http://dx.doi.org/10.1090/pspum/001

FINITE GROUPS PROCEEDINGS OF A SYMPOSIUM IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY

Held in New York April 23-24,1959

Cosponsored by THE INSTITUTE FOR DEFENSE ANALYSIS under contract Nonr 2631 (00) with the Office of Naval Research

Editorial Committee A. A. Albert Irving Kaplansky PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS

VOLUME I

FINITE GROUPS

AMERICAN MATHEMATICAL SOCIETY

PROVIDENCE, RHODE ISLAND 1959 Prepared by the American Mathematical Society under Contract Nonr 2631(00) with the Office of Naval Research

International Standard Serial Number 0082-0717 International Standard Book Number 0-8218-1401-X Library of Congress Catalog Number 50-1183 Copyright © 1959 by the American Mathematical Society Third printing, 1979 Printed in the United States of America

All rights reserved except those granted to the United States Government May not be reproduced in any form without permission of the publishers INTRODUCTION

This volume consists of eleven articles each corresponding to one of the eleven addresses given at the FINITE GROUPS SYMPOSIUM which took place at the Hotel New Yorker on April 23 and 24, 1959. The three sessions of the SYMPOSIUM were presided over by A. A. Albert, I. N. Herstein, and I. Kaplansky and formal discussion was led by Olga Taussky and L. J. Paige, by G. de B. Robinson and D. G. Higman, and by Charles Curtis and Irving Reiner. The table of contents provides a list of the ad• dresses in the order in which they took place.

The sessions were well attended and the discussion was lively. The SYMPOSIUM was a well deserved recognition of an enormous renewed interest in one of the oldest branches of algebra. The three keynote ad• dresses, by John Thompson, Marshall Hall, and Michio Suzuki, called at• tention to major new results in the field and the Symposium should serve to stimulate research activity in the Theory of Groups, one of the most beautiful subjects in Mathematics.

The mathematical community greatly appreciates the support which was given to this SYMPOSIUM by the Communications Division of the In• stitute for Defense Analyses.

A. A. Albert, Symposium Chairman

v CONTENTS

Page

Introduction v

Finite groups with normal p-complements By John G. Thompson 1 Burnside groups and Engel rings By Roger C. Lyndon 4

On the structure of certain solvable groups By Daniel Gorenstein 15

On groups which contain Frobenius groups as subgroups By Walter Feit 22

Current studies on permutation groups By Marshall Hall, Jr 29

Review of some results in collineation groups By Daniel R. Hughes 42

Some finite groups with geometrical properties By Wilhelm Magnus 56

Symmetrical definitions for the binary polyhedral groups By H. S. M. Coxeter 64

Applications of group characters By Michio Suzuki 88

On maximal subgroups By W. E. Deskins 100

On an application of the theory of Lie algebras to group theory. By Hans Zassenhaus 105

vii INDEX

ABA-groups, 15 dicyclic, 66 AMITSUR, S. A., 86 dihedral, 65 Andre V-W planes, 48 finite, 1 Autotopism, 46 generalized quaternion, 66 icosahedral, 65 Binary icosahedral group, 83 Mathieu, 32, 37 Binary octahedral group, 79 metanilpotent, 99 Binary tetrahedral group, 78, 81 octahedral, 65 plane projective, 56 Central collineation, 43 polyhedral, 64 Central expansions, 104 quadruply transitive, 37 Characters, irreducible, 87 quaternion, 78 of subgroups, 87 regular, 18 CLIFFORD, W. K., 72 solvable, 15 Closed subset, 88 supersolvable, 99 Commutator group, 84 tetrahedral, 65 Continuous groups, 68 zero-dimensional, 56 COXETER, H. S. M., 64 Cross polytope, 69 Hall V-W planes, 48 Cyclic group, 79 HAMILTON, W. R., 66 HIGMAN, G., 39 DEHN, M., 56 Homology, 43 Dickson rings, 50 HUPPERT, B., 39 Dicyclic group, 66 HURWITZ, A., 81 Dihedral group, 65 Hyperbolic dedecahedron space, 78 Dimension of a geometry, 58 DIRICHLET, G. L., 70 Icosahedral group, 65 Division ring, 45 Index, complex, 102 Index, normal, 101 Engel condition, 4 Integral quaternions, 81 FEIT, W., 38 KLEIN, F., 87 Finite groups with normal LAVES, F., 87 p-complements, 1 Lens, 70 FOWLER, K. A., 93 Lens space, 76 Frattini subgroup, 100 FROBENIUS, G., 38 Mathieu groups, 32, 37 Frobenius group, 16, 22, 27 Metanilpotent group, 99 kernel, 32 MOSER, W. O. J., 87 problem, 33 reciprocity law, 87, 96 Near-field, 32, 49 theorem, 59 planes, 49 NEUMANN, B. H., 79 Generalized quaternion group, 66 Normal index, 101 Gleason-Andre theorem, 52 Golden section, 84 Octahedral group, 65 Group binary, 78, 79, 81, 83 PARKER, E., 33 characters, 87 Perspectivity, 43 commutator, 84 Plane projective group, 56 continuous, 68 Polyhedral group, 64 cyclic, 79 Projective planes, 42, 48, 49

109 Quadruply transitive group, 37 TITS, J., 38 Quaternion, 65 Trace bilinear form, 104 group, 78 Transfer map, 17 Translations, 43 Radical of a form, 104 Trefoil knot, 81 Regular (#> -group, 18 Truncation, 70 poly tope, 69 Twisted field planes, 49 subgroup, 22 Unit quaternion, 67 SCHLEGEL, V., 67 SEIFERT, H., 75, 78 Semi-nuclear rings, 49 Veblen-Wedderburn planes, 48 Singer's theorem, 52 systems, 45 Solvable group, 150 Spherical dodecahedron space, 78 VINCENT, G., 87 Spherical honeycomb, 70 VOR6NOI, G., 70 S-ring, 30 Steiner triple system, 40 WALL, G. E., 24 SUZUKI, M., 25 WIELANDT, H., 33, 39 WYTHOFF, W. A., 70 Tetrahedral group, 65 THRELFALL, W., 75, 78 ZASSENHAUS, H., 25 TIETZE, H. 76 Zero dimensional group, 56

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