Finite groups with isomorphic group algebras

Abstract

Introduction. Let G denote a finite group and F a field. The group algebra (or group ring) F(G) is the algebra over F with a basis multiplicatively isomorphic with G. Consider the following problem: If G is a group, F a field, find all groups H such that F(G) and F(H) are isomorphic over F. Perlis and Walker [9] solved this problem for Abelian groups over ordinary fields. Deskins [5] solved the problem for Abelian groups over modular fields. In this paper some partial results are obtained in connection with this problem for non-Abelian groups over an ordinary field. In ?1 we see some effects that certain subgroups of G and ideals of F(G) have on F(G). In ?2 a certain class of p-groups is studied. The theorem in ? 2 along with a theorem of S. D. Berman leads to an example of two distinct groups whose group algebras are isomorphic over any ordinary field. Group rings over the complex and real number fields and over the ring of integers are discussed in ? 3. Some examples, including the one mentioned above, are given in the last section. PRELIMINARY REMARKS. The following notations will be used. A (D B denotes the direct sum of algebras A and B over F. A ?F B =A ( B denotes the tensor product (direct product) of A and B over F. If K is an extension field of F, and if A is an algebra over F, then AK is the set K ?F A, considered as an algebra over K. G x H denotes the direct product of groups G and H.