# Cliff-Weiss Inequalities and the Zassenhaus Conjecture

## TL;DR

This paper uses inequalities derived from Cliff and Weiss's work to analyze the Zassenhaus Conjecture, which concerns the conjugacy of units of finite order in integral group rings of finite groups.

## Contribution

It applies Cliff and Weiss's inequalities to advance understanding of the Zassenhaus Conjecture in the context of finite groups with nilpotent normal subgroups.

## Key findings

- Derived linear inequalities on partial augmentations of units
- Applied inequalities to study the Zassenhaus Conjecture
- Provided new insights into units of finite order in group rings

## 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 $\mathbb{Z} G$ of $G$ which maps to the identity under the linear extension of the natural homomorphism $G \rightarrow G/N$. 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 $\mathbb{Z} G$ is conjugate in the rational group algebra of $G$ to an element in $\pm G$.