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.

Let $L := U_3(11)$. In this article, we classify groups with the same order and degree pattern as an almost simple group related to $L$. In fact, we prove that $L$, $L:2$ and $L:3$ are OD-characterizable, and $L:S_3$ is $5$-fold OD-characterizable.