On Class Sums in p-Adic Group Rings

Abstract

In this note we prove that an isomorphism of p-adic group rings of finite p-groups maps class sums onto class sums. For integral group rings this is a well known theorem of Glauberman (see [3; 7]). As an application, we show that any automorphism of the p-adic group ring of a finite p-group of nilpotency class 2 is composed of a group automorphism and a conjugation by a suitable element of the p-adic group algebra. This was proved for integral group rings of finite nilpotent groups of class 2 in [5]. In general this question remains open. We also indicate an extension of a theorem of Passman and Whitcomb. The following notation is used.G denotes a finite p-group.Z denotes the ring of (rational) integers.ZP denotes the ring of p-adic integers.Qp denotes the p-adic number field.