= Solution
Choose a Sylow $p$-subgroup $P$ of a <module vertex> $D$. Its index $[D:P]$ divides the $p'$-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> satisfies
$$
\operatorname{Tr}_P^D\left(\frac{\operatorname{id}_U}{[D:P]}\right)=\operatorname{id}_U.
$$
The <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 \b[every <module vertex> is a $p$-subgroup].
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