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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 satisfieson $H^n(G,M)$. 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, ${\mathrm{Res}}_K^G{\mathrm{Cor}}_K^G$ 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 $n>0$.

For a finite group $G$, Tate cohomology joins group cohomology and group homology through the norm map $N_G={\sum }_{g\in G}g$. For $r\ge 1$ it equals ordinary group cohomology; for $r\le -2$ it equals $H_{-r-1}(G,M)$. The middle terms are ${\hat{H}}^0(G,M)=M^G/N_{G}M$ and ${\hat{H}}^{-1}(G,M)=\ker (N_G)/I_{G}M$, where $I_{G}M$ is generated by $gm-m$. For the trivial module $\mathbb{Z}$, ${\hat{H}}^{-2}(G,\mathbb{Z})=G^{\mathrm{ab}}$. With multiplicative coefficients the norm is the product of the translates.

For an additive left $G$-module $M$, a two-coboundary is a degree-two group cochain obtained from a degree-one cochain $b:G\to M$ byIt is automatically a two-cocycle because ${\delta }^2=0$. Expanding the four terms in $g\cdot (\delta b)(h,k)-(\delta b)(gh,k)+(\delta b)(g,hk)-(\delta b)(g,h)$ makes every contribution cancel.

For trivial action and $G=\mathbb{Z}$, take $M=\mathbb{Z}$ and $b(n)=n^2$. Then $(\delta b)(m,n)=n^2-(m+n{)}^2+m^2=-2mn$, a nonzero cochain representing the zero class in $H^2(G,M)$. Being cohomologically trivial does not require being pointwise zero.

Changing a section in an abelian-kernel group extension changes its extension cocycle by a two-coboundary. This is the mechanism behind quotienting two-cocycles by two-coboundaries when defining second group cohomology.