On the isomorphism of group algebras

Abstract

Let R (G) be the group ring of a finite group G with coefficients from a ring R. The isomorphism problem of group rings, namely, when for a fixed ring Rand two finite groups G and H if R(G) and R(H) are isomorphic so are G and H, remains unsettled in general, even when R is restricted to be the ring of rational integers Z. In case G is abelian, and R is Z, Q (the rational number field), Zp (the field of p elements) or OE (the ring of algebraic integers in an algebraic number field E), the problem has been solved by HIGMAN[7], PERLISand WALKER[8], DESKINS[5] and COHNand LIVINGSTONE [2] respectively. If G is not assumed to be abelian some progress has been made by COLEMAN[3] and COHNand LIVINGSTONE [2]. In this paper we study group algebras of groups with