a more recent result of the author [18] (Theorem 5.3 below). Our purpose is to illustrate some techniques that can be used to prove such theorems without getting bogged down in too much bookkeeping. Theorem 5.1.

(i) There exists a finitely generated metabelian group G1 such that I'(G1) has an unsupported node at some level n > d(G1). (ii) There exists a finitely generated metabelian group G2 such that I'(G2) has a train track of length (at least) one. In light of Proposition 4.1 it suffices t o find metabelian groups GI, G2 that have presentations: (i) F,/R1 2 G1 where n > d(G1) and R1 contains G2 such that R2 contains k but no primitive elements of F,; (ii) F,/R2 not k 1 associated primitive elements of F, for some k with k 1I n - d(G2) . Our strategy is to solve analogous module-theoretic problems and then build groups around the relevant modules. We write d A ( M ) for the minimal number of generators of a A-module M .

+

+

Problem 5.1. Find a ring A1 that has a module M1 with the following property: there exists a presentation L1 ~ - A;" 1 ---H M1 where m > dAl ( M I ) such that L1 does not contain a basis element o f A T . Problem 5.2. Find a ring A2 that has a module M2 with the following AT --h M2 such that L2 contains property: there exists a presentation L2 k elements f r o m a basis of AT but not k 1 elements from a basis of AT for some k with k 1 I m - da, ( M 2 ) . ~f

+

+

We use an idea that goes back to Kaplansky and was extended and used t o great effect by R. Swan [21]. This work of Swan is a good source for information about (i)-(iii) below, as is Husemoller [20]. Let Sn = { ( a l , . . . ,a,+l) E IWn+llaf . . . = l}, the unit n-sphere standardly embedded in IWn+l, and C = C(S") the ring of all continuous real-valued functions on S". Let XI,.. . x,+1 E C be the coordinate functions, so that z i ( ( a 1 , .. . , a,+l)) = ai for all ( a l l . .. ,a,+l) E S", and define a C-module homomorphism 0, : Cn+' -+ C by en((fi7.. . , f n + i ) ) =

+

+

112

+ +

+ +

fix1 . . . fn+lZn+1. NOWO n ( ( X 1 , . . . , ~ , + 1 ) )= X: .. . = 1 and it follows that 0, is an epimorphism. Consequently Cn+l = K, @ Q where K, = ker(6,) and Q E C. We now need some facts that, as far as I know, require topological proofs. They are algebraic versions of well-known results in the theory of tangent bundles on spheres:

(i) K , has a free direct summand of rank T if and only if there exists a set of T orthonormal tangent vector fields on S". (ii) If n # 1 , 3 , 7 then K , is not free. (iii) If n is even then K , is directly indecomposable. The ring C is uncountable and so too big for our purposes. However A = z[zl,... ,xn+1, (21 2)-l ,.. . , (x,+1 2)-l] is a subring of C that is an image of Z[y;', . , . , y:i1], the integral group ring of a free abelian group of rank n + 1. (Here we intend that A be generated as a ring by the

+

+

-

listed functions together with the constant integer-valued functions which A be we identify with Z.)With this notation in place let 0; : An+' the A-homomrphism given by O ; ( ( f l l . . . , f n + l ) ) = fl.1 .. . f n + ~ x n + l . Note that, as above, 0; is an epimorphism and set P, = ker(0;). Now An+' = P, @ Awhere A 2 A. Moreover, P, @A C P K, as a C-module and we deduce that P, is not free if n # 1 , 3 , 7 and is indecomposable if n is even. In particular, on setting n = 4, we obtain the presentation P 4 L--) A5 A where the epimorphism is 0;. Now if P 4 contains an element from a basis of A5 it maps onto A and we have P 4 E A @ W for some submodule W . This is impossible since P 4 is indecomposable and not free. Consequently P4 contains no element from a basis of A5 and, on setting A1 = A = Mi, m = 5 and L1 = P 4 , we have solved Problem 5.1. We next note that there exist four orthonormal tangent vector fields on S15,namely

+ +

-

(52,-51,54,-23,56,-55i~8i-2712101-2912121

- 2 1 1 1 ~ 1 4 1 - ~ 1 3 , ~ 1 6-215) 1

(-z4,

-z3, z2, z11-z8, -z71 z67 z51-z12, - 5 1 1 1 z l 0 1 z91- z l 6 1 -5151 z 1 4 1 z 1 3 )

(-287

-577-267

- 5 5 , Z 4 , 5 3 , 5 2 i Z 1 1 -2161-2151

(-2161 --215~ - 2 1 4 ,

-214,

-~13i21212111z101~9)

- 2 1 3 1 -5121 -2111 -2107 - ~ 9 i 2 8 i ~ ? i ~ 6 , 2 5 i 2 4 i ~ 3 i ~ 2 l 2 1 )

which we call F1, F 2 , F3, F 4 respectively. We let A = Z[zl,. . . , q 6 , (21 2)-',.. . , ( 2 1 6 2)-l], and argue as A and P i 5 = ker(0T5) @ A where A above to deduce that A16 = is not free. Let B = (FT FT FT a 16 x 4 matrix, and consider the module homomorphism y : P I 5 4 A4 given by y ( p ) = p B for all p E P i 5 1 'multiplication by B'. Now F, E PIS for i = 1,.. , , 4 since F, is tangent to

+

FF),

+

113 S15. Moreover F I B = ( F ~ . F ~ , F ~ . F ~ , F ~ . F ~ = , F(1,0,0,0) 1 . F 4 ) = el and similarly FiB = ei for i = 2,3,4. It follows that the above map y is onto . Thus Pi5 = S @ Bwhere B L&' A4 and S = k e r ( y ) . Thus A 1 6 = S @ ( B @ A ) where B @ A E A5. Now A20 = A4 @ A16 = A4 @ S @ ( B @ A ) where A4 @ S E Pi5 is not free. Let D = A4 @ S and note that, by construction, D contains four elements from a basis of A20. Suppose that t l , . . . , t 2 0 is a basis of A20 such that t l , . . . , t 5 E D and let 7r denote the natural projection from A20 = D @ ( B @ A ) onto B @ A , so that D = ker(7r). Now B @ A 2 A5 is 5-generator and so there exist ul,.. . , u5 in ( t 6 , . . . ,t 2 o ) A , the submodule generated by t 6 , . . . ,t 2 0 , such that ~ ( u l ).,. .,7r(u5) generate B @ A . Set si = ti ui for i = 1,.. . , 5 and note that s1,.. . , s5, t 6 , . . . ,t 2 o is a basis of A20. Now there exist u g , . . . ,uz0 E (sl,. . . , s5)A such that 7r(ui) = 7 r ( t i ) for i = 6 , . . . ,20. Let 7-i = ti - ui for i = 6 , . . . , 20 and note that s l , . . . , s5,7-6,.. . ,7-20 is a basis of A20. Since 7 - 6 , . . . ,r20 E ker(7r) = D we have shown that D contains 15 elements from a basis of A20. Let R = (7-6,. . . ,~ 2 0 ) Aand observe that A Z 0 / RE A5 E R2'/D. Since R 5 D and A is Noetherian, we deduce that R = D and so D is free, a contradiction. Thus D does not contain five eleA20 A5 ments from a basis of A20. Note that we have a presentation D in which the epimorphism is 7r. On setting A2 = A, A42 = A;, m = 20 and k = 4 we have solved Problem 5.2. It remains to construct the groups G1 and G2. We shall only construct G1 and sketch a proof of the fact that it has the desired property; the argument for G2 is similar. We have seen that there exists a ring A that is an image of Z(F5/Fi), the integral group ring of the free abelian group of rank 5, and has a short exact sequence of modules P -+ A5 -+ A such that P contains no element from a basis of A5. Let 0 denote the epimorphism in the above sequence, let b l , , . . , b5 be a free basis of H = F5/Fs/, the free metabelian group of rank 5 and let A l l . . . , A5 be a basis of A5. We view A as a ZH-module in the natural way and set G1 = A x H , W = A5 x H . Evidently ( X I , . . . ,X 5 , b l , . , b 5 ) E V(W,10) and v = (e(Al), . . . ,e(A5), bl, . . . ,b5) E V ( G 110). , To complete the proof of Theorem 5.l(i) it suffices to show that v is not Nielsen equivalent to a vector of the form u * 1. Suppose, for a contradiction, that there exists a finite sequence of elementary Nielsen transformations that changes v into a vector ( 1 , 9 2 , . . . ,910) and let u = ( p l , w 2 , . . . ,w10)be the result of applying the same transformations t o ( A l l . . . , As, b l , . . . , b5). Note that p1 E ke7-(0) = P . Using Theorem 6.8 below, it is easy to show that u -+ (PI,. . . , p5, p g b l , . . . , p10b5) for some

+

-

--$)

..

114

p 2 , . . . ,p10 E

R5. There is a natural map from K = ( p s b l , . . . ,p10b5) onto H that has kernel K n A5. However, since K is a 5-generator metabelian group (and so an image of H ) , it is easy to see that the Hopficity of H implies that this kernel is trivial. It follows that p 1 , . . . , p5 generate A and, since A is a commutative ring, we deduce that p 1 , . . . ,p5 is a basis of h5. Since pl E P we have obtained the desired contradiction. Essentially the same techniques we have just used t o prove Theorem 5.1 can be used to establish the following results which are, respectively, those of Noskov and the author referred to above. Theorem 5.2 (17). Let n = 2 m where m 2 3 and m i s odd. T h e n there exists a n ( m 1)-generator metabelian group G that has a n n-generator presentation F,/R such that R contains n o primitive elements of F,.

+

Theorem 5.3 (18). Let k 2 4 be a n integer and let n = 2'--1. T h e n there exists a n ( n k 2)-generator metabelian group G and v E V ( G ,2 n 2 ) such that v * k i s n o t Naelsen equivalent t o a n y vector of the f o r m u*( k + 1) where u E V ( G ,2 n 1).

+

+ +

+

6. A second look at the case n

> d(G)

Recall that a finitely generated module M over a ring R is said t o be stably free if there exists n E N such that M @ R" is free. The main property of the ring of functions C = C ( S n ) that we used in Section 5 is that there exists a non-free stably free C-module if n # 1 , 3 , 7 . In a similar vein, if there exist non-free stably free modules M over the integral group ring of a group H , it is sometimes possible to show that certain extensions of abelian groups by H have interesting stability graphs. To illustate this point we begin with a famous result of R. Swan [21].

Theorem 6.1. Let Q 3 2 denote the generalized quaternion group of order 32. T h e n there exists a ZQ32-module M such that M$z'Q32 z . & 3 2 $ z Q 3 2 but M i s n o t free.

A great deal is now known about stably free modules over integral group rings of finite groups. We refer the reader to Swan [22]for details. With Theorem 6.1 in hand we can prove the following (see [2, Theorem 2.91): Theorem 6.2. There exist a n abelian-by-finite polycyclic group G such that r ( G ) has a n unsupported node at level d ( G ) 1.

+

115

On the other hand, a fairly easy induction on h(G),the Hirsch length of G , as in the proof of [23, Theorem El, establishes:

+

Theorem 6.3. Let G be a soluble minimax group and let n > d(G) 1. Then G has a unique Nielsen equivalence class on n generators. Thus r ( G ) is a bamboo stalk from level d 2.

+

Combining Theorems 6.2 and 6.3 we find that the stabililty graph of a finitely generated soluble minimax is roughly of the form given in Fig. 3. The author hopes to address whether such a graph can have a train track of length one starting at level d in a future work.

i

Fig. 3.

r ( G ) for finitely generated soluble minimax groups G

By exploiting previously-known results on the stable range of Noetherian rings we were able to show the following [2, Theorem 4.91. Theorem 6.4. If G is nilpotent-by-polycyclic then r ( G ) is a bamboo stalk from level d(G) h(G/Fitt(G))+2, i.e. G has a unique Nielsen equivalence class on. n gmerators f o r all n 2 d(G)+ h(G/Fitt(G))+ 2 .

+

The first example of a torsion-free group G such that ZG has a nonfree stably free module is due to M.J. Dunwoody [24] who showed that the integral group ring of G = ( a ,b I a2 = b 3 ) , the fundamental group of the trefoil knot, has a non-free module M with M @ ZG ZG @ ZG. Using this we proved [25]:

116

Theorem 6.5. For each integer n 2 4, there exists an ( n - 1)generator group H that has an n-generator, 2-relator presentation H = ( 5 1 , . . . , 5 , I r l , ra) such that the normal closure of ( r 1 , r z ) in F, does not contain a primitive element of F,. Since the publication of Dunwoody’s paper many more stably free nonfree modules over group rings have appeared in the literature. We draw the reader’s attention t o a wonderful paper of Artamonov [26] which contains many such examples. In particular the following powerful result is a consequence of his work.

Theorem 6.6. Let G be the union of a countable subnormal series 1 = GoaGlaGaa.. . in which the factors Gi+l/Gi are all free abelian. Suppose that the left ideals P of Z G that satisfy P @ Z G FZ Z G e Z G fall into finitely many isomorphism classes. Then G is a free abelian group. With the aid of Artamonov’s modules, we proved a number of results about relatively free groups [23]. We mention one here.

Theorem 6.7. Let 6 j r , d denote the free soluble group of rank r and derived length d. Then, for each r 2 3 and d 2 1 there exists an epimorphism 0 : 6 ; r + l , d + z t e r , d + 2 such that ker(0) is not the normal closure of a single element in 6 ; r + l , d + 2 . It follows easily that r(G;r,d+Z)has an unsupported node ar level r + l for all r and d considered in the theorem. As one might expect, free metabelian groups behave quite differently. Using a famous result of Bachmuth and Mochizuki [28] together with one of Gupta, Gupta and Noskov [29] it is not difficult to prove the next theorem [18, Theorem 31.

Theorem 6.8. Let Mk denote the free metabelian group of rank k and let n 2 k 2 4. Then all elements of V ( M k , n ) are Nielsen equivalent. We remark that M2 has more than one Nielsen equivalence class on 3 generators [30];indeed I’(M2) has an unsupported node a t level 3. Theorem 6.8 should be compared with a classical result of Nielsen, (see, for instance, [31, Chapter 3]), which asserts that Fd, the (absolutely) free group of rank d has a unique Nielsen equivalence class on n generators for all n 2 d. The constructions above involved stably free modules: we conclude this section with a different sort of example [2].

Theorem 6.9. There exists a cyclic ZF2-module M with the following property. For every N 2 1 there exists a module epimorphism $JN :

117

( Z F Z ) -+ ~ M such that ( Z F Z ) cannot ~ be generated by N elements one of which is contained in ker($N). Using this module M we proved a result [2, Theorem 3.71 that implies the following theorem.

Theorem 6.10. There exists a 3-generator abelian-by-free group G such that r ( G ) has unsupported nodes at all levels. Thus F(G) is not eventually a bamboo stalk. 7. Concluding remarks The idea behind our stability graphs r ( G )comes from M.N. Dyer and A.J. Sieradski [32] who use similar graphs in their study of homotopy types of 2-dimensional CW-complexes that have a fixed fundamental group G. Finally, let us pose some problems; to the best of the author’s knowledge they are all open. We begin with a variant of Wiegold’s conjecture.

Problem 7.1. Let G be a finite group and let n > d ( G ) . Does G have a unique Nielsen equivalence class o n n generators? Theorem 6.10 suggests the following.

Problem 7.2. Let G be a finitely generated soluble group. Does there exist N 2 d ( G ) such that G has a unique Nielsen equivalence class on n generators for each n 2 N ? Equivalently, is r ( G ) eventually a bamboo stalk? Contrasting Theorem 3.2(vi) with Theorem 6.2 suggests:

Problem 7.3. Let G be a supersoluble group. Does G have a unique Nielsen equivalence class on d ( G ) 1 generators?

