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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 conjugatein the rational group algebra $\mathbb{Q}G$ to an element of $G$. We prove theZassenhaus 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-solvablegroups for which the Zassenhaus Conjecture has been proved. We also prove thatif $G=\text{SL}(2,p^f)$, with $f$ arbitrary and $u$ is a torsion unit of$\mathbb{Z}G$ with augmentation $1$ and order coprime with $p$ then $u$ isconjugate in $\mathbb{Q}G$ to an element of $G$. By known results, this reducesthe proof of the Zassenhaus Conjecture for this groups to prove that every unitof $\mathbb{Z}G$ of order multiple of $p$ and augmentation $1$ has actuallyorder $p$.

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