Zassenhaus Conjecture on torsion units holds for $\text{SL}(2,p)$ and $\text{SL}(2,p^2)$

TL;DR

This paper proves the Zassenhaus Conjecture for the groups SL(2,p) and SL(2,p^2), marking the first infinite family of non-solvable groups where the conjecture holds, and extends results for units with orders coprime to p.

Contribution

The paper establishes the Zassenhaus Conjecture for SL(2,p) and SL(2,p^2), and reduces the problem for units of order multiple of p to a specific case.

Findings

Proved the conjecture for SL(2,p) and SL(2,p^2).

Extended results to units with order coprime to p.

Reduced the conjecture to a specific case for units of order multiple of p.

Abstract

H.J. Zassenhaus conjectured that any unit of finite order and augmentation $1$ in the integral group ring $\mathbb{Z}G$ of a finite group $G$ is conjugate in the rational group algebra $\mathbb{Q}G$ to an element of $G$. We prove the Zassenhaus Conjecture for the groups $\text{SL}(2,p)$ and $\text{SL}(2,p^2)$ with $p$ a prime number. This is the first infinite family of non-solvable groups for which the Zassenhaus Conjecture has been proved. We also prove that if $G=\text{SL}(2,p^f)$, with $f$ arbitrary and $u$ is a torsion unit of $\mathbb{Z}G$ with augmentation $1$ and order coprime with $p$ then $u$ is conjugate in $\mathbb{Q}G$ to an element of $G$. By known results, this reduces the proof of the Zassenhaus Conjecture for this groups to prove that every unit of $\mathbb{Z}G$ of order multiple of $p$ and augmentation $1$ has actually order $p$.