From examples to methods: Two cases from the study of units in integral group rings

TL;DR

This paper reviews proofs of the Zassenhaus Conjecture for small alternating groups, introduces two methods derived from these proofs, and applies them to larger groups, highlighting open problems and potential research directions.

Contribution

It introduces the HeLP and lattice methods developed from studying specific cases of the Zassenhaus Conjecture and demonstrates their application to larger groups.

Findings

Proved the conjecture for A5 and A6

Developed the HeLP and lattice methods

Applied methods to A7 with one open case

Abstract

In this article, we review the proofs of the first Zassenhaus Conjecture on conjugacy of torsion units in integral group rings for the alternating groups of degree 5 and 6, by Luthar-Passi and Hertweck. We describe how the study of these examples led to the development of two methods -- the HeLP method and the lattice method. We exhibit these methods and summarize some results which were achieved using them. We then apply these methods to the study of the first Zassenhaus conjecture for the alternating group of degree 7 where only one critical case remains open for a full answer. Along the way we show in examples how recently obtained results can be combined with the methods presented and collect open problems some of which could be attacked using these methods.