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

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 $p$.