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

Ischia Group Theory 2004

http://dx.doi.org/10.1090/conm/402

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 J. Bernstein, Tel-Aviv University

M. Katz, Bar-Ilan University

H. Furstenberg, Hebrew University S. Gelbart, Weizmann Institute

B. Pinchuk, Netanya Academic College S. Shnider, Bar-Ilan University 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.

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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

On the number of commutators in groups MARCEL HERZOG, PATRIZIA LONGOBARDI, and MERCEDE MAJ

New results in the theory of finite 2-groups ZVONIMIR JANKO

vii

171

181

193

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.

Professor Herzog has travelled extensively during his academic career. He spent the Winter Semester of 1971 as a Visiting Professor at Aarhus University in Denmark and the years 1974-1977 in Canberra, as a Senior Research Fellow at the Australian Institute of Advanced Studies. He was a Visiting Professor at the University of Southern California in Los Angeles during the academic year 1983-84; and between the years 1988-2001, he spent two and a half years as a Visiting Professor at the University of Hawaii in Honolulu.

Professor Herzog has published about 100 scientific papers, mainly in group theory, but also in combinatorics and number theory. His joint paper with Erdos and SchOnheim in 1970 gives him an Erdos number 1. Most of Professor Herzog's papers deal with finite group theory, with an emphasis on character theory and conjugacy classes. There are also numerous papers dealing with groups of permu-tations and the structure of simple groups. Since 1993, he has also published papers dealing with infinite groups satisfying some finiteness conditions.

Professor Herzog has collaborated and continues to collaborate with a great number of mathematicians from all over the world. His main collaborators are J. Schonheim, C.E. Praeger, K.B. Reid, T. Berger, D. Chillag, Z. Arad, E. Bertram, A. Mann, M. Bianchi, A. Gillio, C.M. Scoppola, P. Longobardi, M. Maj, G. Kaplan and A. Lev.

xiii

Bibliography of Marcel Herzog

1. A characterization of some projective special linear groups, J. Algebra 6 (1967), 305-308.

2. A characterization of the simple groups PSL(2,p),p > 3, Israel J. Math. 5 (1967), 79-85.

3. On finite groups which contain a Frobenius subgroup, J. Algebra 6 (1967), 192-221.

4. On finite group containing a CCT-subgroup with a cyclic Sylow subgroup, Pacific J. Math. 25 (1968), 523-531. .

5. On finite groups with cyclic Sylow subgroups for all odd primes, Israel J. Math. 6 (1968), 206-216.

6. On finite simple groups of order divisible hy three primes only, J. Algebra 10 (1968), 383-388.

7. On centralizers of involutions, Proc. Amer. Math. Soc. 22 (1969), 170-174.

8. On finite groups with independent cyclic Sylow subgroups, Pacific J. Math. 29 (1969), 285-293.

9. (with P. Erdos and J. Schonheim), An extremal problem ·an the set of noncoprime divisors of a number, Israel J. Math. 8 (1970), 408-412.

10. On a problem of E. Artin, J. Algebra 15 (1970), 408-416. 11. On finite groups with a cyclic Sylow subgroup, Illinois J. Math. 14 (1970),

188-193. 12. On groups of order 2"'3i3p-r with a cyclic Sylow 3-subgroups, Proc. Amer.

Math. Soc. 24 (1970), 116-118. 13. Finite groups with a large cyclic Sylow subgroup, Finite Simple Groups,

Chapter V, Academic Press, London, 1971, pp. 199-203. 14. (with R. Donagi), On the additive completion of polynomial sets of inte-

gers, J. Number Theory 3 (1971), 150-154. 15. CBB-groups involving no Suzuki groups, Pacific J. Math. 39 (1971), 687-

689. 16. Groups with strongly self-centralizing 3-centralizers, Israel J. Math. 9

(1971), 507-510. 17. Intersections of nilpotent Hall subgroups, Pacific J. Math. 36 (1971), 331-

333. 18. On Burnside's lemma, Proc. Amer. Math. Soc. 28 (1971), 379-380. 19. (with J. Schonheim), Linear and nonlinear single-error-correcting perfect

mixed codes, Inform. and Control18 (1971), 364-368. 20. On 2-Sylow intersections, Israel J. Math. 11 (1972), 326-327.

