# Partial Augmentations Power property: A Zassenhaus Conjecture related problem

## TL;DR

This paper introduces a new intermediate condition related to the Zassenhaus Conjecture in group rings, proves its validity for certain units, and uses it to verify the conjecture for specific group classes.

## Contribution

It proposes a novel partial augmentations power property, bridging known conjecture versions, and applies it to prove the Zassenhaus Conjecture for particular groups.

## Key findings

- The new condition holds for units mapping to identity modulo a nilpotent normal subgroup.
- The condition simplifies the HeLP Method, making it more effective.
- The Zassenhaus Conjecture is proved for a special class of groups under this condition.

## Abstract

Zassenhaus conjectured that any unit of finite order in the integral group ring $\mathbb{Z}G$ of a finite group $G$ is conjugate in the rational group algebra of $G$ to an element in $\pm G$. We review the known weaker versions of this conjecture and introduce a new condition, on the partial augmentations of the powers of a unit of finite order in $\mathbb{Z}G$, which is weaker than the Zassenhaus Conjecture but stronger than its other weaker versions. We prove that this condition is satisfied for units mapping to the identity modulo a nilpotent normal subgroup of $G$. Moreover, we show that if the condition holds then the HeLP Method adopts a more friendly form and use this to prove the Zassenhaus Conjecture for a special class of groups.