Cliff–Weiss inequalities and the Zassenhaus Conjecture

Abstract

Let N be a nilpotent normal subgroup of the finite group G. Assume that u is a unit of finite order in the integral group ring ZG of G which maps to the identity under the linear extension of the natural homomorphism G→G/N. We show how a result of Cliff and Weiss can be used to derive linear inequalities on the partial augmentations of u and apply this to the study of the Zassenhaus Conjecture. This conjecture states that any unit of finite order in ZG is conjugate in the rational group algebra of G to an element in ±G.