+

Problem 7.4. Does there exist a finitely generated soluble group G and a ZG-module M such that M @ ( Z G ) 22 ( Z G ) 3 but M @ Z G 2 Z G 2 ? Problem 7.5. The Grushko-Neumann theorem about free products A * B can be interpreted in the following way: let v E V ( A* B , c d ) where c = d(A) and d = d ( B ) . Then v -+ (a1,..., ac,bl , . . . ,b d ) for some ( a l , . . . , a c ) E V ( A , c ) and ( b l , . . . ,bd) E V ( B , d ) . If v + ( a ; , . . .,aL,b;,.. . , b & ) where ( a ; , . .. ,a:) E V ( A , c ) and ( b i , . . . , b&) E V ( B , d ) , then does it follow that ( a l , . . . , a c ) -+ (a;, . . . ,a;) and ( b l , . . . , bd) -+ (b;, . . . , b&)?

+

118

Problem 7.6. Suppose that the presentation G = ( X I , .. . , x, I R ) is such that no primitive element of F, = ( X I , ... , x, 1 ) is a relator. Does it follow that the presentation G * C, = ( X I , ... , x,, x,+1 I R ) is such that no primitive element of F,+1 = ( X I , ... ,x,,x,+~ I ) is a relator? Problem 7.7. Let R be a proper characteristic subgroup of F,. Must it be the case that d(F,/R) = n ? Acknowledgments This paper is dedicated t o Akbar Rhemtulla on t h e occasion of his retirement. T h e author would likc to thank t h e conference sponsors for financial support a n d t h e conference organizers for making Ischia Group Theory Conference 2006 a n extremely enjoyable experience.

References 1. R. Gilman, ‘Finite quotients of the automorphism group of a free group’, Canad. J . Math. 29 No.3 (1977) 541-551. 2. M. J. Evans, ‘Presentations of groups involving more generators than are necessary’, Proc. London Math. SOC67 (3) (1993) 106-126. 3. B.H. Neumann and H. Neumann, ‘Zwei Klassen charakteristischer Untergruppen und ihre Faktorgruppen’, Math. Nachr. 4 (1951) 106-125. 4. D. Stork, ‘Structure and application of Schreier coset graphs’, Comm. Pure and A p p l . Math. 24 (1971) 707-805. 5. M.J. Evans, ‘Problems concerning generating sets for groups’, Ph.D. Thesis, University of Wales, (1985). 6. A.M. Brunner, ‘Transitivity systems of certain one-relator groups’, Proc. Conf. Canberra 1973. (Lecture notes in Math., Vol 372, 131-140, Springer 1974). 7. R. Guralnick and I. Pak, ‘On a question of B.H. Neumann’, Proc. Amer. Math. SOC131 NO.7 (2003) 2021-2025. 8. M.J. Dunwoody, ‘On T-systems of groups’, J . Austral. Math. SOC.3 (1963) 172-1 79. 9. M.J. Dunwoody, ‘Nielsen transformations’, in ‘Computational problems in abstract algebra, proceedings of a conference in Oxford 1967’ Pergamon, Oxford (1970) 45-46. 10. M.J. Evans, ‘T-systems of certain finite simple groups’, Math. Proc. Camb. Phil. SOC.113 (1993) 9-22. 11. E.A. Nettles, IT,-systems for the Mathieu group Mil’, Ph.D Thesis, University of Alabama, (1999). 12. D. McCullough and M. Wanderley, ‘Free actions on handlebodies’, J . Pure Appl. Algebra 181, no. 1, (2003) 85-104. 13. I. Pak, ‘What do we know about the product replacement algorithm?’, in:

119

14.

15. 16. 17. 18.

19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

29.

30. 31. 32.

Groups and Computation III (Columbus, OH, 1999), 301-347, Ohio State Univ. Res. Inst. Publ., 8, deGruyter, Berlin, 2001. J.L. Brenner and J. Wiegold, ‘Two-generator groups I,, Michigan Math. J . 22 (1975) 53-64. R.M. Guralnick and A. Shalev, ‘On the spread of finite simple groups’, Combinatorica, 23 (2003) No. 1, 73-87. R.C. Lyndon and P.E. Schupp, Combinatorzal group theory, (Springer, Berlin, 1977). G.A. Noskov, ‘Primitive elements in a free group’, Math. Zametki 30 (4) (1981) 497-500. M.J. Evans, ‘Presentations of groups involving more generators than are necessary, 11’, to appear in Combinatorial Group Theory, Number Theory and Discrete Groups edited by B. Fine, A. Gaglione and D. Spellman. Contemporary Mathematics, A.M.S. R.G. Swan, ‘Vector bundles and projective modules’, Trans. Amer. Math. SOC.105 (1962) 264-277. D. Husemoller, Fibre bundles, (second edition), (Springer-Verlag, New York, 1975). R.G. Swan, ‘Projective modules over group rings and maximal orders’, A n n . of Math.(2) 76 (1962) 55-61. R.G. Swan, ‘Projective modules over binary polyhedral groups’, J. Reine Angew. Math. 340 (1983) 66-171. M.J. Evans, ‘Relation modules of infinite groups’, Bull. London Math. SOC, 31 (1999) 154-163. M.J. Dunwoody, ‘Relation modules’, Bull. London Math. SOC4 (1972) 151155. M.J. Evans, ‘Primitive elements in free groups’, Proc. Amer. Math. SOC.106 (1989) 313-316. V.A. Artamonov, ‘Projective nonfree modules over group rings of soluble groups’, Math. USSR Sbornik 44, No. 2, (1983) 207-217. M.J. Evans, ‘Epimorphisms between the free groups in a variety of groups’, J . Algebra 220 (1999) 492-511. S. Bachmuth and H.Y. Mochizuki, ‘ A u t ( F ) + A u t ( F / F ” ) is surjective for free groups F of rank 2 4’, Trans. Amer. Math. SOC.292, No 1. (1985) 81-101. C.K. Gupta, N.D. Gupta and G.A. Noskov, ‘Some applications of ArtamonovQuillen-Suslin theorems to metabelian inner rank and primitivity’, Canad. J . Math. 46, No. 2, (1994) 298-307. M.J. Evans, ‘Presentations of the free metabelian group of rank 2’, Canad. Math. Bull. 37, No. 4. (1994) 468-472. W. Magnus, A. Karass and D. Solitar, Combinatorial group theory (Interscience, New York, 1966). M.N. Dyer and A.J. Sieradski, ‘Trees of homotopy types of two-dimensional CW-complexes’, Comm. Math. Helv.48 (1973) 31-44.

COUNTING CONJUGACY CLASSES OF SUBGROUPS IN FINITE p-GROUPS, I1 MANUEL EGIZII DI M A R C 0 Dipartimento d i Matematica Pura ed Applicata, Universitci dell 'Aquila, V i a Vetoio, 67010 Coppito (L'Aquila), Italy E-mail:[email protected]

GUSTAVO A. FERNANDEZ-ALCOBER Matematika Saila, Euskal Herriko Unibertsitatea, 48080 Bilbao, Spain E-mail: [email protected]

LEIRE LEGARRETA Matematika Saila, Euskal Herriko Unibertsitatea, 48080 Bilbao, Spain E-mail: leire. [email protected] We study the number w ( G ) of conjugacy classes of normalizer subgroups of a finite pgroup G, and prove the bound w ( G ) 2 ( p - l)[(c- 1)/2] 1 for odd p , where c is the nilpotency class of G. We also propose some open problems.

+

Keywords: Finite p-groups; Conjugacy classes.

In this paper, which is the continuation of [3] (also in this volume), we consider the number w ( G ) of conjugacy classes of normalizer subgroups in a finite p-group G. As mentioned in the introduction of [3], we have the following lower bound for w ( G ) .

Theorem 1. Let G be a finite p-group, where p i s a n odd prime, and let c be the nilpotency class of G . T h e n ,

This result was proved by Egizii di Marco in his Ph.D. thesis [l].The purpose of this paper is to give a short proof of Theorem 1. In order to do

120

121

this, we still borrow the main ideas of [l],but we introduce some particular changes that reduce significantly the length of the proof. First of all, observe that it suffices to see that

for any capable group, and then simply apply this last bound to G / Z ( G ) . Recall that a group G is called capable if there exists a group H such that G G! H / Z ( H ) . An important property of capable pgroups is the following: if Z # 1 is a cyclic subgroup of G and S is the set of all elements of G having a non-trivial power in 2,then S generates a proper subgroup of G (see [5], page 137). We need a couple of lemmas.

Lemma 1. Let G be a finite p-group and suppose that G i s the u n i o n of s proper subgroups H I , . , . ,H,. T h e n s 2 p 1. Furthermore,

+

+

(1) If s = p 1 t h e n all the subgroups H i are maximal in G . (2) If s 5 2p and p is odd, t h e n at least p of the subgroups Hi are maximal

in G.

Proof. Let the order of G be p". By induction on T , it easily follows that the cardinality of the union of r maximal subgroups of G is a t most rp"-' ( r - l ) ~ " -As ~ .a consequence, G cannot be the union of less than p 1 maximal subgroups. Hence s 2 p + 1. Let us now prove (i) and (ii). We may assume that H I , . . . , H , are maximal subgroups of G, and that H,+1,. . . ,H, are not maximal, with r < s. Then

+

5 rpn--l

-

( r - 1)pnp2

+ +

+ (s

-

r)pnW2 = p n p 2 ( r p

+ s - 2r + I).

+

Thus p 2 < r ( p - 2) s 1. Now if s = p 1 and r < s, this inequality does not hold. The same happens if p is odd, s 5 2 p and r < p . 0 Note that we have not stated the previous lemma in its best possible version (it is possible to assure that there are p maximal subgroups for bigger values of s), but only in the form we are going to use it.

Lemma 2. Let G be a non-abelian finite capable p-group, and suppose that [%,GIi s cyclic for some element x E &(G) \ Z(G). T h e n w ( G ) 2 w(G/Z(G))+ p - 1.

122

Proof. Since &(G) lies in. the preimage of every normalizer in G / Z ( G ) ,it suffices to find p - 1 non-conjugate normalizers in G that do not contain Z2(G). Let Z be the subgroup of order p of [z, GI. Since G is capable, the set S = {g E G I 2 5 (9)) is contained in a maximal subgroup M of G. Let C be a maximal subgroup of G containing C G ( ~ ) . Let g E G \ ( M U C). If z E NG((g)), then 1 # [z,g] E (9) and Z I (g), which is a contradiction. Thus z # NG((g)), and N, = NG((g))@(G)is a proper normal subgroup of G. Now since

G=

u c u (UgEG\(MUC) Ng)1

it follows from Lemma 1that a t least p - 1 of the subgroups Ng are different. Then the corresponding normalizers NG((9)) are non-conjugate and do not contain 2 2 (G). 0 With the help of these lemmas, we can determine the following relation between w(G) and w(G/Z;!(G)), for a capable pgroup G. Theorem 2. Let G be a non-abelian finite capable p-group, where p i s a n odd prime. T h e n w(G) 2 w(G/Zz(G)) p - 1.

+

Proof. If & ( G ) / Z ( G ) is cyclic then it suffices t o apply Lemma 2 to G / Z ( G ) .Therefore we assume that & ( G ) / Z ( G ) is not cyclic. Let T be a subgroup of &(G) such that T/Z(G) is elementary abelian of order p 2 , and let T I , . .. ,Tp+lbe the maximal subgroups of T containing Z(G). We consider two types of proper subgroups of G: on the one hand, the centralizers C G ( T ~ and ) , on the other hand, the subgroups Ng = NG((g))@(G)for all g E G such that T $ NG((g)). Let us see that G is the union of all these subgroups. For this purpose, we choose g E G such that T 5 NG((g)) and we prove that g E C G ( T ~for ) some i. Observe that T/CT(g) 5 NT((g))/CT(g) can be embedded in Aut(g). Now, since p is odd, the group Aut(g) has a cyclic Sylow psubgroup. Thus T/CT(g) is cyclic. Since expT/CT(g) 5 expT/Z(G) = p , it follows that IT : CT(g)I 5 p and Ti 5 CT(g) for some i. Consequently g E &(Ti),and we are done. Now we may assume that there are less than p different subgroups of the form N,, since otherwise w(G) 2 w ( G / Z ( G ) )+ p follows as in the proof of Lemma 2. Thus G is the union of at most 2 p proper subgroups. By Lemma 1, a t least p of these subgroups are maximal in G I and therefore one of the subgroups C G ( T ~must ) be maximal in G. Hence we can choose an element y such that G = (y, Cc(Ti)).Let .7: be an element in the difference Ti\Z(G).

123

Then [ x ,GI = ( [ x y, ] ) is cyclic. As a consequence, we can apply Lemma 2 and we are done. 0 Now everything is ready to prove Theorem 1. Proof of Theorem 1. As already mentioned, it suffices t o prove (1) under the assumption that G is capable. We argue by induction on c. The result is clear if c = 1. If c = 2, then G/Zz(G)is the trivial group and w(G/Zz(G))= 1. It follows from Theorem 2 that w(G) 2 p , as desired. Finally, if c 2 3 then w(G/Zz(G))has class c-2 and the result follows from Theorem 2 and the induction hypothesis. 0

Since every conjugacy class of proper normalizers in a group comes from a conjugacy class of non-normal subgroups, it follows that the relation v ( G ) 2 w(G) - 1 holds generally. (Recall from [3] that v(G) stands for the number of conjugacy classes of non-normal subgroups of G.) This makes it tempting, whenever we have a bound for v ( G ) ,to try t o prove that the same bound also holds for w ( G ) if we add 1. In particular, we can consider the following two bounds given in [3]:on the one hand,

v(G) 2 p ( c - 2 )

+1

for all non-Hamiltonian groups, and on the other hand,

v(G) 2 P ( ~ C - 1) + 1, where /G’I = p k , with the exception of Hamiltonian groups and generalized quaternion groups. Thus we suggest the following two problems as an attempt at sharpening Theorem 1.

Problem 1. Does the bound w(G) 2 p ( c - 2 ) + 2 hold f o r all finite p-groups with the exception of Hamiltonian groups? In order to prove this bound for odd primes, we need a twofold improvement of Theorem 1. On the one hand, we have to substitute p for p - 1. A close look a t the proof of Theorem 2 shows that it is enough to sharpen Lemma 2 in order to get this result. On the other hand, we have t o eliminate somehow the 2 in the denominator of the bound. Again, it follows from the proof of Theorem 2 that the case to be studied is when Z z ( G ) / Z ( G )is cyclic and Z ( G ) is not cyclic.

+

Problem 2. Does the bound w ( G ) 2 p ( k - 1) 2 hold f o r all finite p groups with the exception of Hamiltonian groups and generalized quaternion groups?

124 This bound has been proved true in Theorem 4 of [4] for pgroups of maximal class which are not generalized quaternion groups. In the proof, it is important the fact that we have a good knowledge of the structure of groups of maximal class (see for example [2]). However, the answer is negative in general, as the following example shows.

Example 1. Let p be an odd prime and let m 2 1. Then the group

G = ( a ,b I up*"

= bPm =

1, [a,b]

= up")

is a group of class 2 for which k = m, but w ( G ) = m+ 1. More precisely, the subgroups ( u p " , b ) , for 0 5 i 5 m, form a complete system of representatives of the conjugacy classes of normalizer subgroups of G.

It is more likely that the following less ambitious question has a positive answer (recall from [3] that v(G) 2 k holds if G is not Hamiltonian). Problem 3. Let G be a finite p-group. Is it true that w ( G ) 2 k the exception of Hamiltonian groups?

+ 1, with

Acknowledgments The last two authors are supported by the Spanish Ministry of Science and Education, grant MTM2004-04665, partly with FEDER funds, and by the University of the Basque Country, grant UPVO5/99.