XV

xvi

21.

22.

23.

24. 25.

26. 27.

28.

29.

30.

31.

32.

BIBLIOGRAPHY OF MARCEL HERZOG

(with J. Schonheim), Group partition, factorization and the vector cover-ing problem, Canadian Math. Bull. 15 (1972), 207-214. (with J. Schonheim), On certain sets of divisors of a number, Discrete Math. 1 (1972), 329-332. (with J. Schonheim), The Br property and chromatic numbers of general-ized graphs, J. Combin. Theory, Ser. B 12 (1972), 41-49. Central 2-Sylow intersections, Pacific J. Math. 45 (1973), 535-538. On simple groups with a cyclic maximal 2-Sylow intersection, Israel J. Math. 15 (1973), 350-355. On Sylow intersections, Proc. Amer. Math. Soc. 37 (1973), 352-354. Simple groups with cyclic central 2-Sylow intersections, J. Algebra 25 (1973), 307-312. (with E. Shult), Groups with central 2-Sylow intersections of rank at most one, Proc. Amer. Math. Soc. 38 (1973), 465-470. The influence of 2-Sylow intersections on the structure of finite simple groups, Proceedings 2nd International Conference Theory of Groups, Can-berra 1973, Lecture Notes in Mathematics, vol. 372, 1974, pp. 361-365. On simple groups with a quaternion maximal 2-Sylow intersection, Israel J. Math. 19 (1974), 225-227. (with J. Schonheim), An observation and a problem concerning represen-tation of a group as a union of cosets, Canadian Bull. Math. 17 (1974), 150. On groups with extremal blocks, Bull. Aust. Math. Soc. 14 (1976), 325-330.

33. (with C. E. Praeger), Minimal degree of primitive permutation groups, Combin. Math. IV, Proceedings 4th Australian Conference, Adelaide 1975, Lecture Notes in Mathematics, vol. 560, 1976, pp. 116-122.

34. (with C. E. Praeger), On the fixed points of Sylow subgroups of transitive permutation groups, J. Aust. Math. Soc., Ser. A 21 (1976), 428-437.

35. (with C. E. Praeger), On the order of linear groups of fixed finite exponent, J. Algebra 43 (1976), 216-220.

36. (with K. Reid), Number of factors in k-cycle decompositions of permuta-tions, Combin. Math. IV, Proceedings 4th Australian Conference, Ade-laide 1975, Lecture Notes in Mathematics, vol. 560, 1976, pp. 123-131.

37. (with K. Reid), Representation of permutations as products of cycles of fixed length, J. Aust. Math. Soc., Ser. A 22 (1976), 321-331.

38. (with Z. Arad), A classification of groups with a centralizer condition. II, Bull. Aust. Math. Soc. 16 (1977), 55-60.

39. (with Z. Arad), A classification of groups with a centralizer condition. II: Corrigendum and addendum, Bull. Aust. Math. Soc. 17 (1977), 157-160.

40. (with Z. Arad), On fundamental subgroups of order divisible by three, Houston J. Math. 3 (1977), 309-313.

41. Counting group elements of order p modulo p2 , Proc. Amer. Math. Soc. 66 (1977), 247-250.

42. On linear relations between character values, J. Algebra 47 (1977), 154-161.

BIBLIOGRAPHY OF MARCEL HERZOG xvii

43. (with G. I. Lehrer), A note concerning Coxeter groups and permutations, Group Theory, Proceedings Miniconference Canberra 1975, Lecture Notes in Mathematics, vol. 573, 1977, pp. 53-56.

44. (with C. E. Praeger), On character values in finite groups, Bull. Aust. Math. Soc. 17 (1977), 451-461.

45. (with K. Reid), Permutation groups generated by cycles of fixed length, Israel J. Math. 26 (1977), 221-231.

46. (with D. Wright), Characterization of a family of simple groups by their character table, J. Aust. Math. Soc., Ser. A 24 (1977), 296-304.

47. (with T. R. Berger), Criteria for nonperfectness, Commun. Algebra 6 (1978), 959-968.

48. (with T. R. Berger), On characters in the principal2-block, J. Aust. Math. Soc., Ser. A 25 (1978), 264-268.

