Introduction

Let G be a finite group and R an integral domain of characteristic zero, in which no rational prime divisor of |G| is invertible, K is the field of fractions of R. By RG we denote the group ring of G over R, and e : RG → R is the augmentation.

The following is a conjecture of Hans Zassenhaus for R = Z:

(1.1) Let RG = RH as augmented algebras for a finite group H. Then there exists a unit a ∈ KG such that aHa−1 = G. (Such an element a would then automatically normalize RG.)

Since the hypotheses on R guarantee that finite subgroups in V(RG), the units of augmentation one in RG, are linearly independent over R [B], it is enough to assume in (1.1), that H is a finite subgroup in V(RG) with |G| = |H|.

The isomorphism problem asks,

(1.2) whether RG = RH implies that G and H are isomorphic.

A positive answer to the Zassenhaus conjecture would settle the isomorphism problem positively, but it also would give information about the embedding of H in V(RG).

If the isomorphism problem has a positive answer, then the Zassenhaus conjecture is equivalent to:

(1.3) Let a be an augmentation preserving automorphism of RG. Then a is the composition of an automorphism induced from a group automorphism followed by a central automorphism; i.e., an automorphism fixing the centre elementwise. (This is just the Skolem-Noether theorem.)