in this paper, we investigate the zassenhaus conjecture for $psl(4,3)$ and $psl(5,2)$. consequently, we prove that the prime graph question is true for both groups.

we investigate the classical h.~zassenhaus conjecture‎ ‎for integral group rings of alternating groups $a_9$ and $a_{10}$ of degree‎ ‎$9$ and $10$‎, ‎respectively‎. ‎as a consequence of our‎ ‎previous results we confirm the prime graph‎ ‎conjecture for integral group rings of $a_n$ for all $n leq 10$‎.

Using the Luthar–Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the Rudvalis sporadic simple group Ru. As a consequence, for this group we confirm Kimmerle’s conjecture on prime graphs.

For a finite group $G$ and $U: = U(\mathbb{Z}G)$, the of units integral ring $G$, we study implications structure on abelianization $U/U'$ $U$. We pose questions connections between exponent $G/G'$ as well ranks torsion-free parts $Z(U)$, center $U$, $U/U'$. show that originating from known generic constructions in $\mathbb{Z}G$ are well-behaved under projection $U$ to our have positive answer ...