49. (with C. E. Praeger), Direct factors of Sylow groups, Commun. Algebra 6 (1978), 1375-1382.

50. (with C. E. Praeger), On the fixed points of Sylow subgroups of transitive permutation groups: Corrigendum, J. Aust. Math. Soc., Ser. A 26 (1978), 383-384.

51. (with K. Reid), Regularity in tournaments, Theor. Appl. Graphs, Pro-ceedings Kalamazoo 1976, Lecture Notes in Mathematics, vol. 642, 1978, pp. 442-453.

52. (with D. Chillag), Defect groups, trivial intersections and character tables, J. Algebra 61 (1979), 152-160.

53. On the classification of finite simple groups by the number of involutions, Proc. Amer. Math. Soc. 77 (1979), 313-314.

54. (with C. E. Praeger), On characters in the principal 2-block. II, J. Aust. Math. Soc., Ser. A 28 (1979), 100-106.

55. (with Z. Arad and A. Shaki), On finite groups with almost nilpotent max-imal subgroups, J. Algebra 65 (1980), 445-452.

56. (with Z. Arad and A. Shaki), On maximal subgroups with a nilpotent subgroup of index 2, Finite Groups, Santa Cruz Conference 1979, Proc. Symp. Pure Math. 37 (1980), 201-203.

57. (with A. R. Camina), Character tables determine Abelian Sylow 2-subgroups, Proc. Amer. Math. Soc. 80 (1980), 533-535.

58. Character tables, trivial intersections and number of involutions, Finite Groups, Santa Cruz Conference 1979, Proc. Symp. Pure Math. 37 (1980), 425-429.

59. (with D. Wright), Characterization of a family of simple groups by their character table. II, J. Aust. Math. Soc., Ser. A 30 (1980), 168-170.

60. (with Z. Arad and D. Chillag), Classification of finite groups by a maximal subgroup, J. Algebra 71 (1981), 235-244.

61. (with z. Arad and D. Chillag), On a problem of Probenius, J. Algebra 74 (1982), 516-523.

62. (with Z. Arad), Products of conjugacy classes in groups, Representations of Algebras, Proceedings 4th International Conference, Ottawa/Canada 1984, Vol. 2, Carleton-Ottawa Math. Lect. Note Ser. 2, Exp. 2, (1984).

63. (editor with Z. Arad), Products of Conjugacy Classes in Groups, Lecture Notes in Mathematics, vol. 1112, Springer-Verlag, 1985.

xviii

64.

65.

66.

67.

68.

69.

70.

71.

72.

73.

BIBLIOGRAPHY OF MARCEL HERZOG

(with Z. Arad and J. Stavi), Powers and products of conjugacy classes in groups, in Products of Conjugacy Classes in Groups, Lecture Notes in Mathematics, vol. 1112, Springer-Verlag, 1985, pp. 6-51. (with E. A. Bertram), Finite groups with large centralizers, Bull. Aust. Math. Soc. 32 (1985), 399-414. (with Z. Arad and D. Chillag), Powers of characters of finite groups, J. Algebra 103 (1986), 241-255. (with D. Chillag), On character degrees quotients, Arch. Math. 55 (1989), 25-29. (with E. A. Bertram and A. Mann), On a graph related to conjugacy classes of groups, Bull. Lond. Math. Soc. 22{6) (1990), 569-575. (with M. Bianchi, A. Gillio and B. Mauri), Groups with non-central classes of distinct lengths, Rend. Sci. Mat. Appl., A 124 (1990), 157-160. (with L. Brailovsky and G. Freiman), Special elements in groups, Group Theory, Proceedings 2nd International Conference, Bressanone/Italy 1989, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 23, 1990, pp. 33-42. (with D. Chillag), On the length of the conjugacy classes of finite groups, J. Algebra 131 (1990), 110-125. Conjugacy classes in finite groups, Rend. Semin. Mat. Fis. Milano 60 (1990)' 9-14. (with E. A. Bertram), Bounds on character degrees and class numbers of finite non-abelian simple groups, Groups, Vol. 1, Proceedings Interna-tional Conference, St. Andrews/UK 1989, London Math. Soc. Lecture Notes Ser., vol. 159, 1991, pp. 46-51.

74. (with E. A. Bertram), On medium-size subgroups and bases of finite groups, J. Combin. Theory, Ser. A 57 (1991), 1-14.

