Isomorphisms of p-adic Group Rings

Abstract

A long-standing problem, first posed by Graham Higman [15] and later by Brauer [4] is the “isomorphism problem for integral group rings.” Given finite groups G and H, is it true that ZG = ZH implies G 2: H? Many authors have worked on this question, but progress has been difficult [30]. Perhaps the best positive result was that of Whitcomb in 1968 [37], who showed that the implication G = H holds for G metabelian. Dade [9] showed there were counterexamples, even in the metabelian case, if Z were replaced by the family of all fields. Some mathematicians came to believe the problem was a kind of grammatical accident, that counterexamples for Z were surely there, if difficult to find. We ourselves began this investigation looking for counterexamples, but found that they were indeed very difficult to find, much more difficult than we had anticipated. Slowly we began to believe that at least some exciting positive results were possible. In this paper we answer the isomorphism problem for finite p-groups over the p-adic integers Z, = Zcp), and in a very strong way: In the normalized units (augmentation 1) of Z pG there is only one con&gay cZu.ss of groups of order ] G 1. This answers the isomorphism problem in the affirmative (for finite p-groups G, over Z or Zp) and in addition computes the entire Picard group [2] of the category of Z,G-modules in terms of automorphisms of G. Similarly we are able to settle the isomorphism problem for finite nilpotent groups and compute the associated semilocal Picard groups. We also treat more general coefficient domains: namely, integral domains S of characteristic 0 in which no (rational) prime divisor of the group order is invertible, for the SG isomorphism problem, and treat similar semilocal Dedekind domains for the Picard group computations. The Zassenhaus conjecture concerning the rational behavior of group ring automorphisms is verified for the nilpotent case in this general setting (cf. Corollary 3 below). Over Z, we announce a positive answer to the isomorphism