References 1. M. Egizii di Marco, Norm and conjugacy classes of normalizers in finite pgroups, PhD thesis, Universita dell'Aquila, (L'Aquila, Italy, 2005), pp. viif57. 2. G.A. FernBndez-Alcober, An introduction to finite pgroups: regular groups and groups of maximal class, Mat. Contemp. 20, 155-226 (2001). 3. G.A. Fernhdez-Alcober, L. Legarreta, Counting conjugacy classes of subgroups in finite pgroups, I, this volume. 4. N. Gavoli, L. Legarreta, C. Sica and M. Tota, On the number of conjugacy classes of normalisers in a finite p-group, Bull. Austral. Math. SOC.73,219230 (2006). 5 . P. Hall, The classification of prime-power groups, J . Reine Angew. Math. 182,130-141 (1940).

POSITIVE LAWS O N LARGE SETS OF GENERATORS AND ON WORD VALUES GUSTAVO A. FERNANDEZ-ALCOBER Matematika Saila, Euskal Herriko Unibertsitatea, 48080 Bilbao, Spain E-mail: [email protected] PAVEL SHUMYATSKY Department of Mathematics, University of Brasilia 70910 Brasilia DF, Brazil E-mail: [email protected] We survey several questions related to positive laws in groups, with special emphasis on our latest work [5], where we address the following problem: if all values of a word w in a group G satisfy a positive law, does it follow that the whole verbal subgroup w(G) also satisfies a positive law? Keywords: Positive laws; Residually-p groups; Word values; Verbal subgroups.

1. Introduction to positive laws

Let X be an alphabet of symbols and let a and p be two different group words on X . We say that a subset T of a group G satisfies the law a = p if, for every replacement of the symbols of X by elements of T , the value of a is the same as the value of p. A positive law is a law in which both a and ,B are positive words, i.e. words which do not involve any inverses of elements of X. The degree of the positive law is then the maximum of the lengths of a and p. The simplest positive laws, requiring only one symbol, are the exponent laws x e = 1, where e 2 1 is a positive integer. Thus groups of finite exponent, and in particular finite groups, satisfy a positive law. On the other hand, abelian groups satisfy the positive law xy = yx. Now, if G/Z(G)satisfies a law a = p, it is clear that G satisfies ap = ,Ba. By induction on the class, it follows that every nilpotent group of class c satisfies a positive law in two symbols of degree 2=, which is known as the Malcev law M c ( x ,y).

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For example, Mz(x, y) is the law xyyx xyyxyxxy

f yxxy

and Ms(x, y) is given by

= yxxyxyyx.

As a consequence, all nilpotent-by-(finite exponent) groups satisfy a positive law of the form Mc(xe,ye). If a group G satisfies a positive law a: EE ,B, then it also satisfies a positive law in only two symbols. To see this, give new names x l , . . . , x, to the symbols used in a and P, and then perform the change xi H yxi. Clearly, we get different words in two variables if a and ,D are different. Note however that this argument does not work for a subset T of G which is not a subgroup, since yxi need not be in T for all x , y E T . In any case, it is always possible to assume that both words a and ,B have the same length. Indeed, if the length of a is greater than the length of P, say the difference is m, then T satisfies the exponent law zm E 1, and hence also the positive law a = 4zm. 2. Positive laws in residually finite groups and generalizations

As mentioned in the previous section, nilpotent-by-(finite exponent) groups satisfy a positive law. The converse is false in general, as shown by Olshanskii and Storozhev [13], who provide a 2-generated counterexample. However, it is true for most of the classes of groups that arise in the literature. The first result in this direction is due to Shalev, who deals more generally with collapsing groups. A group G is called n-collapsing if for every subset S = (91,. . . , g,} of n elements of G, there exist two different positive words as and ,Dson n symbols such that as(g1,. . . ,gn) = Ps(g1,. . . ,gn). Thus being n-collapsing can be understood as a local version of satisfying a positive law on n symbols. Then Shalev’s main result is as follows (see Theorem A of [IS]).

Theorem 2.1. Let G be a residually finite group. If G is collapsing, then G is an extension of a strongly locally nilpotent group by a group of finite exponent. Here, the meaning of ‘strongly locally nilpotent’ is that, for every fixed d, all d-generated subgroups of G are nilpotent with a uniform bound for the nilpotency class. This theorem follows from a quantitative result valid for every n-collapsing finite group G:it possesses a nilpotent normal subgroup N such that the exponent of GIN is n-bounded and the class of

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every d-generated subgroup of N is { n , d}-bounded. (Throughout the paper, if S is a set of parameters, we use the expression ‘S-bounded’ to mean ‘bounded in terms of the parameters in S’.) Shalev proves this result by ‘means of several reductions, first to the soluble case, then to the nilpotent case (equivalently, to finite p-groups), and finally to the case of powerful pgroups. An interesting consequence of Theorem 2.1 is that a collapsing residually finite group satisfies in fact a positive law. In Theorem 1of [a], Burns, Macedoriska and Medvedev improve Shalev’s result for finite groups by showing that a finite group G satisfying a positive law of degree n has a normal nilpotent subgroup N such that both the exponent of GIN and the class of N are n-bounded. They follow very much the same ideas of Shalev’s, the main difference being a t the final stage of powerful p-groups, where they use Lie ring techniques in order to get the improvement. More precisely, they apply Zelmanov’s theorem stating the nilpotency of a bounded Engel Lie algebra over a field of characteristic zero. Furthermore, Theorem B of [2] also contains an important improvement with respect to Theorem 2.1, in the sense that it applies to a much wider class than that of residually finite groups. Let us say that a group is an SB-group if it lies in some product of finitely many varieties, each of which is either soluble or a restricted Burnside variety (that is, the variety of all locally finite groups of exponent dividing e l for some e ) . Theorem 2.2. Let C be the class of groups which i s obtained from the class of SB-groups by iteration, taking each t t m e all groups that are either locally

or residually in the previous class. If a group G in the class C satisfies a positive law of degree n, t h e n G i s nilpotent-by-(iocally finite of finite exponent), with both the nilpotency class and the exponent n-bounded. The key to the reduction of the finite case to nilpotent groups and powerful p-groups is a good knowledge of the action by conjugation of the elements of G on the abelian normal sections of G. This is the purpose of Proposition 3.4 of [15] and Lemmas 3.1 and 3.2 of [16]. Unfortunately, Lemma 3.2 is not valid as stated, more precisely there is a problem in the proof of Case 2. (See the remark after Lemma 4 in [3] for details.) Thus new arguments have to be provided in order to assure the truth of both Theorems 2.1 and 2.2. This task was undertaken by Burns and Medvedev [3], who make the necessary corrections in order t o justify the veracity of Theorem 2.2. They also extend Theorem 2.2 to the even more general class of locally graded groups, i.e. groups in which every non-trivial finitely generated subgroup has a proper subgroup of finite index.

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It is also worth mentioning that Macedonska [12] has proved that, in the class C defined above, every collapsing group actually satisfies a positive law. 3. Positive laws on large sets of generators

In this and the next sections, we consider several instances of the following general problem: to what extent does a positive law on a set of generators of a group imply a (possibly different) positive law on the whole of the group? Clearly, this question does not have a positive answer in all generality, even for the class of finitely generated residually finite groups: it suffices to consider any one of the well-known residually finite counterexamples to the General Burnside Problem (see [6,9,19] and Section 9 of [7]). These groups are finitely generated and periodic, hence any finite set of generators satisfies an exponent law, but they cannot be nilpotent-by-finite, since they are infinite. Alternatively (and more simply), consider the free product G = P * Q of any two non-trivial finite pgroups P and Q. Then G is a residually-p group, being the free product of two residually-p groups. Now if S and T generate P and Q, respectively, then their union is a set of generators of G which satisfies an exponent law. However, G is not nilpotent-by-finite unless both P and Q have order 2. Of course, the size of the set of generators must also play a role in the problem we have raised. In the end, if we choose G itself as a set of generators then the result holds true! Note that the sets of generators we have used in the examples above are all finite, so they are small if compared with the whole infinite group. One could guess that the situation is different if the set of generators is sufficiently large but, what do we mean by ‘a large set of generators’? Our purpose in the remainder of the paper is to precise some types of large sets of generators for which we get a positive answer to our question, at least for some particular classes of groups. We give two previously known examples in this section and then, in Sections 4 and 5 we comment on other examples from our latest work [5]. As a first attempt, consider the case where the set T of generators is a monoid, i.e. T contains 1 and is closed for products. In this context, G. Bergman [l]posed the following more ambitious question.

Bergman’s question. Let G be a group and suppose that T C G is a monoid generating G. If T satisfies a positive law, does it follow that G satisfies the same positive law? There are (unpublished) examples constructed independently by Ivanov

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and Rips which show that the answer to Bergman’s question is negative in general. On the other hand, since any finite monoid is actually a group, the answer is trivially positive for finite groups and consequently also for residually finite groups. More interesting is the following result of Burns, Macedoliska and Medvedev (see Theorem D in [2]).

Theorem 3.1. Bergman’s question has a n afirmative answer for solubleby-(locally finite of finite exponent) groups.

A different result in the same vein is the following, due to Shumyatsky ~71. Theorem 3.2. Let A be a n e l e m e n t a y abelian finite p-group of order at least p3 acting o n a finite p‘-group G. If CG(a) satisfies a positive law of degree n for every a E A, a # 1, then G satisfies a positive law of { n , p } bounded degree. Of course, the interesting part in the conclusion of this theorem is not the trivial fact that the finite group G satisfies a positive law, but that the degree of the law is {n,p}-bounded. Let us see how Theorem 3.2 relates t o the question we have posed. First of all, according t o Theorem 6.2.4 of [8],we have G = (CG(a) I a E A, a # l ) ,since A is non-cyclic and the action is coprime. Let T be the union of all centralizers CG(a) with a # 1. This set of generators of G can be considered t o be large, since the cardinality m of T , together with the prime p , bound the order of the group G: this follows from Lemma 2.3 in [17]. Since every centralizer CG(a) satisfies a positive law of degree n, it follows from Theorem 2.2 that it also satisfies a law of the form M c ( x e y, e ) , where c and e only depend on n, not on a. Hence, even if we cannot assure that the entire set T satisfies a positive law, it is partitioned in subsets all of which satisfy the same positive law of n-bounded degree. Thus the hypothesis of Theorem 3.2 is weaker than the assumption that T satisfies a positive law of degree n, but nevertheless we get the desired conclusion that the whole of G also satisfies a positive law of bounded degree. 4. Positive laws on commutator-closed normal sets of

generators Now we address a different instance of the problem we proposed in the preceding section. Let us say that a subset of a group is commutator-closed if it is closed under taking commutators of its elements. Then the kind

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of large sets of generators we consider now are commutator-closed normal subsets of the group in question. This choice is inspired by the following problem: if we know that all commutators in G satisfy a positive law, does it follow that the whole derived subgroup G' also satisfies s positive law? Note that the set of commutators is obviously commutator-closed and normal in G. A particular case of this problem is that of an exponent law, i.e. when there exists a positive integer n such that [z,yln= 1 for all z,y E G. If n is a prime-power and G is residually finite, then Shumyatsky [IS] has proved that G' is locally finite. Also, if G is finitely generated then G' has finite exponent, in fact it has { n ,d}-bounded exponent, where d is the number of generators of G. In particular, if G is residually-p for some prime p , then these results are valid for every exponent n without restriction. However, the case of residually finite groups and general exponent n is still open, and if the group is not residually finite then there are examples showing that G' need not even be periodic. For general positive laws, Riley and Shumyatsky [14] have proved that G' satisfies a positive law if G is finitely generated and residually-p, under the stronger condition that the law holds not only for commutators, but also for all products tuk, with t and u commutators and k 2 0. Furthermore, if G is d-generated and n is the degree of the original positive law, then the degree of the positive law satisfied by G' is { n ,p, d}-bounded. Also, the same conclusion is true if we substitute simple commutators of length m for commutators and the subgroup .ym(G)for GI. With these examples in mind, we pose the following question: if G is a finitely generated residually-p group and T is a commutator-closed normal set of generators of G satisfying a positive law, does also G satisfy a positive law? Unless otherwise stated, we assume that G satisfies all these conditions in the remainder of this section. As already mentioned in Section 2, it is important to know the action on any abelian normal section A of G. This information has to be extracted from the positive law satisfied by T . If we write the law in the form Xil

. . . xi,

= X j I . . . xj, ,

with i,,j, E (1,. . . , n } , we are allowed to make substitutions z i H ti, with the ti in T . What we do is to fix a E A and t E TI and then choose ti = tai = tai(lPt). At this point we differ from Proposition 3.4 of [15], in which the elements ti = atpi are used in the case that the whole group G satisfies a positive law. Note that this choice is not valid in our case,

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since at-i need not lie in T . This change of the elements ti will allow us to deal directly with commutators or powers of commutators satisfying a positive law, without needing to impose the law on products tuk with t and u commutators. From the substitution xi H ti, we eventually get the following result. In the statement we reduce to finite pgroups, which is all we need and has the advantage of allowing induction on the order of the group.

Theorem 4.1. Let G be a finite p-group and suppose that T normal subset which satisfies a positive law of degree n. Then:

CG

is a

(i) If A i s a n elementary abelian normal section of G, t h e n [A,,t] = 1 f o r every t E T . (ii) There exists a finite set P ( n ) of primes with the following property: i f p # P ( n ) , t h e n [A,,t] = 1 f o r every abelian normal section A of G and every t E T . (iii) There exist n-bounded positive integers m and k with the following property: if A i s a n abelian normal section of G and T i s power-closed modulo A, then [A,mt k ]= 1 f o r every t E T . Here, we say that T is power-closed modulo a normal section A = K / L of G provided that ti E KT for all i 2 1. Also, in the remainder of the paper, we keep the notation P ( n ) for the finite set of primes, depending only on n, whose existence is assured in (ii) of Theorem 4.1. How do we use this information about the action of T on abelian normal sections? As we explain below, we have t o combine it with four very powerful theorems. Let L,(G) be the Lie algebra over IF, associated to the dimension subgroup series of G , and let L be the subalgebra generated by the image T of T . As a consequence of the ‘Engel action’ of the elements of T on elementary abelian sections given in part (i) of the previous theorem, all commutators of elements of T are ad-nilpotent. Then the following theorem comes into Play.

Theorem 4.2 (Zelmanov [22], page 36). Let L be a PI L i e algebra (i.e. a L i e algebra satisfying a polynomial identity). If L can be generated by a finite set S such that every commutator of elements of S i s ad-nilpotent, t h e n L i s nilpotent. But is the Lie algebra L defined above PI? For this we need this second theorem.

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Theorem 4.3 (Wilson-Zelmanov [21],Theorem 1). If a group G satisfies a law, t h e n the L i e algebra L,(G) i s PI f o r every prime p .

So if we further assume that G satisfies a law, then the algebra L is nilpotent. This, in turn, is known to imply that L,(G) is also nilpotent, of the same class as L . Next we need a result that translates this fact into information about the group G (see Interlude A of [4]). Theorem 4.4. Let G be a finitely generated pro-p group. T h e n the following are equivalent:

(a) L,(G) i s a nilpotent L i e algebra. (ii) G has finite rank, i.e. there exists r such that every closed subgroup of G i s r-generated. (iii) G is isomorphic t o a closed subgroup of GL,(Z,) f o r some n. Now our residually-p group G need not be a pro-p group, but this is not a problem, since G embeds into its pro-p completion and all the arguments above also apply to G,. Thus G is linear over a field, and wc can use the next theorem. A

ep

Theorem 4.5 (The Tits Alternative [20]). Let G be a finitely generated group which i s linear over a field. T h e n either G i s soluble-by-finite or it contains a non-cyclic free subgroup. Since we have also assumed that G satisfies a law, the second possibility in the Tits Alternative cannot happen. Consequently G is soluble-by-finite and, being residually-p, we may conclude that G is soluble. Now all these arguments can be presented in a quantitative fashion, which leads t o the following result.