75. (with M. Bianchi, D. Chillag, A. Gillio, B. Mauri and C. M. Scoppola), Applications of a graph related to conjugacy classes in finite groups, Arch. Math. 58 (1992), no. 2, 126-132.

76. (withY. Berkovich and D. Chillag), Finite groups in which the degrees of the nonlinear irreducible characters are distinct, Proc. Amer. Math. Soc. 115 (1992), 955-959.

77. (with D. Chillag and A. Mann), On the diameter of a graph related to conjugacy classes of groups, Bull. Lond. Math. Soc. 25 (1993), no. 3, 255-262.

78. (with P. Longobardi and M. Maj), On a combinatorial problem in grqup theory, Israel J. Math. 82 (1993), 329-340.

79. (with 0. Manz), On the number of subgroups in finite solvable groups, J. Aust. Math. Soc., Ser. A 58 (1995), 134-141.

80. (with F. Menegazzo), On deficient products in infinite groups, Rend. Semin. Mat. Univ. Padova 93 (1995), 1-6.

81. (with C. M. Scoppola), On deficient squares groups and fully-independent subsets, Bull. Lond. Math. Soc. 27 (1995), 65-70.

82. (with E. Bertram), On regular bases of finite groups, Amer. Math. Monthly 103 (1996), 796-799.

83. (with S. Adami, M. Bianchi and G. B. Mauri), On graph-complete groups, Istit. Lombardo Accad. Sci. Lett. Rend. A 130 (1996), no. 1-2, (1997), 295-307.

BIBLIOGRAPHY OF MARCEL HERZOG xix

84. (with L. Brailovsky), Counting squares of two-subsets in finite groups, Ars Combin. 42 (1996), 207-210.

85. (with P. Longobardi, M. Maj and A. Mann), Finite groups containing many subgroups are solvable, J. Algebra 179 (1996), 894-904.

86. (with K. H. Dovermann), Gap conditions for representations of symmetric groups, J. Pure Appl. Algebra 119 (1997), 113-137.

87. Two new elementary results in finite groups, Proceedings of the First Jamaican Conference on Group Theory and its Applications (Kingston, 1996), pp. 64-66.

88. (with E. W. Ellers and N. Gordeev), Covering numbers for Chevalley groups, Israel J. Math. 111 (1999), 339-372.

89. Covering numbers for groups, Groups St. Andrews 1997 in Bath, Selected papers of the International Conference, Bath, UK, July 26-August 9, 1997, Vol. 1 (C. M. Campbell et al., eds.), Cambridge University Press, Lond. Math. Soc. Lect. Note Ser., vol. 260, 1999, pp. 353-355.

90. New results on subset multiplication in groups, Structure Theory of Set Addition (Jean-MarcDeshouillers et al., eds.), Societe Mathematique de France, Asterisque, vol. 258, 1999, pp. 309-315.

91. (with M. Bianchi, A. Gillio, B. Mauri and L. Verardi), On finite solvable groups in which normality is a transitive relation, J. Group Theory 3(2) (2000), 147-156.

92. (with P. Longobardi and M. Maj), On an ascending series of types of groups, Groups - Korea '98, Proceedings of the 4th International Confer-ence, Pusan, Korea, August 10-16, 1998 (Young Gheel Baik et al., eds.), Walter de Gruyter, 2000, pp. 181-187.

93. (with P. Longobardi, M. Maj and A. Mann), On generalized Dedekind groups and Tarski super monsters, J. Algebra 226 (2000), 690-713.

94. (with E. Bertram), Powers of cycle-classes in symmetric groups, J. Com-bin. Theory, Ser. A 94 (2001), 87-99.

95. (with G. Kaplan), Large cyclic subgroups contain non-trivial normal sub-groups, J. Group Theory 4 (2001), 247-253.

96. (with P. Longobardi and M. Maj), Elements of finite length in groups, J. Group Theory 5 (2002), 317-323.

97. (with E. A. Bertram and A. Lev), A correspondence between conjugacy classes and irreducible characters of finite groups, Algebra Colloq. 10 (2003), 205-208.

98. (with G. Kaplan and A. Lucchini), On subgroups containing non-trivial normal subgroups, Israel J. of Math. 137 (2003), 183-188.

