  We show that if the Sylow $p$-subgroup of a finite group $G$ is of order $p$,
then the normalized unit group of the integral group ring of $G$ contains a
normalized unit of order $pq$ if and only if $G$ contains an element of order
$pq$, where $q$ is any prime. We use this result to answer the Prime Graph
Question for most sporadic simple groups and some simple groups of Lie type,
including a new infinite series of such groups. Our methods are based on the
understanding of blocks of cyclic defect and Young tableaux combinatorics.