Theorem 4.6. Let G be a d-generated residually-p group which satisfies a certain law v = 1. Suppose further that G can be generated by a commutatorclosed normal subset T satisfying a positive law of degree n. T h e n G has { n , p , d , v}-bounded rank and i s soluble of { n , p , d , v}-bounded derived length. We need a final ingredient, which can be proved by induction on the derived length.

Theorem 4.7. Let G be a finite p-group of rank r and derived length ,!f and let T be a commutator-closed normal set of generators of G . If [ A l n t k= ] 1 f o r every abelian characteristic section A of G and f o r every t E T ,

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then G has a characteristic subgroup whose nilpotency class and index are {n, k,r , !}-bounded. Now it suffices to combine this last theorem with Theorem 4.1, which assures that T acts nicely on abelian sections, and with Theorem 4.6, which bounds the rank and derived length of the group. Thus we can give the following answer to the question raised at the beginning of this section.

Theorem 4.8. Let G be a d-generated residually-p group which satisfies a certain law v = 1. Suppose that G is generated by a commutator-closed normal subset T satisfying a positive law of degree n . Then:

(a) If p 6 P ( n ) , then G i s nilpotent of {n,p, d , v}-bounded class. (ii) If T i s power-closed, then G contains a characteristic nilpotent subgroup of {n, p, d , v}-bounded class and index. It is important to remark that the set of primes P ( n ) is a real obstruction in part (i) of the last theorem. In fact, for every prime p it is possible t o construct (though the construction is quite involved) a metabelian 2generated residually-p group which does not satisfy a positive law, even if it can be generated by a commutator-closed normal subset satisfying a positive law. All the material in this and the next section can be found in full detail in [5] by the authors. 5. Positive laws on word values

The machinery we have set up in the previous section can now be applied to a more general problem than the original one about commutators satisfying a positive law. Given a finitely generated residually-p group G and a word w,suppose that the set G, of all values of w in G (w-values, for short) satisfies a certain positive law. Does it follow that the verbal subgroup w(G)also satisfies a positive law? Note that the set of all w-values in G can be thought of as a large set of generators of w(G). However, the application of Theorem 4.8 to this situation, with w(G) playing the role of G and G, the role of T , is not straightforward. More precisely, the following questions arise: (i) Obviously, G, is a normal subset of G and in particular of w(G). But is it commutator-closed? This imposes a restriction on the possible words to which our results apply.

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(ii) If we want t o use Theorem 4.8, we need w(G)t o bc finitely generated, but we are assuming instead that G is finitely generated. How can we overcome this obstacle? (iii) Does w(G)satisfy a law? Yes, in fact, the whole of G satisfies a law in this case: if the positive law holding on G, is given by a = p, then the composition of w and ap-l is 1 in G. In order t o get rid of the problem in (i), we take the easiest solution: we suppose from the outset that the set G, is commutator-closed. If that is the case, we say that the word w is commutator-closed in G.The good news is that many common words are commutator-closed in any group: of course, the commutator word [x,y], but also the simple commutator of length m, [ X I , .. . , z m ] ,and the word dm(zl,.. . , ~ 2 defining the m-th derived subgroup G(,). Recall that 6, is defined recursively by means of 6 1 ( x 1 , ~ )= [xl,x2] and 6 m ( ~ 1 , . . . , x 2 m= ) [6m-1(x1,.. . , ~ 2 ~ - 1 )6m-1(x2m-~+l,.. , . , x p ) ] .More generally, if we compose a commutator-closed word with a simple commutator, then the resulting word is also commutator-closed. In particular, iteration of simple commutators gives commutator-closed words. Note that the corresponding verbal subgroup is of the form rm,(rm,(...(rm,(G))...)) . As for (ii), we argue in the following way. Let a and b be any two w values of G ,and consider an arbitrary quotient ?? = GIN which is a finite pgroup. Since G is finitely generated, all these quotients are d-generated for somc d. Thcn si and 6 can be cxprcsscd in terms of somc clcmcnts, say k, of Let be the subgroup generated by these k elements. Then is also generated by n a set which is commutator-closed (since we are assuming w t o be commutator-closed) and normal in Thus we can apply Theorem 4.8 and deduce that satisfies a law of the type Mc(xe, ye). Furthermore, if k can be bounded as a, b and N vary, then we can take c and e independent of a , b and N.Since G is residually-p, we conclude that w(G)satisfies the positive law M c ( x e ,ye), as desired. The way t o assure that k will be bounded as a , b and N vary is again to impose a restriction on the word w. Let us say that a word w has width at most k in a group G if every element of w(G)can be written as a product of no more than k elements or inverses of elements of G,. According t o the last paragraph, we need words that have bounded with in the class of all d-generated finite pgroups. Fortunately, this is the case of the simple commutator of length m, which has width at most d"-l. A different possibility is t o use words that have finite width in the particular residually-p group we are working with. For example, we can use a result of Jaikin-Zapirain,

cw. z

cw z,

z

r.

~

)

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Theorem 1 of [lo], assuring that all words have finite width in a p a d i c analytic group. This way we get the following results.

Theorem 5.1. Let G be a d-generated residually-p group. Then:

(i) If all simple commutators of length m in G satisfy a positive law of de) a positive law of { n , p , d,m}gree n and P # P ( n ) , then T ~ ( Gsatisfies bounded degree. (ii) If all powers of simple commutators of length m in G satisfy a positive law of degree n, then y,(G) satisfies a positive law of { n , p , d , m } bounded degree. Theorem 5.2. Let G be a p-adic analytic pro-p group, and let w be any word which is commutator-closed in G . Suppose one of these two conditions holds:

(i) All w-values in G satisfy a positive law of degree n, and p (ii) All powers of w-values in G satisfy a positive law.

# P(n).

Then w(G)also satisfies a positive law. Is the set P ( n ) a real obstruction in part (i) of these theorems? We know that we need to avoid this set in Theorem 4.8, but this does not mean that the same happens in these other situations. As a matter of fact, the example we have mentioned a t the end of Section 4 is not valid for the case of a positive law on simple commutators. On the other hand, the set of simple commutators is not only closed under taking commutators of its elements, but also under taking commutators with any element of the group. Thus we have stronger conditions than those in Theorem 4.8, which gives some hope that the set P ( n ) can be deleted at least from Theorem 5.1.

Problem 5.1. Let G be a d-generated residually-p group. If all sample commutators of length m in G satisfy a positive law of degree n, does T ~ ( G ) always satisfy a positive law of {n,p , d , m}-bounded degree? In order to give further evidence that the answer t o this question could be ‘yes’, we deal with the case when the verbal subgroup rym(G)is finitely generated, in contraposition to the previous assumption that the whole of G is finitely generated. We then get the following result.

Theorem 5.3. Let G be a residually-p group, and suppose that all simple commutators of length m in G satisfy a positive law of degree n. If T ~ ( G ) is d-generated, then it satisfies a positive law of { n ,P , d, m}-bounded degree.

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The main problem we have to face in order to prove Theorem 5.3 is that we cannot assure now a nice action of the simple commutators of length m on the abelian normal sections of G (to be precise, we have to consider the quotients of G which are finite p-groups). At this point, it is fundamental that part (iii) of Theorem 4.1 applies not only when T is power-closed, but also when it is power-closed modulo A . Another important ingredient is the use of powerful p-groups. For these groups, we prove the following extension of P. Hall’s criterion saying that a group G is nilpotent if both N and GIN’ are nilpotent for some normal subgroup N of G . Recall that this criterion has a quantitative version: if N has class c and GIN‘ has class d , then the class of G is { c , d}-bounded (see Theorem 3.26 of [ll]).

Theorem 5.4. Let G be a powerful p-group and let N be a powerful normal subgroup of G . If is nilpotent of class c and N f is nilpotent of class d, then Gef is nilpotent of {c, d}-bounded class. Hence we replace ‘nilpotent’ in Hall’s criterion by ‘nilpotent-by-(finite exponent)’. Even if this result is easy to obtain, it is a fundamental tool in our proof, since it provides the grounds for an induction argument. We finish by stating another problem for further research in this subject.

Problem 5.2. Find more families of words for which Theorem 5.1 or Theorem 5.3 hold true. Alternatively, give examples of families of residually-p groups for which the kind of results we seek are valid f o r a wide class of words (as happens with p-adic analytic groups). Acknowledgments This work is supported by the joint project CAPES/MECD 065/04 of the Brazilian and Spanish Governments. The first author is also supported by the Spanish Ministry of Science and Education, grant MTM2004-04665, partly with FEDER funds, and by the University of the Basque Country, grant UPVO5/99.

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TWO APPLICATIONS OF THE HUGHES SUBGROUP OF FINITE p-GROUPS N. GAVIOLI Universith degli studi - L'Aquila Dipartimento d i Matematica Pura ed Applicata Via Vetoio - 67010 Coppito ( A & ) Italy A. MANN Einstein Institute of Mathematics Hebrew University Givat Ram, Jerusalem 91 904 Israel C. M. SCOPPOLA Universith degli studi - L'Aquila Dipartimento d i Matematica Pura ed Applicata Via Vetoio - 67010 Coppito (A&) Italy DEDICATED T O AKBARRHEMTULLA In this paper we study the class of p-groups in which every maximal cyclic subgroup of order larger than p is self centralized as well as the class of pgroups in which the elements of order p2 generate an extraspecial proper subgroup. In both cases the Hughes subgbroup plays a crucial role.

Keywords: Finite p-groups, Hughes subgroup

1. Introduction Recently Z. Janko has determined the structure of some classes of 2-groups, thus answering, for the case p = 2, questions posed by Y.Berkovich (out of his list of more than 1500 questions about p-groups appearing in [l]); these questions ask for the determination of some extreme situations. In this note we give some results on p-groups, p odd, satisfying similar assumptions. The common theme to our results, besides the Berkovich-Janko connection, is

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the application of the Hughes subgroup. Recall that if p is a prime, the Hughes subgroup Hp(G) of a group G is the subgroup generated by all elements of order not p . The Hughes conjecture that ( G : Hp(G)I 5 p whenever exp(G) # p , while holding for finite groups which are not pgroups, does not always hold for pgroups, though it holds for many classes of pgroups, in particular for p = 2 and for p = 3. One of the classes of groups that we consider is finite p-groups G in which the elements of order p 2 generate a proper subgroup E which is extraspecial. Janko determined completely the case p = 2 in [6], and for odd primes he proved that E = Hp(G).Here we show that his result holds without the assumption p = 2: in pgroups of odd order the elements of order p 2 never generate an extraspecial proper subgroup. In particular, no pgroup of odd order has a proper extraspecial Hughes subgroup. The second class, with which we start, is: p-groups G such that whenever H is a maximal cyclic subgroup of G of order greater than p , then H is selfcentralizing in G. Let us call such groups SCC-groups. Janko determined all these groups when p = 2 in [7]:they are cyclic, elementary abelian, of maximal class, or one specific group of order 32. Here we show that for each odd prime only finitely many SCC-groups are not cyclic or of exponent p . These finitely many exceptions always exist, and are determined completely for p = 3. The examples we have are all groups of maximal class, and we start by discussing them. 2. SCC-groups

Notation 2.1. All groups are finite p-groups. Z = Z(G) and Zi = Zi(G) are the centre and the i-th centre of G, yi = ri(G) is the i-th term of the lower central series, and G' = 72(G). The subgroup generated by the p-th powers is denoted by GP, and exp(G) is the exponent of G.

A group G of order pn is of maximal class, if its nilpotency class cl(G) is n - 1. The basic theory of these groups, due t o N. Blackburn, is developed in [5, 111.14.1 In such a group, let Gi = yi(G) for 2 5 i 5 n - 1 and GI = C G ( G ~ / G * )Then . ( G : GI1 = (Gi : Gi+ll = p for 1 5 i 5 n - 1. The elements outside G1 have order p or ~ 2 If. n 5 p + 1, then GP 5 Z(G), with equality for n = p 1, and in that case also (G1)P = Z(G), unless G is the wreath product of two groups of order p , in which case G1 is the elementary abelian base group. If n > p 1, then exp(G1) > p . G is non-exceptional if G1 = CG(&(G)), and exceptional otherwise. If G is exceptional, we write C = C ~ ( 2 2 )Then . IG : CI = p . Exceptional groups exist if p 2 5, n is

+

+

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+

even, and 6 I n I p 1, and only for these values of p and n. The group G/Z(G) is always non-exceptional. Finally, a pgroup G is of maximal class iff G has an element x such that I c ~ ( x )= l p2. We need also some properties of regular and of powerful p-groups. These can be found in [5, III.lO], and in chapter 2 of [2].

Proposition 2.1. A group G of maximal class and order pn, p > 2, is SCC either exp(G) = p , or exp(G) = p2, exp(G1) = p, and exp(C) = p (if C exists). If G is a n SCC-group of maximal class, then n 5 p 1, and SCC-groups of maximal class and exponent p2 exist for all n in the range 35n p , and let x be an element of GI of maximal order. Since elements outside GI have order p 2 a t most, x generates a maximal cyclic subgroup, which is then self-centralizing. If G is non-exceptional, then x centralizes 2 2 , an elementary abelian subgroup of order p2, a contradiction. If G is exceptional, then exp(G) = p 2 , therefore 3: has order p2, and G / Z is non-exceptional, implying ] C G ( X2) ~ICG,Z(XZ)I2 p3, again a contradiction. Thus exp(G1) = p, implying exp(G) = p2. If G is not exceptional, then all elements outside G I have a centralizer of order p2, but if G is exceptional, then the elements of C - G1 have centralizers of order p3, so t o be an SCC-group it is necessary that these elements do not have order p 2 , i.e. exp(C) = p. If n 2 p+2, then G is non-exceptional and exp(G1) > p, and so G is not an SCC-group. For n = pf 1, the wreath product of two p-cycles is an SCCgroup of order p". Let 3 5 n 5 p , and let H be a non-exceptional group of maximal class, order p", and exponent p (e.g. we can take H = K / Z , where K is of maximal class and order p"+l). Let .7: E H - H I , and form a cyclic extension G of H1 by an element y inducing on H1 the same automorphism as 2, but such that y* is a non-identity element of Z ( H ) . Then G is an SCC-group of maximal class, order pn, and exponent p2. 0

Theorem 2.1. Let G be an SCC-group, and assume that exp(G) > p, and that G is neither cyclic nor of maximal class. Then G contains a normal elementary abelian subgroup A of orderp2. Put K = CG(A).Then IZ(G)I = p, A = Zz(G), IG : KI = p , exp(K) > p, HP(K)= GP, and IG : GPI 2 9. Proof. By assumption, G contains self-centralizing cyclic subgroups, therefore Z(G) is cyclic. It is well known that a subgroup like A exists if G is

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neither cyclic nor a 2-group of maximal class (see [5 111.7.61). Suppose that Z ( G ) = ( z ) has order greater than p , and let u be a non-central element in A. Note that A = ( An Z ( G ) ) ( u )Since . z u and z have the same order, we can embed zu in a cyclic self-centralizing subgroup H . But then H contains also z , therefore H contains u and A, a contradiction. Thus IZ(G)I = p (this argument is identical to Janko's for p = 2). Since A is not central, G / K Z C,. If exp(K) = p , then exp(G) = p 2 , and G contains self-centralizing subgroups of order p 2 , and is of maximal class. Thus exp(K) > p . Let x E K have order o(z) > p . Since x centralizes A, its centralizer is not cyclic, therefore z = y p for some y . Conversely, let t be any non-identity p t h power. Then t is a power of some element y which is not a p-th power, and y generates a self-centralizing subgroup. If y has order p 2 , then G is of maximal class. Thus y p , an element of K , has order at least p 2 , and so y p and t lie in H p ( K ) .Therefore H p ( K )= GP. If IG : GPI 5 p p - ' , then G is regular, and then K = H p ( K ) ,which means that IG : GPI = p and G is cyclic. Let x E K - GP, and let xy = yx. If y E K , then either y or xy lies outside H,(K), therefore it has order p , and so does y . If y $ K , then y is not a p-th power, and y commutes with both x and Z ( G ) , and therefore o(y) = p . Thus exp(C,(x)) = p . If GI 5 GP, then G is powerful, and then GP is powerfully embedded in G , and if GP 5 H 5 K , with IH : GPI = p , then H is powerful (see Lemmata 2.4(i) and 2.2(iii) in [2]). But then H = H p ( H ) ,which is another contradiction. Thus there exist commutators in K - GP, and these commutators have order p and centralize Zz(G). By the previous paragraph, ex~(Zz(G)= ) P. Let t generate a maximal cyclic subgroup T of order > p . Commutation by t induces a homomorphism of 2 2 into 2.Here the image has order p , and so does the kernel 22 n T , therefore 1221 5 p 2 , which implies 2 2 = A. 0

Corollary 2.1. If G is a n SCC 3-group, then G is cyclic, of exponent 3, the non-abelian group of order 27 and exponent 9, or the wreath product of order 81 of two groups of order 3. Indeed, for p = 3 the Hughes subgroup, if not trivial, has index at most p (see [ll]), so the previous theorem shows that if G is neither cyclic nor of exponent 3, it has maximal class, and Proposition 2.1 applies.