99. (with P. Longobardi and M. Maj), Elements and groups of finite length, Groups St. Andrews 2001 in Oxford, Selected Papers from the Interna-tional Conference, Oxford, UK, August 5-18, 2001, Vol. I (C. M. Campbell et al., eds.), Cambridge University Press, Lond. Math. Soc. Lect. Note Ser., vol. 304, 2003, pp. 249-255.

100. (with P. Longobardi and M. Maj), Groups with all elements of finite length, J. Algebra 263 (2003), 177-187.

101. (with P. Csorgo), On supersolvable groups and the nilpotator, Commun. Algebra 32 (2004), 609-620.

xx BIBLIOGRAPHY OF MARCEL HERZOG

102. (with G. Kaplan and A. Lev), On the commutator and the center of finite groups, J. Algebra 278 (2004), 494-501.

103. (with G. Kaplan and A. Lev), Representation of permutations as products of two cycles, Discrete Math. 285 (2004), 323-327.

104. (with M. Bianchi, A. Gillio, G. Qian and W. Shi), Characterization of non-nilpotent groups with two irreducible character degrees, J. Algebra 284 (2005), 326-332.

105. (with P. Longobardi and M. Maj), On the number of commutators in groups, this volume.

Conference Program

Wednesday, March 31

09.20

Chair: W. HERFORT

09.30 M. ISAACS 10.20 Z. ARAD

11.30 D. RoBINSON

12.20 B. HUPPERT

Chair: A. CARANTI

15.30 0. KEGEL 16.20 A. CAMINA

Chair: A. CARANTI

17.30 18.20

19.10

P. NEUMANN G. FREIMAN

A. PREVITALI

Thursday, April 1

Welcome Greetings

Numbers of classes of p--groups of given orders The history of the classification of finite groups with a CC-subgroup Chain conditions for subgroups of infinite order or index Character-degrees of simple groups

Diagonal groups and algebras Implications of conjugacy class size for finite groups

Orbital equations in primitive permutation groups The notion of isomorphism of subsets in additive number theory, group theory and probability the-ory Toward a uniqueness proof of the sporadic Thompson group

Chair: C. TIBILETTI MARCHIONNA

09.30 10.20

11.30 12.20

Z. JANKO D. CHILLAG

M. VAUGHAN-LEE F. MENEGAZZO

New results in the theory of finite 2-groups Algebras with positive bases, commutators and covering numbers 4-Engel groups Complements in simple and almost simple groups

xxi

xxii CONFERENCE PROGRAM

Chair: M. C. TAMBURINI

15.30 J. THOMPSON 16.10 F. FUMAGALLI 16.30 R. BRYANT

Chair: G. TRAUSTASON

17.40 A. JUHASZ

18.30 S. MATTAREI 19.00 H. HEINEKEN

Friday, April 2

Chair: L. DI MARTINO

15.30 A. RHEMTULLA

16.20 ED BERTRAM

Chair: M. BILIOTTI

17.30 H. SMITH 18.20 A. BIALOSTOCKI 19.10 N. GAVIOLI

Saturday, April 3

Chair: G. ZACHER

09.30 10.20 11.30

12.20

C. PRAEGER A. LEV A. REGEV

G. KAPLAN

Chair: M. HERZOG

15.30 16.20

16.40

A. LUCCHINI E. SPINELLI

A. MANN

Finite projective planes The Quillen complex of a solvable group Groups acting on free Lie algebras

On elements of finite order in certain quotients of amalgamated free products and HNN-extentions Lie algebras with a nonsingular derivation Central series and norm

Ordered groups in which all convex jumps are cen-tral Conjugacy classes and centralizers in finite groups

Embedding groups in simple groups On the Erdos-Ginzburg-Ziv theorem Just infinite periodic Lie algebras and pro-p-groups

Limits of finite vertex-primitive graphs On the covering numbers of finite groups On Hook number identities and some characters of the symmetric group The existence of normal and characteristic sub-groups in finite groups

The X-Dirichlet polynomial of a finite group Modular group algebras with maximal lower Lie-nilpotency indices Probabilistic Z-functions for profinite and arith-metic groups

List of Participants

Erhard AICHINGER Johannes Kepler University Linz, Austria

Zvi ARAD Bar-Ilan University and Netanya Academic College, Israel

Marina AVITABILE U niversita di L' Aquila, Italy

Yiftach BARNEA Imperial College, U.K.

