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Contemporary Mathematics 402 CONTEMPORARY MATHEMATICS 402 Israel Mathematical Conference Proceedings Ischia Group Theory 2004 Proceedings of a Conference in Honor of Marcel Herzog March 3l-April3, 2004 Naples, Italy Zvi Arad Mariagrazia Bianchi Wolfgang Herfort Patrizio Longobardi Mercede Maj Carlo Scoppola Editors http://dx.doi.org/10.1090/conm/402 Ischia Group Theory 2004 Israel Mathematical Conference Proceedings (IMCP) Israel Mathematical Conference Proceedings (IMCP) is a publication, part of the Contemporary Mathematics Series, devoted to the proceedings of conferences, symposia and seminars. Collections of papers focusing on a certain subject will also be published. Prospective volumes may be submitted to any member of the editorial board. Each volume has an editor (or editors) responsible for its prepara- tion. In order to ensure inexpensive and timely distribution, authors are requested to submit to the Editor of the volume an electronic 'JEX file of their manuscript in AMS-~'JEX, using the Contemporary Mathematics style file which can be down- loaded at http: I /www. ams. org/tex/author-info .html. For further information, contact the Managing Editor, IMCP, Department of Mathematics, Bar-Ilan Uni- versity, Ramat-Gan 52900, Israel; e-mail: rowen(Omacs. biu. ac. il. CoNTEMPORARY MATHEMATICS 402 Israel Mathematical Conference Proceedings Ischia Group Theory 2004 Proceedings of a Conference in Honor of Marcel Herzog March 31-April 3, 2004 Naples, Italy Zvi Arad Mariagrazia Bianchi Wolfgang Herfort Patrizio Longobardi Mercede Maj Carlo Scoppola Editors American Mathematical Society Providence, Rhode Island Bar-llan University Ramat Gan, Israel EDITORIAL COMMITTEE Dennis DeThrck, managing editor George Andrews Carlos Berenstein Andreas Blass Abel Klein Editorial Board of Israel Mathematical Conference Proceedings Louis Rowen, Bar-Ilan University, managing editor Z. Arad, Netanya Academic College M. Katz, Bar-Ilan University J. Bernstein, Tel-Aviv University B. Pinchuk, Netanya Academic College H. Furstenberg, Hebrew University S. Shnider, Bar-Ilan University S. Gelbart, Weizmann Institute L. Small, University of California at San Diego V. Gol'dshtein, Ben-Gurion University L. Zalcman, Bar-Ilan University Miriam Beller, Technical Editor 2000 Mathematics Subject Classification. Primary 20C15, 20D15, 20D20. Library of Congress Cataloging-in-Publication Data Ischia Group Theory 2004 (2004 : Naples, Italy) Ischia Group Theory 2004: proceedings of a conference in honor of Marcel Herzog, March 31- April 3, 2004, Naples, Italy / Zvi Arad ... [et a!.], editors. p. em. - (Contemporary mathematics, ISSN 0271-4132 ; 402) (Israel mathematical confer- ence proceedings) Includes bibliographical references. ISBN 0-8218-3711-7 (alk. paper) 1. Representations of groups-Congresses. 2. Nilpotent groups-Congresses. 3. Sylow subgroups-Congresses. I. Arad, Z. (Zvi), 1942- II. Herzog, M. (Marcel), 1935- III. Title. IV. Series. V. Contemporary mathematics (American Mathematical Society) ; v. 402. QA176.I83 2006 5121.22-dc22 2006040728 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Managing Editor, IMCP, Department of Mathematics, Bar-Ilan University, Ramat-Gan, 52900 Israel. Requests can also be made by email to rowen«<macs. biu. ac. il. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) © 2006 by Bar-Ilan University. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http: I /www. ams. org/ 10 9 8 7 6 5 4 3 2 1 11 10 09 08 07 06 Dedicated to Marcel Herzog Contents Preface ix Sponsors xi Marcel Herzog xiii Bibliography of Marcel Herzog xv Conference Program xxi List of participants xxiii The history of the classification of finite groups with a CC-subgroup ZVI ARAD and WOLFGANG HERFORT 1 Structure of finite p-groups with given subgroups YAKOV BERKOVICH and ZVONIMIR JANKO 13 Lower bounds for the number of conjugacy classes in finite groups EDWARD A. BERTRAM 95 Monounary simple algebras MARIAGRAZIA BIANCHI, ANNA GILLIO, and LIBERO VERARDI 119 Algebras with positive bases, commutators and covering numbers DAVID CHILLAG 133 On certain group theoretical properties generalizing commutativity COSTANTINO DELIZIA and CHIARA NICOTERA 143 Probabilistic non-generators in profinite groups ELOISA DETOMI and ANDREA LUCCHINI 149 Limits of vertex-transitive graphs MICHAEL GIUDICI, CAI HENG LI, CHERYL E. PRAEGER, AKos SEREss, and VLADIMIR TROFIMOV 159 Group rings with simple augmentation ideals RUDIGER GOBEL and OTTO H. KEGEL 171 On the number of commutators in groups MARCEL HERZOG, PATRIZIA LONGOBARDI, and MERCEDE MAJ 181 New results in the theory of finite 2-groups ZVONIMIR JANKO 193 vii viii CONTENTS The existence of normal and characteristic subgroups in finite