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 conference 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

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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; classification 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 contributions; 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 permutations 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 BIBLIOGRAPHY OF MARCEL HERZOG

21. (with J. Schonheim), Group partition, factorization and the vector covering problem, Canadian Math. Bull. 15 (1972), 207-214. 22. (with J. Schonheim), On certain sets of divisors of a number, Discrete Math. 1 (1972), 329-332. 23. (with J. Schonheim), The Br property and chromatic numbers of generalized graphs, J. Combin. Theory, Ser. B 12 (1972), 41-49. 24. Central 2-Sylow intersections, Pacific J. Math. 45 (1973), 535-538. 25. On simple groups with a cyclic maximal 2-Sylow intersection, Israel J. Math. 15 (1973), 350-355. 26. On Sylow intersections, Proc. Amer. Math. Soc. 37 (1973), 352-354. 27. Simple groups with cyclic central 2-Sylow intersections, J. Algebra 25 (1973), 307-312. 28. (with E. Shult), Groups with central 2-Sylow intersections of rank at most one, Proc. Amer. Math. Soc. 38 (1973), 465-470. 29. 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. 30. On simple groups with a quaternion maximal 2-Sylow intersection, Israel J. Math. 19 (1974), 225-227. 31. (with J. Schonheim), An observation and a problem concerning representation of a group as a union of cosets, Canadian Bull. Math. 17 (1974), 150. 32. 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 permutations, 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 maximal 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 BIBLIOGRAPHY OF MARCEL HERZOG

64. (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. 65. (with E. A. Bertram), Finite groups with large centralizers, Bull. Aust. Math. Soc. 32 (1985), 399-414. 66. (with Z. Arad and D. Chillag), Powers of characters of finite groups, J. Algebra 103 (1986), 241-255. 67. (with D. Chillag), On character degrees quotients, Arch. Math. 55 (1989), 25-29. 68. (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. 69. (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. 70. (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. 71. (with D. Chillag), On the length of the conjugacy classes of finite groups, J. Algebra 131 (1990), 110-125. 72. Conjugacy classes in finite groups, Rend. Semin. Mat. Fis. Milano 60 (1990)' 9-14. 73. (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 subgroups, 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 Welcome Greetings

Chair: W. HERFORT

09.30 M. ISAACS Numbers of classes of p--groups of given orders 10.20 Z. ARAD The history of the classification of finite groups with a CC-subgroup 11.30 D. RoBINSON Chain conditions for subgroups of infinite order or index 12.20 B. HUPPERT Character-degrees of simple groups

Chair: A. CARANTI

15.30 0. KEGEL Diagonal groups and algebras 16.20 A. CAMINA Implications of conjugacy class size for finite groups

Chair: A. CARANTI

17.30 P. NEUMANN Orbital equations in primitive permutation groups 18.20 G. FREIMAN The notion of isomorphism of subsets in additive number theory, group theory and probability theory 19.10 A. PREVITALI Toward a uniqueness proof of the sporadic Thompson group

Thursday, April 1

Chair: C. TIBILETTI MARCHIONNA

09.30 Z. JANKO New results in the theory of finite 2-groups 10.20 D. CHILLAG Algebras with positive bases, commutators and covering numbers 11.30 M. VAUGHAN-LEE 4-Engel groups 12.20 F. MENEGAZZO Complements in simple and almost simple groups

xxi xxii CONFERENCE PROGRAM

Chair: M. C. TAMBURINI

15.30 J. THOMPSON Finite projective planes 16.10 F. FUMAGALLI The Quillen complex of a solvable group 16.30 R. BRYANT Groups acting on free Lie algebras

Chair: G. TRAUSTASON

17.40 A. JUHASZ On elements of finite order in certain quotients of amalgamated free products and HNN-extentions 18.30 S. MATTAREI Lie algebras with a nonsingular derivation 19.00 H. HEINEKEN Central series and norm

Friday, April 2

Chair: L. DI MARTINO

15.30 A. RHEMTULLA Ordered groups in which all convex jumps are central 16.20 ED BERTRAM Conjugacy classes and centralizers in finite groups

Chair: M. BILIOTTI

17.30 H. SMITH Embedding groups in simple groups 18.20 A. BIALOSTOCKI On the Erdos-Ginzburg-Ziv theorem 19.10 N. GAVIOLI Just infinite periodic Lie algebras and pro-p- groups

Saturday, April 3

Chair: G. ZACHER

09.30 C. PRAEGER Limits of finite vertex-primitive graphs 10.20 A. LEV On the covering numbers of finite groups 11.30 A. REGEV On Hook number identities and some characters of the symmetric group 12.20 G. KAPLAN The existence of normal and characteristic subgroups in finite groups

Chair: M. HERZOG

15.30 A. LUCCHINI The X-Dirichlet polynomial of a finite group 16.20 E. SPINELLI Modular group algebras with maximal lower Lie- nilpotency indices 16.40 A. MANN Probabilistic Z-functions for profinite and arithmetic groups List of Participants

