Finite-group cohomology is annihilated by the group order
= Finite-group cohomology is annihilated by the group order
If $G$ is finite and $n>0$, multiplication by $|G|$ annihilates $H^n(G,M)$. Indeed, restriction to the trivial subgroup is zero in positive degree, while the restriction-corestriction composite is multiplication by $|G|$.