Lecture Notes in

Edited by A. Oold and B. Eckmann

1185

Group Theory, Beijing 1984 Proceedings of an International Symposium held in Beijing, Aug. 27-Sep. 8, 1984

Edited by Tuan Hsio-Fu

Springer-Verlag Berlin Heidelberg New York Tokyo Editor

TUAN Hsio-Fu Department of Mathematics, Peking University Beijing, The People's Republic of China

Mathematics Subject Classification (1980): 05-xx, 12F-xx, 14Kxx, 17Bxx, 20-xx

ISBN 3-540-16456-1 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-16456-1 Springer-Verlag New York Heidelberg Berlin Tokyo

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1986 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210 PREFACE

From August 27 to September 8, 1984 there was held in Peking Uni• versity, Beijing an International Symposium on Theory. As well said by Hermann Wey1: "Symmetry is a vast subject signi• ficant in art and nature. Whenever you have to do with a structure endowed entity, try to determine the group of those transformations which leave all structural relations undisturbed." This passage underlies that the group concept is one of the most fundamental and most important in modern mathematics and its applications. Group theory is indeed a vast branch of mathematics. The chief topics of the Symposium were finite groups and their connections with , classical groups, alge• braic groups and Lie groups (with emphasis on connections with algebra) . The Symposium was sponsored by our Ministry of Education. It was conducted by an Organization Committee consisting of seven Chinese pro• fessors of group theory: Tuan Hsio Fu (Duan Xuefu) (Peking University), Chairman Cao Xihua (East China Normal University), Vice Chairman Wan Zhexian (Institute of System Science, Academia Sinica), Vice Chairman Yen Zhida (Nankai University) Zhang Yuanda (Wuhan University) Zeng Kencheng (Graduate School, China University of Science and Technology) Wang Efang (Peking University) Professor Hua Loo­Keng and Professor S.S.Chern kindly gave their advice and support to the Symposium. The purpose of the Symposium was to provide a timely forum for the exchange of information and encouragement of international dialogue. For this purpose, we invited eight well­known from abroad: John G. Thompson (University of Cambridge, U.K.) Michael Aschbacher (California Institute of Technology, U.S.A.) Charles W. Curtis (University of Oregon, U.S.A.) Donald Livingstone (University of Birmingham, U.K.) Alexander Hahn (University of Notre Dame, U.S.A.) E.R. Kolchin (Columbia University, U.S.A.) T.A. Springer (University of Utrecht, Netherlands) J.C. Jantzen (University of Bonn; University of Hamburg, FRG) Each of them delivered a series of six one­hour lectures presenting an exposition and (or) providing an overview of recent advances on topics of current interest. IV

All together, there were fifty registered Chinese participants and twenty others. In general, registered participants and a few others gave a lecture (of half an hour to one hour) about their recent works either in a general meeting or in one of three sessions, emphasizing respectively on finite groups and combinatorics, classical groups, algebraic groups and Lie groups. In total, forty five papers were pre• sented, including six joint works and two papers not read in person. On invitation, Professor Hua Loo­Keng gave an one­hour lecture entitled Non­negative square matrices and planning economy. In the Proceedings of the Symposium, contributions of seven of our eight foreign guests form Part A, while 9 papers of Chinese participants form Part B. In order to keep the Volume with a reasonable size, other papers of our Chinese participants are not included, and most of these have already appeared or will appear in other mathematical journals. However, a list of other Chinese participants with their affiliations and titles of their lectures will be given as an appendix for reference. It is believed that the Symposium has achieved its purpose of inter• national academic communication and would promote the research of group theory in China. Thanks are due to Springer­Verlag for consenting to publish the Proceedings of the Symposium as a Volume in Lecture Notes in Mathematics.

Hsio­Pu Tuan (Duan Xuefu) June 12, 1985 TABLE OF CONTENTS

Preface Part A Michael Aschbacher Finite simple groups and their subgroups. Charles W. Curtis Topics in the theory of representations of finite groups. 58 A. Hahn Algebraic K-theory, Morita theory and the classical groups. 88 J. C. Jantzen Modular represencations of reductive groups. 118 E. R. Kolchin Differential algebraic groups. 155 T. A. Springer Conjugacy classes in algebraic groups. 175 John G. Thompson Some finite groups which appear as Gal L/K where Regular Galois extensions of Q (x); Bilinear forms in characteristic p and the Frobenius-Schur indicator. 210

Part B Chen Yu Homomorphisms from linear groups over division rings to algebraic groups. 231 Chen Zhijie A prehomogeneolls vector space of characteri- stic 3. 266 Dong Chongying On full subgroups of twisted groups. 277 Huang Jianhua A characterization of the 2-local subgroups in and Higman-Sims group. 289 Li Huiling On 2-connectedness of geometries. 308

Shi Shengming On the number of p-blocks with given defect groups and some applications of the p-power homomorphism. 314 Wang Efang The equi-type and quasiequi-type decompositions of arbitrary permutations by permutations of order 2 or 3. 340 Wu Xiaolong On the extensions of abelian varieties by affine groups schemes. 361 Ye Jiashen On the first Cartan invariant of the groups SL (3 , pn) and SU (3 , pn) . 388 Appendix List of other Chinese participants and titles of their lectures. 401