The Group of Units of the Integral Group Ring ZS3

Abstract

We denote by ZG the integral group ring of the finite group G. We call ±g, for g in G, a trivial unit of ZG. For G abelian, Higman [4] (see also [3, p. 262 ff]) showed that every unit of finite order in ZG is trivial. For arbitrary finite G (indeed, for a torsion group G, not necessarily finite), Higman [4] showed that every unit in ZG is trivial if and only if G is(i) abelian and the order of each element divides 4, or(ii) abelian and the order of each element divides 6, or(iii) the direct product of the quaternion group of order 8 and an abelian group of exponent 2.