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Corollary 2.2. Let G be a n SCCp-group, p 2 5 . If either G is metabelian, o r cl(G) 5 4p - 3, then G i s either cyclic, or of exponent p, o r of maximal class.

Proof. If G is metabelian, the Hughes subgroup has index at most p (see 141). Assume that cl(G) 5 4p - 3 and, by way of contradiction, that G is neither cyclic nor of exponent p nor of maximal class. Then, since by Theorem 2.1 we have &(G) 5 Z ( K ) ,we also have cl(K) 5 4p-4. Theorem 7.4.18 (p. 216) in [8], states that if cl(K/Hp(K)) = c and a minimal set of generators for K / H , ( K ) has d elements, then cl(K) 2 ( p - l ) ( d - 1 c(c 1)/2) 1. Together with the previous inequality this implies that c = 1 and d 5 3. Then K’ 5 H,(K), which shows that Q ( K )5 H P ( K ) therefore , IK : HP(K)I 5 p 3 . By Theorem 2.1 H P ( K )= GP, therefore IG : GPI 5 p4. On the other hand this contradicts the final inequality of the same theorem.

+

+ +

A little more can be said if we assume that p is large enough. When exp(G) > p 2 7, and G is neither cyclic nor of maximal class, thc same method shows that cl(K) 2 5p - 4. This is because jG : GPJ 2 p p , so if c = 1 we have d 2 6, and if c = 2 we have d 2 3 (by looking at K/Gp). Thus if cl(G) 5 5p - 4, then G is of known type. If we assume that p is even bigger, we get better bounds, but computations become more and more complicated. Next we use deep results of E. I. Khukhro, which in turn depend on Kostrikin’s solution of the Restricted Burnside Problem for exponent p, to derive the following

Theorem 2.2. For a given prime p , there are only finitely m a n y SCC p-groups which are n o t cyclic o r of exponent p .

Proof. By Proposition 2.1, we may assume that G is not of maximal class, and that Theorem 2.1 applies. Using the notations of that theorem, we have IK : Hp(K)I > p . By Theorem 7.3.4 of [8], the exponent of K is bounded by some function of p. Then the exponent of G is also bounded. Thus the order of any maximal cyclic subgroup T is bounded. Taking T to be selfcentralizing, we see that IG/G’I 5 ICG(T)I = IT( is bounded, which in particular bounds the number of generators of G. This in turn bounds the number of generators of K , which, finally, bounds IK) and IGl, by Theorem 7.3.5 of [8]. 0

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3. p-groups whose elements of order p2 generate a proper extraspecial subgroup We now turn to the other application, which is an extension of a result in 161:

Theorem 3.1. Let G be a p-group in which the elements of order p 2 generate a proper subgroup H . If H i s extraspecial, t h e n p = 2 and G i s the semidihedral group of order 16. Proof. If p = 2, then Janko has shown in [6] that G is the semidihedral group of order 16. Therefore it is enough t o show that for p odd there are no examples a t all. Let G be a minimal counterexample. We divide the proof in several steps. Step 1. H = H,(G).

This is shown in [GI. Step 2. IG : HI = p .

Write 2 = Z ( H ) . Then exp(H/Z) = p , and therefore exp(G/Z) = p . Suppose that G contains a subgroup L 2 H such that IL : HI = p 2 . Then IL/Zl is a metabelian group of exponent p , and therefore of class at most p (see [lo]). But then cl(L) 5 p 1, and the result of [9] yields the contradiction IL : HI = IL : H,(L)I 5 p . Thus IG : HI = p .

+

Step 3. cl(G) = p .

By Step 2, G / Z is a group of exponent p containing an abelian maximal subgroup, therefore cl(G/Z) < p (see [5] 111.10.10) a.nd cl(G) 5 p . Since groups of class less than p are regular, and equal to their Hughes subgroups (if the latter are not trivial), we have cl(G) = p . Notation 3.1. Let a E G - H, so that a has order p , and G = ( a )H. Also, for h E H, we define hl := h and inductively hi := [hi-l, a]. Step 4. For z, y E H we have 1 = [z, a , y] [z, [y, a]][[z,a],[y, a ] ] .

This is an immediate consequence of the fact that [z, y] E Z(H) = Z ( G ) , and therefore [z, y] = [z, y]" = [z",y"] = [z[z,a],y[y, a]]. Step 5. For every h E H and every i we have z i ( h ) := [h,-i, hi] = 1.

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We use P. Hall's Commutator Collection, as in Section 12.3 of [3]. We assume the reader familiar with the construction of an ordered set of basic commutators of increasing weight in the generators a ~a 2, , . . . a,, and we only recall Theorem 12.3.1 on [3],page 182, which states that, if c, denotes the m-th basic commutator in the generators a l l a 2 , . . .a,, after collecting it we have the equality (41a.2., . a,) n

= a;.;.

m

..a : ( n

c,"")d,

(1)

i=l

where a! is a product of uncollected basic commutators later than c, in the order, and, if wi is the weight of ci,

for certain integers b j ( i ) that depend on ci but do not depend on n. In our context, we will compute the value of (1) with T = 2, a1 = a , a2 = h i , for h E H , and n = p . We have that = 1, and since ahj $ H , by Step 1 we also have that the left hand side of (1)is trivial, in our case. The basic commutators of weight less than p appear in (1) with an exponent divisible by p , and since we know that G' has exponent p , we can cancel. By Step 3, we are left with h j p and powers of basic commutators of weight p , that are central, and linear in all their arguments. Furthermore, since our subgroup H is normal of class 2, all the basic commutators that contain more than two instances of hi among their arguments are trivial in G. Thus the only basic commutators that we need to deal with in the right hand side of (1) in our case are: [hj, a;p - 11 and

for i = 1 , .. . , ( p - 1)/2. Now we compute the exponents of these basic commutators in Hall's formula. For each of them, since all the binomial coefficients that appear are divisible by p except the last one, it clear that we only need to compute the value of b,. This is done explicitly on [3]. For the convenience of the reader, we summarize the computation here. We label each occurrence of a generator as shown in formula 12.3.1 on page 179 of [3], and we start the collection, so that each occurrence of a basic commutator that is formed in the process will be naturally labelled by the set of labels of the generators that actually occur in it. This set of labels must satisfy some obvious partial order conditions, that arise from the coilection procedure. The number b, is the number of the possibilities

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of forming such a poset of labels, using all the labels: in other words, b, is the number of order preserving maps from that poset, that depends only on the basic commutator we are dealing with, onto the set P of the integers from 1 to p with their natural order. The simplest example, among the commutators we are interested in, is [ h j , a ; p - 11. When this commutator is formed in the collection process, we write it, in labelled form, as [ h j ( l l )a(l2), , . . . , a ( l p ) ]and , the way we collect (at each step we collect t o the left the leftmost uncollected occurrence of the basic commutator we are collecting at that time) guarantees 11 < 12 < . . . < 1,. Now there is obviously only one possible choice of labels, i.e. only one order preserving map from the poset of the labels onto P : the identity map. This confirms the well known fact that [hj,a ; p - 11 has exponent 1 in the collection. For the ki's the argument is similar, but slightly more complicated. Set the labels as [ [ h j ( r ) , a ( l l.).,. ,a(l,-i-l)], [ h j ( s ) , a ( j l .).,. ,a(ji--l)]]. The way we collect guarantees r < 11 < . . . < l,-i-l, and s < j l < . . . < ji-1. Furthermore, r 5 s. If r = s the set of labels is too small t o map onto P , so this possibility does not contribute t o b,. Therefore we may assume that r < s. Now it is easy to count the order preserving maps: there are such maps. It is well known that is congruent t o (-l)i modulo p . Thus the exponent of ki in Hall's formula applied to our context is really

("i')

Ti')

(-1)i.

Observe that by linearity we have [ h j , a ; p- 11 = h;, and, for i = 1 , .. . , ( p - 1)/2, ki = [h,-i, hi]j2 = ~ i ( h ) j ' . Now Step 4, applied to z = h,-i,y = hi-1, gives us zi(h)-l = zi-l(h), and therefore, recursively, z i ( h ) = zl(h)(-')'-', and thus ki = Zl(h)j2(-1)t-1. Summarizing, we are now able to write (1) as 1 = (&)P

=

(hPhP ) j z 1 ( h ) 4 ( P - 1 ) / 2 .

Since all the elements involved are central and this equality has to hold for every j , we deduce that hph, = 1 and that zi(h) = zl(h)(-l)'-' = 1 as claimed.

Step 6. For every h E H we have h,-1 E Z ( H ) . Let h , k E H . By Step 5 1 = Zl(hk) = Zl(h)Zl(k)[hp-l, kl[k,-l, hl = [hp-1,kI[kp-l, hl = [hp-l,kI 2 ,

where the last equality follows by an iterated use of Step 4.

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Final contradiction. G has class at most p - 1 in contradiction with Step 3. Since H is extraspecial we have t h a t Z ( H ) = Z ( G ) . For all h E H we have t h a t h,-l E Z ( G ) a n d t h a t H / Z ( H ) is abelian so t h a t 7,-1(G), being generated by left normed commutators of weight p - 1, is contained in Z ( G ) . 0

References 1. Y. Berkovich and Z. Janko, Groups of prime power order, in preparation. 2. J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic pro-p groups, second ed., Cambridge Studies in Advanced Mathematics, vol. 61, Cambridge University Press, Cambridge, 1999. 3 . Marshall Hall, Jr., The theory of groups, Chelsea Publishing Co., New York, 1976, Reprinting of the 1968 edition. 4. Guy T. Hogan and W. P. Kappe, O n the Hp-problem for finite p-groups, Proc. Amer. Math. SOC.20 (1969), 450-454. 5. B. Huppert, Endliche gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, vol. 134, Springer-Verlag, Berlin, 1967. 6. 2. Janko, Elements of order at most 4 in finite 2-groups. 11, J. Group Theory 8 (2005), no. 6, 683-686. 7. Z. Janko, Finite 2-groups all of whose maximal cyclic subgroups of order > 2 are self centralizing, preprint, March 2006. 8. E. I. Khukhro, Nilpotent groups and their automorphisms, de Gruyter Expositions in Mathematics, vol. 8, Walter de Gruyter & Co., Berlin, 1993. 9. I. D. Macdonald, Solution of the Hughes problem for finite p-groups of class 2 p - 2 . , Proc. Amer. Math. SOC.27 (1971), 39-42. 10. H. Meier-Wunderli, Metabelsche Gruppen, Comment. Math. Helv. 25 (1951), 1-10, 11. E. G. Straus and G. Szekeres, O n a problem of D. R. Hughes, Proc. Amer. Math. SOC.9 (1958), 157-158.

FREE PRODUCTS AND HIGMAN-NEUMANN-NEUMANN TYPE EXTENSIONS OF LATTICE-ORDERED GROUPS A . M. W . GLASS Queens’ College Cambridge CB3 SET, England and Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences Wilberforce Rd., Cambridge CB3 OWB, England E-mail: amwgQdpmms. cam. ac.uk

To Akbar H. Rhenztulla with thanks for years of enjoyable colluboraliori We prove the lattice-ordered group analogues of two easy results from group theory. Theorem A Let G, H be lattice-ordered groups with soluble word problem. Then the free product of G and H (in the category of lattice-ordered groups) has soluble word problem. Theorem B Let G be a lattice-ordered group and H a convez sublattice subgroup of G. Then G can be !-embedded in L , where L has presentation (G, t : t - l h t = h (h E H ) ) in the category of lattice-ordered groups. If g E G, then in L , [t,g] = 1 iff g E H , and i f f , g are finite subsets of G (which may overlap), then w ( f , g ) # 1 in G implies w(t-’ft,g) # 1 in L.

T h e proofs use permutation groups, a technique of Holland and McCleary, and the ideas used to prove t h e lattice-ordered group analogue of the Boone-Higman Theorem. Keywords: free products, HNN-extensions, lattice-ordered groups, presentations, permutation groups, representations, soluble word problem.

AMS Classification: 06F15, 20F60, 20B27.

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1. Introduction Since the category of lattice-ordered groups is equationally defined, free lattice-ordered groups on any set of free generators exists. So does the free product (in this category) of any set of lattice-ordered groups; moreover, each lattice-ordered group in this set is naturally embeddable in this free product and we will identify it with its image. If G I , Gz are lattice-ordered groups, then their free product G1 * L G in ~ this category is defined (to within isomorphism between lattice-ordered groups) by the standard property: if L is a lattice-ordered group and cpj : G j 4 L ( j = 1 , 2 ) are ehomomorphisms (in this category), then there is a unique l?-homomorphism 'p : G1 * L Gz 4 L whose restriction to G j is cpj ( j = 1 , 2 )

Cautions: (1) The subgroup of G * L H generated by G U H is general the group free product of G and H .

not in

(2) If a lattice-ordered group A is C-embeddable in GI and G z , there is not usually a lattice-ordered group L in which GI and Gz can be eembedded to make the diagram commute; thus HNN-extensions do not in general exist in the category of lattice-ordered groups. For more details, see the next section Despite these cautions, we can prove very special analogues of the corresponding group theoretic results.

T h e o r e m A Let G, H be lattice-ordered groups with soluble word problem. Then the free product of G and H (in the category of lattice-ordered groups) has soluble word problem. T h e o r e m B Let G be a lattice-ordered group and H a convex sublattice subgroup of G. Then G can be e-embedded in L , where L has presentation ( G , t : t-lht = h ( h E H ) ) in the category of lattice-ordered groups. If g E G , then in L , [ t , g ] = 1 iff g E H, and i f f , g are finite subsets of G (which may overlap), then w ( f , g )# 1 in G implies w ( t - l f t , g ) # 1 in L. More general results should be true. 2. Background and notation

Throughout we will use N for the set of non-negative integers, Z+for the set of positive integers, Q for the set of rational numbers and R for the set

149

of real numbers. The only order on the usual one.

Q and R that

we will consider will be

If X and Y are totally ordered sets, let X z Y be the set X x Y totally ordered by: ( x , y ) < (x',y') if either (y < y' in Y ) or (y = y' in Y & x < x' in X ) . We assume that the reader has a minimal knowledge of recursive function theory (see [IS]). In any group G we write [ f , g ] for f - l g - l f g . If H is a subgroup of G, we write [ g ,H ] for { [ g ,h] : h E H } .

