finite group homology: $nH_k(G;M)=0$ for $n=|G|$?

Question

Let $G$ be a finite group. Is there a simple proof (if any) that the order of $G$ annihilates the Eilenberg-MacLane homology $H_k(G;M)$ for all $k\geq1$?

A simple proof of the statement for cohomology can be found in Grillet, Abstract Algebra, 12.9.2.