On the characterization of finite groups by characters

Summary

Abstract

The first object is a survey on the isomorphism problem of integral group rings and the Zassenhaus conjectures related to this question. Then it is shown how automorphisms of character tables may be used to get information about automorphisms of integral group rings. This permits the proof of one of the Zassenhaus conjectures for series of simple or almost simple groups. Finally a short proof is given that {1, 2, 3}-characters determine a finite group up to isomorphism.

Introduction

R. Brauer in his famous lectures in 1963 on representations of finite groups posed more than 40 problems which have had a big influence on this topic since then [7]. Many of these problems concern the question of which properties of a finite group G are reflected by its character table. The basic problem is to determine what kind of information about G additional to its character table is needed in order to determine G up to isomorphism. We shall consider two aspects of these questions.

The first one is the isomorphism problem of integral group rings of finite groups. This is an old and still open problem. It was considered for the first time in G. Higman's thesis in 1939 [38, p. 100]. The question is whether ℤG ≅ ℤH implies that G ≅ H. It is a result of G. Glauberman that finite groups G and H with isomorphic integral group rings ℤG and ℤH have the same character table [16, (3.17)]. Thus the isomorphism problem fits well into the context of Brauer's problems. Note that K. W. Roggenkamp and A. Zimmermann recently constructed two non-isomorphic infinite polycyclic groups G and H whose integral group rings are Morita equivalent.