 Solution

Choose a Sylow $p$-subgroup $P$ of a module vertex $D$. Its index $[D:P]$ divides the $p^{\prime }$-part of $|G|$, so it is a unit in $R$ under the hypothesis. For every $RD$-module $U$, the induction counit from $P$ splits by averaging: equivalently, the relative trace satisfiesThe D. Higman criterion therefore makes every $RD$-module relatively $P$-projective. Since $M$ is relatively $D$-projective, transitivity of induction makes it relatively $P$-projective. Minimality of the module vertex now forces $D=P$. Thus every module vertex is a $p$-subgroup.