Claudio BARTOLONE Universita di Palermo, Italy

Yakov BERKOVICH University of Haifa, Israel

Ed BERTRAM University of Hawaii, U.S.A.

Arie BIALOSTOCKI University of Idaho, U.S.A.

Mariagrazia BIANCHI Universita di Milano, Italy

Mauro BILIOTTI Universita di Leece, Italy

Adalbert BOVDI University of Debrecen, Hungary

Victor BOVDI University of Debrecen, Hungary

Roger BRYANT University of Manchester, U.K.

Daniela BUBBOLONI Universita di Firenze, Italy

Alan CAMINA University of East Anglia, U.K.

Andrea CARANTI Universita di Trento, Italy

Francesco CATINO Universita di Leece, Italy

Rara CELENTANI Universita di Napoli Federico II, Italy

Vladimir CEPULIC University of Zagreb, Croatia

David CHILLAG Technion - Israel Institute of Technology, Israel

Marco COSTANTINI Universita di Trento, Italy

Mauro COSTANTINI Universita di Padova, Italy

Francesca DALLA VOLTA Universita di Milano-Bicocca, Italy

Erika DAMIAN Universita di Brescia, Italy

Alma D' ANIELLO Universita di Napoli, Italy

Ulderico DARDANO Universita di Napoli Federico II, Italy

Costantino DELIZIA Universita di Salerno, Italy

Eloisa DETOMI Universita di Padova, Italy

Lino DI MARTINO Universita di Milano-Bicocca, Italy

xxiii

xxiv PARTICIPANTS

Cristina DI PIETRO Universita di L'Aquila, Italy

Silvio DOLFI Universita di Firenze, Italy

Bayanbat ENKHGEREL Verebely L'aszl'o Technikum, Hungary

Clara FRANCHI Universita Cattolica del Sacra Cuore, Italy

Gregory FREIMAN Tel-Aviv University, Israel

Francesco FUMAGALLI Universita di Firenze, Italy

Norberta GAVIOLI Universita di L'Aquila, Italy

Anna GILLIO Universita di Milano, Italy

Theo GRUNDHOFER Universitiit Wiirzburg, Germany

Hermann HEINEKEN Universitiit Wiirzburg, Germany

Wolfgang HERFORT Technische Universitiit Wien, Austria

Marcel HERZOG Tel Aviv University, Israel

Bertram HUPPERT Universitiit Mainz, Germany

Marty ISAACS University of Wisconsin-Madison, U.S.A.

Zvonimir JANKO Universitiit Heidelberg, Germany

Arye JUHASZ Technion - Israel Institute of Technology, Israel

Gil KAPLAN The Academic College of Tel-Aviv-Jaffo, Israel

Otto KEGEL Albert-Ludwigs Universitiit, Germany

Antonella LEONE Universita di Napoli Federico II, Italy

Arieh LEV The Academic College of Tel-Aviv-Jaffo, Israel

Patrizia LONGOBARDI Universita di Salerno, Italy

Andrea LUCCHINI Universita di Brescia, Italy

Maria Silvia LUCIDO Universita di Udine, Italy

Mario MAIN ARDIS Universita di Udine, Italy

Mercede MAJ Universita di Salerno, Italy

A vinoam MANN The Hebrew University, Israel

Sandra MATTAREI Universita di Trento, Italy

Peter MAYR Johannes Kepler Universitii Linz, Austria

Federico MENEGAZZO Universita di Padova, Italy

Maddalena MICCOLI Universita di Leece, Italy

Giuseppe MICELLI Universita di Leece, Italy

Maria Elena MILITO PAGLIARA Universita di Padova, Italy

Valerio MONTI Universita di Roma, Italy

Alessandro MONTINARO Universita di Leece, Italy

Primoz MORAVEC IMPM, Ljubljana, Slovenia

Marta MORICI Universita di Bologna, Italy

PARTICIPANTS XXV

Misha MUZYCHUK Netanya Academic College, Israel

Chiara NICOTERA Universita di Salerno, Italy

Emanuele PACIFIC! Universita di Udine, Italy

Elisabetta PASTOR! Universita di Firenze, Italy

Cheryl PRAEGER University of Western Australia, Australia

Andrea PREVITALI Universita Insubria, Italy

Amitai REGEV The Weizmann Institute of Science, Israel

Akbar RHEMTULLA University of Alberta, Canada

Derek ROBINSON University of Illinois, U.S.A.

