= Restriction-corestriction identity in group cohomology
Let $K$ be a <finite-index subgroup> of a <group> $G$ and $M$ a $G$-<module>. The <restriction map in group cohomology> followed by the <corestriction map in group cohomology> satisfies
$$
\operatorname{Cor}_K^G\operatorname{Res}_K^G=[G:K]\,\mathrm{id}
$$
on $H^n(G,M)$. The <corestriction map in group cohomology> sums the coset contributions, and the inner-conjugation maps on the original <group cohomology> are the identity.
The opposite composite has a different domain. When $K$ is normal, $\operatorname{Res}_K^G\operatorname{Cor}_K^G$ on $H^n(K,M)$ is the sum of the conjugation actions of the cosets in $G/K$. Those actions need not be trivial. For finite $G$ and $K=1$, positive-degree <group cohomology> of $K$ vanishes, so the first composite proves that $|G|$ annihilates $H^n(G,M)$ for $n>0$.
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