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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 anormalized 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 GraphQuestion 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 theunderstanding of blocks of cyclic defect and Young tableaux combinatorics.

Orders of units in integral group rings and blocks of defect $1$