Paolo SANTANIELLO Seconda Universita di Napoli, Italy

Carlo Maria SCOPPOLA Universita di L'Aquila, Italy

Luigi SERENA Universita di Firenze, Italy

Carmela SICA Universita di Salerno, Italy

Salvatore SICILIANO Universita di Leece, Italy

Howard SMITH Bucknell University, U.S.A.

Ernesto SPINELLI Universita di Leece, Italy

Chiara TAMBURINI Universita Cattolica del Sacra Cuore, Italy

John THOMPSON University of Florida, U.S.A.

Cesarina TIBILETTI MARCHIONNA Universita di Milano, Italy

Maria TOTA Universita di Salerno, Italy

Gunnar TRAUSTASON Lund University, Sweden

Luca URSELLI Universita di Leece, Italy

Maria Alessandra VACCARO Universita di Palermo, Italy

Michael VAUGHAN-LEE Oxford University, U.K.

Evgenii VDOVIN Universita di Padova, Italy

Thomas WEIGEL Universita di Milano-Bicocca, Italy

Giovanni ZACHER Universita di Padova, Italy

Titles in This Series

402 Zvi Arad, Mariagrazia Bianchi, Wolfgang Herfort, Patrizia Longobardi, Mercede Maj, and Carlo Scoppola, Editors, Ischia group theory 2004, 2006

401 Katrin Becker, Melanie Becker, Aaron Bertram, PaulS. Green, and Benjamin McKay, Editors, Snowbird lectures on string geometry, 2006

400 Shiferaw Berhanu, Hua Chen, Jorge Hounie, Xiaojun Huang, Sheng-Li Tan, and Stephen S.-T. Yau, Editors, Recent progress on some problems in several complex variables and partial differential equations, 2006

399 Dominique Arlettaz and Kathryn Hess, Editors, An Alpine anthology of homotopy theory, 2006

398 Jay Jorgenson and Lynne Walling, Editors, The ubiquitous heat kernel, 2006 397 Jose M. Munoz Porras, Sorin Popescu, and Rubi E. Rodriguez, Editors, The

geometry of Riemann surfaces and Abelian varieties, 2006 396 Robert L. Devaney and Linda Keen, Editors, Complex dynamics: Twenty-five years

after the appearance of the Mandelbrot set, 2006 395 Gary R. Jensen and Steven G. Krantz, Editors, 150 Years of Mathematics at

Washington University in St. Louis, 2006 394 Rostislav Grigorchuk, Michael Mihalik, Mark Sapir, and Zoran Sunik, Editors,

Topological and asymptotic aspects of group theory, 2006 393 Alec L. Matheson, Michael I. Stessin, and Richard M. Timoney, Editors, Recent

advances in operator-related function theory, 2006 392 Stephen Berman, Brian Parshall, Leonard Scott, and Weiqiang Wang, Editors,

Infinite-dimensional aspects of representation theory and applications, 2005 391 Jiirgen Fuchs, Jouko Mickelsson, Grigori Rozenblioum, Alexander Stolin,

and Anders Westerberg, Editors, Noncommutative geometry and representation theory in mathematical physics, 2005

390 Sudhir Ghorpade, Hema Srinivasan, and Jugal Verma, Editors, Commutative algebra and algebraic geometry, 2005

389 James Eells, Etienne Ghys, Mikhail Lyubich, Jacob Palis, and Jose Seade, Editors, Geometry and dynamics, 2005

388 Ravi Vakil, Editor, Snowbird lectures in algebraic geometry, 2005 387 Michael Entov, Yehuda Pinchover, and Michah Sageev, Editors, Geometry,

spectral theory, groups, and dynamics, 2005 386 Yasuyuki Kachi, S. B. Mulay, and Pavlos Tzermias, Editors, Recent progress in

arithmetic and algebraic geometry, 2005 385 Sergiy Kolyada, Yuri Manin, and Thomas Ward, Editors, Algebraic and topological

dynamics, 2005 384 B. Diarra, A. Escassut, A. K. Katsaras, and L. Narici, Editors, Ultrametric

functional analysis, 2005 383 Z.-C. Shi, Z. Chen, T. Tang, and D. Yu, Editors, Recent advances in adaptive

computation, 2005 382 Mark Agranovsky, Lavi Karp, and David Shoikhet, Editors, Complex analysis and

dynamical systems II, 2005 381 David Evans, Jeffrey J. Holt, Chris Jones, Karen Klintworth, Brian Parshall,

