Units in Blocks of Defect 1 and the Zassenhaus Conjecture

TL;DR

This paper develops a method to analyze units of finite order in blocks of group rings with defect 1, proving special cases of the Zassenhaus conjecture and constructing potential counterexamples.

Contribution

It introduces a new approach to determine the existence of finite order units in blocks of group rings, advancing understanding of the Zassenhaus conjecture.

Findings

Units of order p in certain blocks are conjugate to elements of ±G

New results on units in Z(PSL(2,q)) for specific q

Constructed a unit of order 15 as a counterexample

Abstract

Building on previous work by Caicedo and the second author, we develop a method that decides the existence of units of finite order in blocks of $\mathbb{Z}_p G$ of defect 1. This allows us to prove that if $p$ is a prime and $G$ is a finite group whose Sylow $p$-subgroup has order $p$, then any unit $u\in \mathbb{Z} G$ of order $p$ is conjugate to an element of $\pm G$. This is a special case of the Zassenhaus conjecture. We also prove some new results on units of finite order in $\mathbb{Z} \mathrm{PSL}(2,q)$ for certain $q$, and construct a unit of order $15$ in $V(\mathbb{Z}_{(3,5)}\mathrm{PSL}(2,16))$ which is a $3$- and $5$-local counterexample to the Zassenhaus conjecture, raising the hope that our methods may lead to a global counterexample amongst simple groups.