A lattice-ordered group is a group which is also a lattice that satisfies the identities x(y A z)t = xyt A xzt and x(y V z)t = xyt V xzt. Throughout we write x y as a shorthand for x V y = y or x A y = x, and l-group as an abbreviation for lattice-ordered group. A sublattice subgroup of an l-group is called an t-subgroup. An l-group that is totally ordered is called an 0-group.


2 and c E G / F - {l}, then c and c-l are distinct from each other and conjugate in G I F . Hence G / F contains an element of even order, a contradiction. So G / F is finite in this case. Suppose now that k ( G / F ) = 3. If the exponent of G / F is 2, 3, 4 or 6, then GIF is known t o be locally finite and since it is a union of conjugate finite subgroups generated by two elements, it is finite by Theorem 2 in [l]. So assume that the exponent of G / F is not 2 , 3 , 4 or 6. Our aim is to reach a contradiction. If the exponent of G I F equals p for some prime p , then p 2 5 and if c E G - F is of order p , then c, c-l, c2 are different from each other and hence c is conjugate to c", where u E (-1, 2). But then c" = cg for some g E NG,F(< c >) - C G / F ( < c >), which is impossible, since g is of order p .

179

If the exponent of G I F equals p q for some odd primes p , q, p 5 q , then an element of order p is conjugate to its inverse and hence G I F contains an element of even order, a contradiction. So G I F - (1) consists of one class of elements of order two and another class of elements of prime order p , p 2 5. If c E G / F is of order p , then it is conjugate t o c2. If c2 = c" for some u E G I F , then v E N G / F ( < c >) - C G / F ( ) and hence u must be of order 2. But then c4 = c"* = c, contradicting p 2 5. The proof of Proposition 1 is complete. 0

We continue now with results, which will be used in the proofs of Theorems A, B and C. For each of them we supply either a proof or a reference. Lemmas 2.1, 2.2 and 2.4 deal with finite groups.

Lemma 2.1. Let G be a finite group and let F be a normal subgroup of G. Suppose that [G : F ] = 2 and k(G - F ) = r for some positive integer r . If x1,x2,.. . ,xr are representatives of the conjugacy classes of G lying outside F and cj = ICp(zj)I for j = 1 , . . . , r , then

-1+ - +1. . . + - 1 c1

c2

= ]

Cr

Proof. This is a special case of Lemma 4 in [2].

0

Lemma 2.2. Let G be afinite 2-group and suppose that there exists F 5 G , F # 1, such that [G : F ] = 2 and k(G - F ) 5 2. Then either G is a group of order 4, or it is one of the following groups: dihedral, semidihedral and general quaternion. Conversely, i f G is either a group of order 4 or the dihedral group of order 8 or the quaternion group of order 8 and F is a n y subgroup of G of index 2, then k ( G - F ) = 2; if G is a group of order 2n+1 ( n 2 3) which is either dihedral, or semidihedral or generalized quaternion and F is a cyclic subgroup of G of order 2n, then k(G - F ) = 2. Proof. Suppose that [G : F ] = 2, k(G - F ) 5 2 and let z E G - F . Then G = F (x)and CG(Z)= (z)CF(z).By Lemma 2.1, either I C F ( ~= ) ~1 or I C F ( ~=) 2. ~ But 1 < Z ( G )n F 5 C F ( ~so ) ,ICF(Z)I= 2 and ICG(Z)I= 4. Thus either G is an abelian group of order 4, or it is non-abelian and by [3] (111,14.23) it is of maximal class. Hence [G : G'] = 4 and by [3] (111,ll.g) it is one of the groups listed above.

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Conversely, if G is either of order 4 or dihedral of order 8 or quaternion of order 8, it is easy t o see that k(G - F ) = 2 for any subgroup F of G of index 2. Suppose now that G = (c, x ) is either dihedral, or generalized quaternion or semidihedral, with IGI = 2n+1 ( n 2 3 ) and IcI = 2n. Set F = (c). Then c" = c-l and for any g E G - F , we have IcG(g)I = 4 (see [3] (111,14.23)). Then ICF(g)I = 2 for any g E G - F and k(G - F ) = 2 , by Lemma 2.1.

Lemma 2.3. Let G be a group and suppose that G = F ( x ) , where F is a subgroup of G of index 2 and x E G. Assume that there exists S I Z(F), such that S2 = S , s" = s-' for each s E S and k ( G / S - F / S ) = 2. T h e n k(G F ) = 2 . ~

Proof. Let ( X S ) ~ and / ~( ( u x ) S ) ~ /be ' the two classes in G I S - F I S , with a E F . For every s E S there exists t E S such that s = t2.As [ t , x ]= t P 2 , it follows that s = t2 = [x,t]and hence x s = xt. Since S 5 Z(F),also s = [ a x , t ] ,which implies that ( u x ) s = ( u x ) ~ . for suitable Now let g E G - F . Then either g = x d s or g = d E F and s E S . Thus either g = x d s = ( x s ) = ~ xtd for a proper t E S and g E x G , or g = ( a x ) d s = ( a x s ) d = ( a x ) t d for a proper t E S and g E ( a x ) G . Since xG # ( u x ) ~it, follows that k(G - F ) = 2. Finally

Lemma 2.4. Let G be a non-abelian finite group and let F be a normal subgroup of G . Suppose that [G : F ] = 2, k(G - F ) = 2 and denote by P a Sylow 2-subgroup of G. T h e n one of the following statements holds: (a) G has an abelian normal 2-complement. (b) G has a non-abelian normal 2-complement and P E &8. (c) G has an abelian non-normal 2-complement and P &.

Proof. This is a special case of Theorem 2.2 in [8].

0

3. Proofs of Theorems A, B and C

Proof of Theorem A. Suppose that G E D2,[G : F ] = 2 and y2 E Z ( F ) for each y E G - F . Let x E G - F and a E F be such that xG and ( u x ) are ~ the two infinite conjugacy classes in G - F . Then G = F (x)and U = ( x 2 ,( a x ) 2 )I Z ( G )n F . For any f E F , either f x = xd or f x = ( a x ) d for some d E F . In any case (fx)' E U and f f" E U . Therefore f" = f-' (mod U ) for each f E F and hence F / U is abelian. Thus F' 6 U 5 Z ( G ) .

181

Since G E Dz, it follows from Theorem 5 in [2] that x16 = 1 = (uz)16 and consequently c8 = 1 for each c E U . Assume, first, that there exists c E U such that zc E ( U Z ) ~ .Since c E U , c8 = 1. Denote C = ( c ) ;clearly C 9 G , G / C = ( F / C )( x C ) and G / C E D1. Since C is finite, F C ( G / C )= F / C and as G / C E D1, it follows by Theorem 1 in [2] that F / C is abelian, (FIG)' = F / C and j" = f-' (mod C ) for each f E F . Consequently, F' 5 C 5 Z ( G ) , F4C = F and for any f , h E F there exist s , t E F and C U , E ~ Z, such that f = s4ca, h = t4c0. Since c8 = 1, it follows that [ f , h ]= [s4, t4]= [s,t]16= 1 and hence F is abelian. Now F = F 2 ( c ) , which implies, since F is abelian, that F s = F16 (c8) = F16 and F = Fa ( c ) . Write S = F8. Then S a G , S2 = S and G = F (x)implies that G = ( S ,c, x) = S (c,x). For any f E F we have f" = f-'cY for some y E Z,so = f-', which implies that s" = s-l for each s E S. Finally G / S (c ,x) /( (c,x)n S ) is a finite abelian 2-group satisfying k ( G / S - F / S ) 5 2 and by Lemma 2.2, G / S is either cyclic of order 4 or the Klein 4-group. In the first case G / S = (US) is cyclic of order 4 for some w E (c ,x). Since c E F n Z ( G ) and x @ S , we must have G / S = (xS)and (1) holds. In the second case, G / S = ( c S , x S ) with c E Z ( G ) ,and G / S is of type ( 2 ) . In order to conclude the proof, it suffices to deal with the following case: for each c E U , xc E xG. Write K = { k E F I xk = xd for some d E F } . By our assumption F' 2 U K . We claim that K is an abelian subgroup of F which is normal in G and k" = k-' for each k E K . First we prove that K is a subgroup of F . If k l , kz E K , then x k l k ~= xd1k2 for some d l E F and since F' 2 Z ( G ) n K , there exist dZ,d3 E d-l

F such that x k l k z = (xk,' ) d l = ( x k 2 [ k ~ , d ~ l = ])~ ( ~l ~ ~ [ k 2 , d = r~])~l ( ~ [ k dz ;, l ] ) d z d l= xd3d2d1. Hence k1k2 E K . Moreover, if k E K , then xk = xd for some d E F and z = z d k - l = ( ~ ( k - l ) ~ -= ' ) (~z k - l [ k - l , d - l ] ) d . Since [k-',d-']-l E F' 5 Z ( G ) n K , we get xk-l = x d - ' [ k - l , d - l ] - l = (x[k- l ,d - l ] - ' ) d - ' = xs for some s E F , which implies that k-l E K . It follows that K is a subgroup of F . Moreover, F' 5 K implies that K 9 F . Since by our assumptions x2 E Z ( F ) , for each d E F we have 1 = [x2,d] = [x,d ] " [ zd, ] . But each k E K can be written as k = [x,d ] for some d E F , so k" = IC-' for each k E K . Since conjugation by x inverts the elements of K , it follows that K is an abelian normal subgroup of G. The proof of our claim is complete. We claim, next, that F = K ( u ) , u2 E K , [F : K ] = 2 and F is abelian. Indeed, for any d E F , either d E K or zd = for some d l E F . In the latter case xd = ( z ~ [ u , z ]and ) ~ ld = [ z , d ~ ] u [ ~ , d ~ ] Since [ a , x F' ]~5 ~ .K

182

and [ x , d ~ []x, , a ] E K by thc definition of K , it follows that d E Ka. Thus F = K U K a , which implies that F = K ( a ) and a2 E K . Since xa = ( a x ) " , it follows that a $ K and hence [F : K ] = 2. Moreover, if k E K , then either ( a x ) k = (ax)dor ( a z ) k = xd for some d E F. If ( a x ) k = xd,then ax = (~(k-')~-')~and since (k-l)d-' E K , a r = '2 for some s E F and ax E x G , a contradiction. Hence (ax)k = and thus any k E K can be written as k = [ a x i d ]for some d E F . Since ( a x ) 2 E Z ( F ) , previous arguments yield k"" = k-' for each k E K . But (k-l)z-l = k, so it follows that k" = 5 and a E CF(-Y). As F = K ( a ) and K is abelian, a E C F ( K )implies that F is abelian. The proof of our claim is complete. Since [ F : K ] = 2, it follows that F 2 5 K . Hence f " = f-' for each f E F 2 and F 4 5 K 2 . But, for any k E K , we have k = [ d , x ] for some d E F , so F being abelian implies that k2 = [ d 2 , x ]= d-2(d2)z = d-4 E F 4 and hence F4 = K 2 . If la\ is finite, then la\ = 2'7- for some odd integer T . Since a $ K and a2 E K , it follows that i > 0, b = ar $ K and b2' = 1. Hence F = K ( b ) and F being abelian implies that F2' = K2' ( b 2 ' ) = (K2)Ztp1= ( ~ 4 ) 2 ' - l =

FZZ+'.Set S

=

F2'.Then S2 = S , S 5 Z ( F ) n F 2 5 K , \ F / S J> 1 and

S. Moreover, we claim that G / S is a finite 2-group. Clearly it suffices to show that G / S is finite. Let A = ( a , a " ) = ( a G ) ;then A 5 F , A a G and IA\ is finite. So G / A = ( F / A )( x A ) ,F / A = F C ( G / A ) , G / A E D1 and by Theorem 1 in [2], (F/A)' = F / A . Thus F = F 2 A and since S = F2' for some i > 0, it follows that F = S A . Hence F / S is finite, implying that G/S is finite, as claimed. Moreover, since G = F (x) and F = S A , we have G / S = ( a S , x S ) and it follows by Lemma 2.2, that either G / S is cyclic of order 4 or it is one of the following groups: dihedral s" = s-l for each s E

(including the Klein 4-group) , semidihedral and generalized quaternion. If

G / S is cyclic of order 4, then [ F : S] = 2 and as S 5 K , we have S = K . But then a2 E S , which implies that G / S = (zS)and (1) holds. In the other cases, G / S is of order 2n+' ( n 2 1) and it has a cyclic subgroup CIS = (cS)of order 2". But CF,s(xS) has order at most 2 by Lemma 2.1, and hence xS is of order at most 4. Similarly, each element not in F / S has order at most 4. Hence either n 2 3 and CS is in F / S , in which case (3) holds, or n 5 2, in which case G / S = (US,xS) and ( 2 ) holds. Now assume that la1 is infinite. Since each b E F - K satisfies xb = ( b ~ $) xGI ~ it follows that ( b ~ =) (ax)G ~ and taking into account the previous paragraph, we may also assume that (bl is infinite for each b E

183

nnEN

F - K . Write S = F2".Since S 5 F 2 5 K , we have s" = s-' for each s E S. We claim, first, that every 2-element t of F belongs t o S . For assume that t E F is of order 2n for some n E N.Then, by our assumption, t E K and hence t = [ d , x ] for some d E F . Since [F : K ] = 2, we have d2" E K 2n+1 and hence (d2")" = d-'". Thus 1 = t2" = [d2",x] = d&'"dP2" = d, implying that d E K . Hence t = [ d , z ] = d-ld" = d-2 E K 2 = F4, and we prove now, by induction, that t E F2" for each n N. Suppose that t = r2mE F2mfor some r E F and m E N. Then r is a 2-element, r E K and r = [ f ,x ] for some f E F . Hence, as shown above, f E K and consequently r = f - l f " = f - 2 . So t = (f-1)2m+1 E F2m+1and the inductive proof is complete. It follows that t E S , as claimed. Since x 2 and ( u x ) are ~ 2-elements of F , it follows that z2, ES and hence a"S = ~ - ~ a - ~ ( a x=) a~- lSS . If d E F = K ( a ) ,then d = kaZ for some k E K and i E {O,l}, and d"S = k"(a'))"S = k-'u-'S = d-'S. Hence d " S = d P 1 S for each d E F . Thus x S is of order 2 in G / S and it inverts, by conjugation, each element of F / S . If k E K , then k = [ d , x ] = d-'d" for some d E F and kS = dC2S. Thus K / S 5 ( F / S ) 2and since F 2 5 K , it follows that K / S = ( F / S ) 2 . Therefore F / S = ( F / S ) 2 and hence F / S = (F/S)'" ( a s ) for each i E N. It follows that F/F2' E ( F / S ) / ( F 2 ' / S = ) ( F / S ) / ( F / S ) 2 is ' a cyclic group of order 22 for each i E N. But S = nnENF2", so F / S embeds in the inverse limit of cyclic groups of order 2n, n E N and hence it is a subgroup of the group of 2-adic integers. We claim, next, that S2 = S. In fact, if s E S , then s E F 2 and s = d2 for some d E F . But, for each i E N, S 5 F2' and hence s = f2' for some f E F . Thus d2 = f2' and ( d - 1 f 2 " - 1 ) 2 = 1. But every 2-element of F belongs to S , so d - l f 22-1 E S and as S 5 F2'-l, we get d E F2'-l. This being true for each i E N,it follows that d E S and hence s E S2. Thus S2 = S , as claimed. It follows from our results that if la1 is infinite, then case (4) of Theorem A holds with D = F . This completes the proof of Theorem A in one direction. Conversely, suppose that there exists an infinite normal abelian subgroup S of G , such that S2 = S and one of (l),(2), (3) or (4) holds. Write F = F C ( G ) . Then it is easy to see that F = ( S , x 2 )if (1) holds, F = ( S ,c , x 2 ,( c x ) ' ) if (2) or (3) holds and F = D if (4) holds. In any case, G = F ( x ) ,x 2 E F and S 5 Z ( F ) . By Lemma 2.3 it suffices to show that k ( G / S - F / S ) = 2. This is obvious if (1) holds. Assume that (2) holds.