groups GIL KAPLAN 197 On the covering numbers of finite groups: Some old and new results ARIEH LEV 201 Normal subgroups in the subgroup lattices of finite p-groups MARIO MAINARDIS 211 On characters-classes duality and orders of centralizers AVINOAM MANN 215 Bijections for identities of multisets of hook numbers AMITAI REGEV 219 Hall subgroups of finite groups DANILA 0. REVIN and EVGENII P. VDOVIN 229 Preface The papers of the present volume represent the proceedings of the Conference entitled "Ischia Group Theory 2004," which took place at the Jolly Hotel, Ischia (Naples, Italy), from March, 31st to April, 3rd 2004, in honour of MARCEL HERZOG. The organizing committee consisted of ZVI ARAD, MARIA- GRAZIA BIANCHI, PATRIZIA LONGOBARDI, MERCEDE MAJ and CARLO SCOPPOLA. The local committee was formed by COSTANTINO DELIZIA, CHIARA NICOTERA, CARMELA SICA and MARIA TOTA, all from the Universita di Salerno. The articles in this volume, most of which are close to areas of research of Marcel Herzog, are contributions by speakers and participants of the conference. The following topics are represented by the articles: combinatorial questions, such as counting the number of conjugacy classes of a group, or investigating the covering number of a simple group; the (sub)group structure of groups, such as recognizing the structure of a p-group when there are restrictions on certain of its subgroups; groups with a bound on the number of commutator in groups; classi- fication of Hall subgroups of almost simple groups; classification of groups with a CC-subgroup; algebras, mostly connected to or motivated from character theory; table algebras; group rings with a simple augmentation ideal; probabilistic methods (mainly for profinite groups); graphs with groups of automorphisms acting primi- tively on vertices; certain graphs connected with groups, such as the commutativity graph or the prime graph. A poster session on various topics connected with themes of the conference augmented the scientific program. It is our pleasure to thank the organizers and particularly our colleagues from the Universita di Salerno, who made our stay and work extremely pleasant; the referees of the papers for spending many hours reviewing papers and providing valuable feedback to the authors; the authors of all submitted papers for their con- tributions; the staff of the JOLLY HOTEL for having been very accommodating and supportive; the AMERICAN MATHEMATICAL SOCIETY and BAR-ILAN UNIVERSITY for their kind support for producing these proceedings in a professional and timely manner; MIRIAM BELLER for her expert assistance in compiling this volume; and, of course, MARCEL HERZOG for allowing us to celebrate with him. The Editors ix SPONSORS • Programma di Cofinanziamento MIUR "Teoria dei Gruppi e Applicazioni" • Programma di Cofinanziamento MIUR "Algebre di Lie graduate e pro-p- gruppi: rappresentazioni, periodicita e derivazioni" • Universita degli Studi di Salerno • Universita degli Studi di Milano • Universita degli Studi di L'Aquila • Dipartimento di Matematica e Informatica dell'Universita di Salerno • Dipartimento di Matematica "F. Enriques" dell'Universita di Milano • Facolta di Scienze Matematiche, Fisiche e Naturali dell'Universita di Salerno • Regione Campania • Provincia di Salerno xi MARCEL HERZOG Marcel Herzog was born in Krakow, Poland, on June 6, 1935. He spent the years of World War II in the Soviet Union. In 1946 he returned to Poland; and in 1948, he emigrated to Israel. From 1949 until1953, he worked days and studied in an evening high school. During the years 1953-1955, he served in the Israeli Army. He started his academic studies in 1955, at the Hebrew University in Jerusalem and in 1961 was awarded the M.Sc. in Mathematics. In 1962, he started his Ph.D. studies at Cornell University in Ithaca, N.Y, where his Ph.D. advisor was Professor Walter Feit. He was awarded the Ph.D. degree by Cornell in 1965. Professor Herzog's academic career started with a two year position at the University of Illinois in Champagne-Urbana, followed by one year spent at the University of California in Santa Barbara. In 1968, he returned to Israel, where he was appointed a Senior Lecturer in the Department of Mathematics of Tel-Aviv University. He was promoted there to Full Professor in 1980. During the years 1989- 1991, Professor Herzog served as the head of the School of Mathematical Sciences at Tel-Aviv University. He has supervised eight Ph.D. students, and the ninth is about to submit his Ph.D. thesis. Among his Ph.D. students were Zvi Arad, Arie Bialostocki, Arieh Lev and Gil Kaplan. Professor Herzog retired in 2003 and was appointed Professor Emeritus at Tel-Aviv University.
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