Olivier Pfister, and Harold N. Ward, Editors, Coding theory and quantum computing, 2005

380 Andreas Blass and Yi Zhang, Editors, Logic and its applications, 2005 379 Dominic P. Clemence and Guoqing Tang, Editors, Mathematical studies in

nonlinear wave propagation, 2005 378 Alexandre V. Borovik, Editor, Groups, languages, algorithms, 2005

TITLES IN THIS SERIES

377 G. L. Litvinov and V. P. Maslov, Editors, Idempotent mathematics and mathematical physics, 2005

376 Jose A. de Ia Peiia, Ernesto Vallejo, and Natig Atakishiyev, Editors, Algebraic structures and their representations, 2005

375 Joseph Lipman, Suresh Nayak, and Prarnathanath Sastry, Variance and duality for cousin complexes on formal schemes, 2005

374 Alexander Barvinok, Matthias Beck, Christian Haase, Bruce Reznick, and Volkmar Welker, Editors, Integer points in polyhedra-geometry, number theory, algebra, optimization, 2005

373 0. Costin, M.D. Kruskal, and A. Macintyre, Editors, Analyzable functions and applications, 2005

372 Jose Burillo, Sean Cleary, Murray Elder, Jennifer Taback, and Enric Ventura, Editors, Geometric methods in group theory, 2005

371 Gui-Qiang Chen, George Gasper, and Joseph Jerome, Editors, Nonlinear partial differential equations and related analysis, 2005

370 Pietro Poggi-Corradini, Editor, The p-harmonic equation and recent advances in analysis, 2005

369 Jaime Gutierrez, Vladimir Shpilrain, and Jie-Tai Yu, Editors, Affine algebraic geometry, 2005

368 Sagun Chanlllo, Paulo D. Cordaro, Nicholas Hanges, Jorge Hounie, and Abdelhamid Meziani, Editors, Geometric analysis of PDE and several complex variables, 2005

367 Shu-Cheng Chang, Bennett Chow, Sun-Chin Chu, and Chang-Shou Lin, Editors, Geometric evolution equations, 2005

366 Bernheim Boof3-Bavnbek, Gerd Grubb, and Krzysztof P. Wojciechowski, Editors, Spectral geometry of manifolds with boundary and decompositon of manifolds, 2005

365 Robert S. Doran and Richard V. Kadison, Editors, Operator algebras, quantization, and non-commutative geometry, 2004

364 Mark Agranovsky, Lavi Karp, David Shoikhet, and Lawrence Zalcman, Editors, Complex analysis and dynamical systems, 2004

363 Anthony To-Ming Lau and Volker Runde, Editors, Banach algebras and their applications, 2004

362 Carlos Concha, Raul Manasevich, Gunther Uhlmann, and MichaelS. Vogelius, Editors, Partial differential equations and inverse problems, 2004

361 Ali Enayat and Roman Kossak, Editors, Nonstandard models of arithmetic and set theory, 2004

360 Alexei G. Myasnikov and Vladimir Shpilrain, Editors, Group theory, statistics, and cryptography, 2004

359 S. Dostoglou and P. Ehrlich, Editors, Advances in differential geometry and general relativity, 2004

358 David Burns, Christian Popescu, Jonathan Sands, and David Solomon, Editors, Stark's Conjectures: Recent work and new directions, 2004

357 John Neuberger, Editor, Variational methods: open problems, recent progress, and numerical algorithms, 2004

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstoref.

Experts in the theory of finite groups and in representation theory provide insight into various aspects of group theory, such as the classification of finite simple groups, char-acter theory, groups with special properties, table algebras, etc.

ISBN 0- 8218- 3711- 7

9 780821 837115