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

277 Pages·2007·13.38 MB·English

by  Trevor Hawkes

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 Hawk es
University of Warwick, UK
Patr izia Longo ba rdi
University of Salerno, Italy
Mercede Maj
University of Salerno, Italy
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World Scientific
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Library of Congress Cataloging-in-PublicationD ata
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-PublicationD ata
A catalogue record for this book is available from the British Library.
Copyright 0 2007 by World Scientific Publishing Co. Re. Ltd
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Contents
Preface vii
Sponsors viii
Akbar Rhemtulla ix
Bibliography xii
Conference Program xvii
List of Participants xxi
A Combinatorial Property of Burnside Varieties of Groups 1
Alireza ABDOLLAHI
A Note on Ends of Operator Algebras 7
Tullio G. CECCHERINI-SILBERSTEIN and
Aryeh Y.S AMET- VAILLANT
Minimal Quasinormal Subgroups of Groups 13
John COSSEY, Stewart STONEHE WER 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
Camela 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 83
Gustavo A. FERNANDEZ-ALCOBER and
Leire LEGARRETA
V
vi
On Certain Classes of Generalized Periodic Groups 93
Gerard ENDIMIONI
Nielsen Equivalence Classes and Stability Graphs of Finitely
Generated Groups 103
Martin J. EVANS
Counting Conjugacy Classes of Subgroups in Finite pgroups, I1 120
Manuel EGIZII DI MARCO, Gustavo A.
FERNANDEZ-A LCOBER and Leire LEG AR RET A
Positive Laws on Large Sets of Generators and on Word Values 125
Gustavo A. FERNANDEZ-ALCOBER and
Pave1 SHUMYATSK Y
Two Applications of the Hughes Subgroup of Finite pgroups 138
Norbert0 GAVIOLI, Avinoam MANN and
Carlo M. SCOPPOLA
Free Products and Higman-Neumann-Neumann Type Extensions
of Lattice-Ordered Groups 147
Andrew M. W. GLASS
Attaching to a Profinite Space 166
Wolfang HERFORT and Wolfram HOJKA
On Groups with Two Infinite Conjugacy Classes 175
Marcel HERZOG, Patrizia LONGOBARDI and Mercede MAJ
Centralizers of Nonabelian Irreducible Equivalent G-groups 190
Paz JIMENEZ-SERA L
Some New Methods for Almost Regular Automorphisms 200
Evgeny I. KHUKHRO
Non-Properness of Amenable Actions on Graphs with Infinitely
Many Ends 227
Soyoung MOON and Alain VALETTE
Groups with Finitely Many Maximal Normalizers 234
Carmela SICA and Maria TOTA
Locally Graded Groups with Few Non- (Torsion-by-Nilpotent)
Subgroups 243
Nadir TRABELSI
Preface
The papers of this volume represent the proceedings of the Conference
entitled Ischia Group Theory 2006 which took place at 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:
all the speakers and the partecipants,
0
the authors for their contributions,
0
the referees, who gave generously of their time to review the papers and
0
offer valuable feedback to the authors,
the staff of the Jolly Hotel for being so accommodating and supportive,
the friendly and patient staff at World Scientific Publishing for their
help and advice, and for producing these proceedings in a professional
and timely manner,
all the sponsors,
0
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 to celebrate with him.
Trevor Hawkes, Patrizia Longobardi and Mercede Maj
July 2007
vii
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 Appli-
cazioni
(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
viii
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 pri-
mary school and the Government secondary school up to 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 undergrad-
uate at 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 Quantum Mechanics and
Group Theory as “special topics”, and he subsequently won a research schol-
arship 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
ix