(as)

184

Then F / S is of index 2 and k ( G / S - F / S ) = 2 by Lemma 2.2. In case (3), F / S = (cS) and k ( G / S - F / S ) = 2, again by Lemma 2.2. Now assume that case (4) holds. Then D / S is a subgroup of the 2-adic integers and hence [DIS : ( D / S ) 2 ]= 2. Write D / S = (US)( D / S ) 2 .Then, for any gS E G / S - D / S we have either gS = xd2S or gS = (xu)d2Sfor some d E D. Since D' 5 S and x-ldxS = d-lS, it follows that d2S = [x,d]S = [xu,d]S and hence, in the first case, gS = x [ x , d ] S= x d S and in the second case, gS = xa[xu,d]S = ( ~ u ) ~Since S . ( G I s ) ' I ( D / S ) 2 ,xd2S and (xu)d2Sare not conjugate in G / S , implying that k (G/S - D / S ) = 2, as required. 0

Proof of Theorem B. Since x E G - F and [G : F ] = 2, we have G = F ( x ) and x 2 E F - Z ( F ) .Let xG and ( u x ) be ~ the two infinite conjugacy classes in G - F , where a 6 F . Since x 2 E F , we have [G : C G ( X ~ 0, the set {g E G I d ( x o , g x o ) I R} is finite. ---f

+

The aim of this note is to give a short proof of the following result:

Theorem 1.1. Let X = (V,E ) be a locally finite graph with infinitely m a n y ends. Let 7 = V U ax be the end compactification. Let G be a group of automorphisms of X . Assume that the action of G o n V is amenable and there exists z o E V such thot the closure of Gzo contains ax. T h e n there is a unique G-fixed end in ax,and the action of G (as a discrete group) o n V is not proper.

A deep result of Stallings [4] says that G has infinitely many ends if and only if G is an amalgamated free product rl * A rz or HNN-extension

227

228

H N N ( r ,A, cp) with A finite (with min{[rl : A], [r2 : A]} 2 2, not both 2, in the amalgamated product case; and min{[r : A], : cp(A)]} 2 2, not both 2, in the H N N case). In particular, if G has infinitely many ends, it contains non-abelian free subgroups, hence is non amenable. Tullio Ceccherini-Silberstein asked whether non-amenability of G could be proved without appealing to Stallings’ theorem. Since a finitely generated group G with infinitely many ends acts properly and transitively on its Cayley graph, our result shows that G is not amenable.

[r

Remarks (1) The density assumption of Theorem 1.1is satisfied when G has finitely many orbits in V . This assumption is necessary; for example the action of Z on IF2 = ( a ,b ) defined by n . g = ang, V n E Z, Vg E IF2 is amenable and proper. (2) Except for the non-properness statement, our result is contained in a result of Woess (see Theorem 1 in [6]): if X = ( V , E )is a locally finite graph and G admits an amenable action on V , then either G fixes a nonempty finite subset of V , or G fixes an end of X, or G fixes a unique pair of ends which are the fixed points of some hyperbolic element in

G. (3) There are results on strong isoperimetric inequalities for graphs with infinitely many ends satisfying extra conditions (see Theorem 10.10 in [S]): these give alternative answers to Ceccherini’s question. (4) A stronger question is to prove without appealing to Stallings’ result that a finitely generated group with infinitely many ends, contains a free group on two generators. Such constructions can be found in the work of Woess (Theorem 3 in [7]), Karlsson and Noskov (Proposition 3 in [ 3 ] ) ,and Karlsson (Theorem 1 in [2]). (5) For a finitely generated group with infinitely many ends, Abels shows, using Stallings’ theorem, that for G a finitely generated group with infinitely many ends, the compact set of ends is actually a minimal G-space (Theorem 1 in [l]). This is false for compactly generated, non discrete groups. Abels indeed gives the example of the group of affine mappings (z H arc+b) over Q,. This group G is H N N ( I < ,K , cp), where K is the group of affine mappings over Z,and cp : K + K is given by (z H a z + b ) H (z H a z p b ) . So G has infinitely many ends, but has a unique fixed point on its space of endsa, which is therefore not

+

=This can be seen directly; it also follows from our result, as G is amenable as a discrete group.

229

G-minimal. Acknowledgments

We thank T. Ceccherini-Silberstein for suggesting the question, F. Krieger for pointing out a mistake in a previous version, and A. Karlsson for recommending useful references. 2. Proof of the theorem

Let X be a countable, discrete set. A compactification of X is a compact space = X U d X in which X is an open dense subset. If G is a group of permutations of X , we say that is a G-compactification if the action of G on X extends to an action of G on by homeomorphisms. When X is a locally finite graph (identified with its set of vertices), we will take for d X the set of ends of X . In this case, we say that =X U is the end-compactification of X (it is an Aut(X)-compactification).

x

x

x

x

ax

Lemma 2.1. A s s u m e that G admits a n amenable action without finite

x

be a orbits, o n a countable set X . Let p be G-invariant m e a n o n X . Let G-compactification of X . T h e n f o r every subset A of X with p ( A )= 1, the n d X i s not empty. set ( ngEGgA) Proof. By compactness of d X , it is enough to show that the family ( a n has the finite intersection property. For 91, . . . , gn E G , we have

giA) = 1, while p ( F ) = 0 for every finite subset F c X since G has no finite orbit. So giA is infinite. Therefore giA) nax # 0. p(

n:=l

A fortiori

flyzl ( g nax)# 0.

0

The proof of Theorem 1.1 will follow from the four claims below: Claim 1. Let K be a finite, connected subgraph of X . Let A be an unbounded connected component of X \ K . Then gK c A for infinitely many g in G.

By the assumption, any G-orbit in X has infinite intersection with A (indeed, the assumption implies that Gx is dense in for every vertex x in V since G acts by isometries on X ; therefore the intersection of Gx and A is infinite since 2 is a neighborhood of all cnds contained in it). So for

x

230

x E K , one finds a sequence (g,),>l in G such that gnx are pairwise distinct vertices in A. Since d(g,x, x ) -+ 03 for n 03, we have g,K n K = 0 for n sufficiently large. Then g,K is a connected subset of X\K, and g,KnA # 0. By maximality of A among connected subsets of X \ K , this implies that gnK C A. -+

If K is a finite connected subgraph of X , we shall say that K is good if every connected component of X \ K is infinite. Let K be an arbitrary finite connected subgraph of X. Denote by the union of K and the finite connected components of X \ K ; then is a good subgraph of X .

k

Claim 2 . Let K be a good subgraph of X such that X \ K has at least 3 connected components. Let p be G-invariant mean on V . Then there exists a unique connected component CK of X \ K such that ~ ( C K=)1. Indeed, let A l l . . . , A, be the connected components of X \ K with n 2 3. Without loss of generality, we may assume that p(A1) 5 p ( A i ) , QZE {l,...,n}. By claim 1 , we can find h E G such that hK n K = 0 and hK c A l . Since hA1,. . . , hA, are the connected components of X\hK, and K is connected, there exists a unique Ic E ( 1 , . . . , n } such that K c hAk, so that hAi C All V i # k . Hence u i # k hAi C A1 (see Figure 1). Then by minimality of p(A1),

Hence p(A1) = 0 since n 2 3, and p(Ai) = 0, Vz finite subsets of X , we have 1 = p ( X ) = p ( K U Ak = C K .

# k. Since

uy=lA j )

p is zero on

= p(Ak).We

set

Let zo be a base-vertex in V . Denote by BN the ball of radius N centered at zo. Let NO be such that, for N 2 N o , the complement X \ has at least 3 connected components. Set

&

By Lemma 2.1, DN # 0, and ( D N ) N > Nform ~ a decreasing family of closed non-empty subsets of So by compactness, E = DN is nonempty, and obviously G-invariant.

ax.

231

Fig. 1.

Claim 3. The set E is reduced to one point, and G has no other fixed point in ax. Indeed, if w E E and w' E d X with w # w', then for N large enough w and w' are not in the same closure of a connected component of X \ BN. So w E and w' $ which means w' $ E. Let us show that gw' # w'for a suitable g E G. Recall (see e.g. Theorem 4 and 9 in [ 5 ] )that an automorphism h E Aut(X) is of exactly one of 3 possible types: h

q,

0 0 0

elliptic, if h stabilizes some finite subset of V . parabolic, if h is non-elliptic and fixes exactly one end. hyperbolic, if h is non-elliptic and fixes exactly two ends. Let A'

# CG be a connected component of X \

% with

x.

w' E Let A be a connected component of X \ distinct from A' and CG. By claim 1,we can find g E G such that gBN C A. All connected components of X \% will be mapped into A by g, except one. This exceptional connected component is necessarily CG because p(Cg--) = 1 and p is G-invariant. In

232

particular, gA c A, and this inclusion is strict. So g m A c A, 'dm 2 1. The sequence gmxO possesses a subsequence gmkxo which converges t o an end E in 3. It is obvious that g fixes E ; therefore g is hyperbolic fixing exactly E and w. In particular, gw' # w', as was t o be shown.

Claim 4. The action of G (endowed with the discrete topology) on V is not proper. The proof is inspired by a nice observation due to Karlsson and Noskov (Proposition 4 in [3]; see also Proposition 5 in [2]). As in claim 3, we can find h E G such that hmA' c A', 'dm 2 1 so that h is hyperbolic and fixes exactly one end r] in apart from w. With the same g as in Claim 3, let y , = hngh-". We claim that yn # ym, 'dn # m. Suppose by contradiction that there is n # m such that hngh-" = hmgh-m; so there exists k # 0 such that hkg = ghk. Then hkgr]= ghkq = gr] since h fixes r]. Since hk fixes the same ends as h, gr] has to be r] or w. But this is not possible since q , E and w are all distinct. Now, it remains for us t o prove that the set {ynxo : n E N} is bounded. Indeed, for y a hyperbolic automorphism, let [(y) =: min{d(ykv,v) : k E Z\{O},v E V } be the translation length of y, and let L, =: {v E V : d(yv, w) = [(y)} be the axis of y (this is a line in X ) . We will use one more result of Halin [5]: the end w, being a fixed end of some hyperbolic automorphism, is thin, i.e. for N >> 1 the set C B contains ~ finitely many disjoint rays. As a consequence, the rays L h n c s and L , n C Z stay within finite distance, i.e. there exists R > 0 such that, for every x E Lh n C z , one can find x' E L, n C z with d(x,x') 5 R. To prove that {ynxO : n E N} is bounded, we may clearly assume that xo E Lh. For n large enough, we have h-,xO E C B ~so, we can find x , E L, with d(h-nxg,x,) 5 R. Then,

x,

d(YnxO,20) = d(gh-nxo, h-"xo)

I d(ghPnxoigxn) + d(gxn,xn) + d(xn,h-nzo) I 2R + q g ) ; this concludes the proof.

References 1. H. Abels, O n a problem of Freudenthal's, Compositio Mathematica, 35 (1977), no. 1, 39-47.

233 2. A. Karlsson, Free subgroups of groups with nontrivial Floyd boundary, Comm. Algebra, 31 (2003) 5361-5376. 3. A. Karlsson and G. A. Noskov, Some groups having only elementary actions o n metric spaces with hyperbolic boundaries, Geom. Dedicata, 104 (2004) 119137. 4. J. R. Stallings, O n torsion-free groups with infinitely m a n y ends, Ann. of Math. (2) 88 (1968) 312-334. 5 . R. Halin, Automorphisms and endomorphisms of infinite locally finite graphs, Abh. Math. Sem. Univ. Hamburg. 39 (1973), 251-283. 6. W. Woess, Amenable group actions o n infinite graphs, Math. Ann. 284,(1989), 251-265. 7. W.Woess, Fixed sets and free subgroups of groups acting o n metric spaces, Math. Z . 214 (1993), no. 3, 425-439. 8. W. Woess, Random walks on infinite graphs and groups, Cambridge tracts in mathematics 138, Cambridge university press, 2000.

GROUPS WITH FINITELY MANY MAXIMAL NORMALIZERS CARMELA SICA and MARIA TOTA Dipartamento d i Matematica e Infomatica, Universitci di Salerno, via Ponte don Melillo, Fisciano (SA), 84084, Italy E-mail: [email protected], [email protected] In this paper, we will study groups with finitely many maximal normalizers and we will investigate properties of groups with a fixed number of maximal normalizers. Keywords: Normalizers; Maximal

Introduction We say that a group G has t maximal normalizers (G E N t ) if there exist non normal subgroups HI,.. . ,Ht of G such that for every subgroup K of G, either N G ( K ) = G or N G ( K ) 5 N G ( H ~ ) for , some i E (1,. . . , t } and NG(Hi) $ NG(Hj), if i # j . In such a situation, N G ( H ~ ). ,. . ,N G ( H ~are ) the so called maximal nomnalizers and the notation is standard ( [ 5 ] , [ll],[12]). We say that a group G has finitely many maximal normalizers ( G E N*) if G E Nt for some t E N. Here, of course, nl* = U N,* and all Dedekind iEN

groups lie in N;. Groups with conditions on the number of normalizers and conjugacy classes frequently appear in literature ([l],[a], [4], [7], [lo], [14]) as well as groups with finiteness conditions on conjugates ([3], [6], [S]). In particular, by studying groups with finitely many maximal normalizers, it is helpful to look at what has been already done concerning groups with finitely many ‘ordinary’ normalizers (G E N ) . Such groups have been studied by M. Tota in [14]. Slightly diverging from the notation in [14] we say that a group The authors wish to thank Dr. Giovanni Cutolo for interesting discussions and for useful suggestions.

234

235

G has n normalizers (G E N,) if there exist K1 = G, K2,. . . , K, < G , Ki # Kj if i # j such that {NG(H) I H 5 G} = { K l , . . . , K,}. A group G has a finite number of nomnalizers (G E N ) if G E N, for some n 2 1. Hence, groups which have one normalizer are the Dedekind groups and N,*.More generally, if n > 1, the groups belonging to N, are groups in N& for some 0 < m < n. We will show in Example 1.6 that n - m can be arbitrarily large. Anyway, several results holding in the ordinary case still hold in the maximal case. More precisely, in [14] the author proved that groups in N2 are locally finite, nilpotent of class two and central-by-finite. Similarly, in Section 1 of this paper we will obtain that groups in N; are locally finite and nilpotent of class two (Proposition 1.1). In order t o characterize the groups in N;, we will get an useful decomposition (Theorem 1.3) that will allow us t o restrict our attention on pgroups in N; and it will turn out that groups in N; are central-by-finite (Theorem 1.5). To better understand the structure of groups in N,*,we will take into account that M.D. Pkrez-Ramos, in [9] classified groups in N2, in the finite case and we will get the expected, corresponding structure result (Theorem 1.9) for p-groups of finite exponent in N;. Going further, in Section 2, we will deal with groups in N; that will be compared to groups in N3. It is known from [14]that all groups in N3 are locally finite, nilpotent of class at most three and central-by-finite. Similarly, as before, we will find out that groups in N; are nilpotent, metabelian, with finite derived subgroup. But we will show (Example 2.3) that the nilpotency class of a group in N; cannot be bounded. So, all groups in N; u N; are nilpotent and locally finite. In contrast, Example 2.4 and Example 2.5 will show that groups in N; are not necessarily nilpotent nor locally finite. Moreover, both groups in such examples will be metabelian. In fact, they will lie in N4. However, whereas all groups in N4 are metabelian (see [14]) there exist groups in N; which are not (Example 2.6).

NI

c

1. Groups in

n/;

Let G be a group with one maximal normalizer. If M is such normalizer, then it is clear that (z)aG and G/(z) is a Dedekind group for all z E G\M.