Erhard AICHINGER Alan CAMINA Johannes Kepler University Linz, University of East Anglia, U.K. Austria Andrea CARANTI Zvi ARAD Universita di Trento, Italy Bar-Ilan University and Netanya Francesco CATINO Academic College, Israel Universita di Leece, Italy Marina AVITABILE Rara CELENTANI U niversita di L' Aquila, Italy Universita di Napoli Federico II, Italy Yiftach BARNEA Vladimir CEPULIC Imperial College, U.K. University of Zagreb, Croatia Claudio BARTOLONE David CHILLAG Universita di Palermo, Italy Technion - Israel Institute of Technology, Israel Yakov BERKOVICH University of Haifa, Israel Marco COSTANTINI Universita di Trento, Italy Ed BERTRAM University of Hawaii, U.S.A. Mauro COSTANTINI Universita di Padova, Italy Arie BIALOSTOCKI University of Idaho, U.S.A. Francesca DALLA VOLTA Universita di Milano-Bicocca, Italy Mariagrazia BIANCHI Universita di Milano, Italy Erika DAMIAN Universita di Brescia, Italy Mauro BILIOTTI Alma D' ANIELLO Universita di Leece, Italy Universita di Napoli, Italy Adalbert BOVDI Ulderico DARDANO University of Debrecen, Hungary Universita di Napoli Federico II, Italy Victor BOVDI Costantino DELIZIA University of Debrecen, Hungary Universita di Salerno, Italy Roger BRYANT Eloisa DETOMI University of Manchester, U.K. Universita di Padova, Italy Daniela BUBBOLONI Lino DI MARTINO Universita di Firenze, Italy Universita di Milano-Bicocca, Italy

xxiii xxiv PARTICIPANTS

Cristina DI PIETRO Antonella LEONE Universita di L'Aquila, Italy Universita di Napoli Federico II, Italy Silvio DOLFI Arieh LEV Universita di Firenze, Italy The Academic College of Bayanbat ENKHGEREL Tel-Aviv-Jaffo, Israel Verebely L'aszl'o Technikum, Hungary Patrizia LONGOBARDI Clara FRANCHI Universita di Salerno, Italy Universita Cattolica del Sacra Cuore, Andrea LUCCHINI Italy Universita di Brescia, Italy Gregory FREIMAN Maria Silvia LUCIDO Tel-Aviv University, Israel Universita di Udine, Italy Francesco FUMAGALLI Universita di Firenze, Italy Mario MAIN ARDIS Universita di Udine, Italy Norberta GAVIOLI Universita di L'Aquila, Italy Mercede MAJ Universita di Salerno, Italy Anna GILLIO Universita di Milano, Italy A vinoam MANN The Hebrew University, Israel Theo GRUNDHOFER Universitiit Wiirzburg, Germany Sandra MATTAREI Universita di Trento, Italy Hermann HEINEKEN Universitiit Wiirzburg, Germany Peter MAYR Johannes Kepler Universitii Linz, Wolfgang HERFORT Austria Technische Universitiit Wien, Austria Marcel HERZOG Federico MENEGAZZO Tel Aviv University, Israel Universita di Padova, Italy Bertram HUPPERT Maddalena MICCOLI Universitiit Mainz, Germany Universita di Leece, Italy Marty ISAACS Giuseppe MICELLI University of Wisconsin-Madison, Universita di Leece, Italy U.S.A. Maria Elena MILITO PAGLIARA Zvonimir JANKO Universita di Padova, Italy Universitiit Heidelberg, Germany Valerio MONTI Arye JUHASZ Universita di Roma, Italy Technion - Israel Institute of Technology, Israel Alessandro MONTINARO Universita di Leece, Italy Gil KAPLAN The Academic College of Primoz MORAVEC Tel-Aviv-Jaffo, Israel IMPM, Ljubljana, Slovenia Otto KEGEL Marta MORICI Albert-Ludwigs Universitiit, Germany Universita di Bologna, Italy PARTICIPANTS XXV

Misha MUZYCHUK Cesarina TIBILETTI MARCHIONNA Netanya Academic College, Israel Universita di Milano, Italy Chiara NICOTERA Maria TOTA Universita di Salerno, Italy Universita di Salerno, Italy Emanuele PACIFIC! Gunnar TRAUSTASON Universita di Udine, Italy Lund University, Sweden Elisabetta PASTOR! Luca URSELLI Universita di Firenze, Italy Universita di Leece, Italy Cheryl PRAEGER Maria Alessandra VACCARO University of Western Australia, Universita di Palermo, Italy Australia Michael VAUGHAN-LEE Andrea PREVITALI Oxford University, U.K. Universita Insubria, Italy Evgenii VDOVIN Universita di Padova, Italy Amitai REGEV The Weizmann Institute of Science, Thomas WEIGEL Israel Universita di Milano-Bicocca, Italy Akbar RHEMTULLA Giovanni ZACHER University of Alberta, Canada Universita di Padova, Italy 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. 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, character theory, groups with special properties, table algebras, etc.

ISBN 0- 8218- 3711- 7

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