Proposition 1.1. Let G be a group in N;. Then the following statements hold:

236

(i) G i s nilpotent of class two; (ii) G is a n FC-group; (iii) G i s locally finite.

Proof. Let M be the maximal normalizer of G. Then G = (G\ M ) and GI 5 CG(S)for all x E G \ M , because (x)a G. Therefore GI 5 Z ( G ) and G is nilpotent of class two. Let F be the FC-centre of G and x E G.If (x)aG,then x E F . If (x)fl G, then x E N G ( ( ~ )5 ) M . Hence G = F U M . It follows that G = F is an FC-group. Let T = {x E G I o ( x )is finite}. Since G is an FC-group, T is a subgroup of G containing GI.Let z E G \ T . If (x)a G, then x E Z(G)5 M , otherwise (x2)= [x,GI 5 GI. If (x) fl G, then x E N G ( ( ~ ) 5 ) M . So, G = T U M . Hence G = T is locally finite. 0 Remark 1.2. Let G be a group in N;. If M is the maximal normalizer, then M is a characteristic subgroup of finite index. Theorem 1.3. Let G be a group in N:. T h e n G = P x Q where:

P i s a p-group lying in N;, Q i s a Dedekind p'-group. Moreover, let M = NG(H) be the maximal normalizer, then IG : MI i s a power of p . Proof. Since G is a locally finite, nilpotent group, G = Dri Pi, where Pi is the pi-Sylow subgroup of G. By hypothesis, there exists x E G such that (x) fl G. Without loss of generality, we may assume x E PI. Write P = PI and Q = Dr+1 Pi. Then NG((z)) = Np((x)) x Q 5 M and for all K 5 Q we have NG(K) $ M , as P 5 NG(K), so that K a G and Q is a Dedekind group. Moreover, Np(L) 5 M n P for all L fl P and M n P = N p ( H n P) since H P fl G. Thus, P E MT. 0 Furthermore, IG : MI 5 IG : N G ( ( Z ) )=~ p a for some a 2 1. Lemma 1.4. Let G be a p-group in NT and M be the maximal normalizer. If x E G \ M and C := C G ( X )t,h e n C is a normal, abelian subgroup of G and i s cyclic. Proof. Clearly, (x)a G and C is a normal subgroup. Moreover, (c) a G, for all c E C , since x E CG(C) 5 NG((c)) and x $! M . Hence C is a Dedekind

237

group. This means that either C is abelian or p = 2, C is hamiltonian and expZ(C) = 2. But, z E Z(C) and o(z) 2 4, since z $ Z(G). Hence C is abelian. Say o(x) = p a , then is isomorphic to a subgroup of Aut(x) 2 Z;-. Hence, either is cyclic or p = 2 and there exists g E G such that xg = 2-l. In the latter case, x 2 = [ g , x ] E Z ( G ) , since G is nilpotcnt of class two. Thus 0 1 = [g,x2]= [g,.I2 = x4 and G is cyclic, since pa = 4.

5

Theorem 1.5. Let G be a group in JV-T. T h e n IG : Z(G)I i s finite. Proof. In view of the previous theorem, we can assume that G is a pgroup. Let M be the maximal normalizer and z E G \ M so that (x)4 G. If C := CG(Z),then C is a normal, abelian subgroup and is cyclic, by Lemma 1.4. Let = (yC), then G = (C,y) and C n CG(Y)I Z ( G ) . It follows that IG : Z(G)I is finite, since G is an FC-group. 0

8

5

We noticed in the Introduction that the classes N, and NA are in some sense comparable. However the following example shows that there exists no positive integer t such that all finite N;-groups lie in U,,,,,N~. _ _

Example 1.6. Let p # 2, m > 1 and G = (x,y

1 zpZm= 1 = y p m ,

zy = z I+P”)

Then the set of normalizers of G is {(Y,ZPi)

10 5 i I m>

and consequently G E N,+, n NT. Since Nz-groups are in particular NT-groups we remind the following theorem:

Theorem 1.7 (M.D. Pbrez-Ramos [9]). Let G be a finite p-group. T h e n G E N2 i f and only i f G = A x B where: A i s abelian, B = (x)>a (y), xp‘ exp(A1Y) < p a ; e p = 2 + aL3.

. 0

= 1= yp

a

,

zy

= zl+pa-l,

In order to characterize groups belonging to result:

a > 1, p

N i l we

E

N,

need the following

238

Lemma 1.8. Let X = (u)(v)be a metacyclic p-group of class 2, with ( u )a X and o ( u ) 2 o(v). Then, either X QS or there exists w E X such that X = ( u )x ( w ) . Proof. Let pa = IX : I).( s E N such that vpb =

and p b = l X / ( u ) l ,so that b 5 a and there exists and p does not divide s. Let u1 = u - ~ P ~ -

uSPa

pb-l2

= [v,u]- , p a

$p b - 1

1.

Then v t = u - ~ P ~[v, ~ u] P ~ If vyb = 1, then X = (u)x (vl). Let vTb # 1. Then, p = 2 and a = b, since [v,u]Pb= [vPb,u]= 1, and ~ ( v i=) 2a+1. SO X = ( u ) ( v l )and X = (u)x (v1) unless p = 2 and o(u)= o(v1) = 2a+1. By the above, vfa = [ v , u ] ~and ~ -hence ~ o([w,u])= 2 a , that is, I ( u ) / ( [ v , u ] )=I 2. 0 It follows o(u) = 4, since X has class two, and X Qs.

Theorem 1.9. Let G be a p-group of finite exponent. Then G E N; if and only if G = A B where: A 5 Z(G), 0

B = (x)x ( y ) , xPa expG = p a , expA 5 pm;

0p=2

+

=

1 = y p0 ,

ZY

= xl+Pm,

p 5 m < a 5 p + m,

m>2.

Proof. Let G be a group in N;.Denote by M the maximal normalizer and let x be an element of maximal order in G \ M . Let o(z) = p a , then (x)a G and a > 1 because z f Z(G). If g E G, then either g E G \ M and o(g) 5 o(x) or g E M and gx f M . Pol ( P a- 1 In the latter case 1 = (gx)Pa = gPaxPa[x,g ] 2 since G is nilpotent of class 2 and g P a = 1 because ( x )a G implies ( [ x , g ]< ) ( x ) and [x,g]Pa-'= 1. Hence exp G = p a . Let C := C G ( X )then , H a G for all H 5 C, C is abelian and is cyclic, by Lemma 1.4. Thus, there exists y E G such that G = C(y) and xY = xl+Pm, 0 < m < a. Now, y acts by conjugation on C as a power automorphism. Since C is abelian and o(x) = expC, then cY = c1+Pm for all c E C. Moreover, there exists A 5 C such that C = ( x ) x A . If 2 is not abelian, then p = 2 and it is hamiltonian. It follows A 2 (x)

34 f Z ( 8 ) and expA (2)

= 4. If c E

A and

c(z)

2(8), then o(c) = 4

and cY = c-l. Hence c ~ =+c-l which ~ ~ implies m = 1. But, a 5 2m since G is nilpotent of class two, then a = 2 , that means o ( x ) = 4 and y acts as

~ ~ .

239

-1 on C. Now, [y, C] = [xy,C] = C2 and C2 is not cyclic since it contains x2 and c2. Thus C 2 f (y) and C2 f (xy) and so (y) #I G and (xy) +I G. Hence y,xy E M which is a contradiction since x # M . This means that is abelian and GI 5 (x).Then A 5 Z(G) since A is normal in G and A n GI = 1. In particular, AP" = [A, y] = 1. Thus far we have proved that G = ( z , y ) A , where A 5 Z ( G ) and xu = x'+~". Without loss of generality, by Lemma 1.8, we may assume (2,y) = (x) x (y) unless p = 2 and (x,y) 2 Q8, which would imply m = 1, hence A2 = 1. But then G would be hamiltonian that is a contradiction. So, G = ((x) x (y))A where A 5 M and y E M . Let o(y) = p p . Then 4 1 = [x,yp ] = [x,y]P' = xpnfP and a 5 m + ,B. Next, xp" = [x,xy] E (xy) since xy @ M and (xy) a G. Hence there exists s(a-1) p m s (s- 1) s E N such that xPm = (xy)" = x s y s [ y , x ] ~P.u t r = , then xPm = xSysz-T and ys E (y) n (x) = 1 so that p p divides s. If p # 2, then pm+O divides r ; since a 5 m ,B it follows that xPT=l and xpm = xs. Hence s 3 pm (mod p a ) and so p p divides p m , that means ,B 5 m, since ,B 5 a: and p p divides s. Similarly, if p = 2 then x - ~ ' = 1. On the other hand, x2" # 1 # xs and hence x2m = xSx-' implies o(x2") = o(xs). In particular, 2m is the largest power of p dividing s and ,B 5 m in this case as well. Let p = 2 and m = 1. Then a = 2 since 2 5 a 5 2m and xu = x-'. Thus (yx) +I G, in fact [yx,y] = x2 $ (yx) since (yx)' = y2. It follows yx E M , which is a contradiction since x $ M . Hence p = 2 implies m 2 2. So we have proved that the stated conditions are necessary. Conversely, if these conditions hold, [x,y]Pm = [zp", y] = 1, since a 5 2m. Hence G is nilpotent of class two and exp GI 5 pm. Let g, h E G. Then P"(Pm-1 (gh)pm = gPmhpm[h,g] 2 and y acts as the power automorphism g H glfpm of G unless p = 2 and a! = 2m.. Let p : 2 and a! = 2m. Then, 20-1 cp : g E G H g E G is an endomorphism of G, since m 2 2, and so a - 1 > m and G20-' = ( x ~ ~ - ' Let ) . N := G2a'-1.Then GIN has the same structure as G with the exception that I(x)/NJ= 2a-1 and a - 1 < 2m, so that, by the previous case, the automorphism g N H gYN is a power automorphism of G I N . Thus HY = H for every H 5 G such that N 5 H . On the other hand, let H 5 G such that N $ H . Then H"-l = 1 so that H 5 Kercp = ((x2) x (y))A. By the same argument used for G I N , all subgroups of Ker cp are normalized by y, hence HY = H in this case too. It follows that, for any value of p , y E Norm(G) and -E Norm(G)n(z) (x)

6

~

+

'

is cyclic. Hence G E N;.

0

240

2. Groups in

R/2*

and N2

Let G be a group with two maximal normalizers. If M I ,A 4 2 are such normalizers, then it is clear that (z) a G and G / ( z ) is a Dedekind group for all z E G \ (Mi UM2). We will make use of the fact that a group G is the union of three of its proper subgroups HIK and L if and only if H , K and L have index 2 and their intersection N has index 4 in G (so that GIN V4),from [13].

Theorem 2.1. Let G be a group in hold: (i) (ii) (iii) (iv)

NZ.Then the following

statements

G is metabelian; G is nilpotent and has a subgroup of class at most 2 and index 2; G’ is finite;

G is locally finite.

Proof. Let M I ,M2 be the maximal normalizers. Then G = CG(G’) U A41 U A 4 2 and IG : CG(G’)I 5 2. Hence, G’ 5 CG(G’) and G is metabelian. If G = CG(G’), then G is nilpotent of class c(G) 5 2. Let G # CG(G’),then IG : MII = IG : M21 = 2. If z E G \ (A41 U M 2 ) , then G / ( z ) is a Dedekind group and [G‘, GI I (4. If p is a prime, p # 2, then x p $ M I U M2 and [G’, GI I (9). So, (z)is finite and without loss of generality l(z)l = 2’ for some s E N. It follows that (z) 5 Z8(G). So, in any case, G is nilpotent. In particular, if z still lies in G \ (MI u M 2 ) , then there exists i such that zz E Z(G), and z 4 Z(G). Let a E G such that za # z. If z has infinite = z-i that is a contradiction. order, za = 2-l and zi= Thus (z)is finite. Since G / ( z ) is a Dedekind group, G‘(z)/(z) is finite and G’ is finite. Let T = {y E G 1 o(y) is finite}. Since G is an FC-group, T is a subgroup of G containing G’. Let y E G \ T . If (y) a G, then y E Z(G) I MI n M2, otherwise ( y 2 ) = [y,G] 5 G’. If (y) 56 G, then y E N G ( ( ~ )

248

G/yi+l ( N ) (respectively, G/y,+l (N)) has finitely many conjugacy classes of non-RN (respectively, non-ON,) subgroups. It follows by Lemma 2.4 (ii) that G / y i + l ( N ) (respectively, G/yc+l( N ) )satisfies the minimal condition on non-ON (respectively, non-ON,) subgroups. Since G/yi+l (N) (respectively, G/yc+l(N))is finitely generated and belongs to NF (respectively, NCF), we deduce by Lemma 2.3 that G / y i + l ( N )E FN (respectively, G/y,+l(N) E FN,), so G E ( R F ) N (respectively, G E (RF)N,). (ii) In the general case, let N be a normal subgroup of finite index in G such that N E L(ON) (respectively, N E L(RN,)). If T is the torsion subgroup of N , then T E L ( R F ) and G / T E L(N)F(respectively, G / T E L(N,)F).So G / T E L ( N F ) (respectively, G / T E L(N,F)).It follows that G / T is a locally graded group locally satisfying the maximal condition on subgroups. Since G / T has finitely many conjugacy classes of non-ON (respectively, non-RNc) subgroups, then by Lemma 2.4 (ii) G / T satisfies the minimal condition on non-ON (respectively, non-RN,) subgroups. We deduce from Lemma 2.3 that G / T E L(F)N (respectively, G / T E L(F)N,), hence G E ( L ( O F ) L ( F ) ) N (respectively, G E (L(RF)L(F))N,), that is G E L ( O F ) N (respectively, G E L(OF)N,). 0 Proof of Proposition 1.2. Suppose that G is finitely generated and has finitely many conjugacy classes of non-RN (respectively, non-ON,) subgroups. From Lemma 2.4 (i), we have G E ( R N ) F (respectively, G E (RN,)F). So Lemma 2.5 permits us t o conclude that G E ( R F ) N (respectively, G E (RF)N,). Suppose now that G is not finitely generated and first consider the case where G has finitely many conjugacy classes of non-RN subgroups. From Lemma 2.4 (i), G is locally in the class ( O N ) F . So the set S of finitely generated (ON)F-subgroups of G is a local system of G . Set S = S1 U 3 2 , where 31 = {H E S : H E R N } and 3 2 = { H E S : H $ ON}. From [9, Lemma 1.A.101, at least one between S1 and 3 2 is a local system of G. If 91 is a local system of G , then G E L(RN). It follows by Lemma 2.5 that G E L ( R 3 ) N . Suppose now that the set 32 is a local system of G . Since the number of conjugacy classes of subgroups in 9 2 is finite, and these subgroups are in (RN)F,it easily follows that there exist integers k and n such that each subgroup H E 3 2 has a RNk-subgroup of index at most n.It follows, by Lemma 2.4 (iii), that there is a normal subgroup N of finite index in G such that N has a local system consisting of finitely generated RNk-subgroups. Thus G E L(RN)F,so by Lemma 2.5 we obtain that G E C ( O F ) N . Consider now the case where G has finitely many conjugacy classes of non-ON, subgroups. Hence G has finitely many

249

conjugacy classes of non-RN subgroups. Then G E C(OF)N,as before. Let T be the torsion subgroup of G. Then T E L(OF) and G / T E N . Thus G / T satisfies locally the maximal condition on subgroups. Since t h e number of conjugacy classes of non-RNc subgroups of G / T is finite, then by Lemma 2.4 (ii) we deduce t h a t G / T satisfies the minimal condition on non-RN, subgroups. From Lemma 2.3, we obtain t h a t G / T E L(F)Nclso G / T E N, as it is torsion-free, hence G E C(RF)N,. 0

Acknowledgments

I would like t o thank the referee whose comments improved the exposition of this paper.

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