Kunneth formula for group cohomology

Statement

For trivial group action

Suppose $G_{1},G_{2}$ are groups and $A$ is an abelian group. We have the following formula for the cohomology groups for trivial group action of $G_{1}\times G_{2}$ on $A$ in terms of the cohomology groups for trivial group action of $G_{1}$ and $G_{2}$ respectively on $A$:

$H^{n}(G_{1}\times G_{2};A)\cong \left(\sum _{i+j=n}H^{i}(G_{1};A)\otimes H^{j}(G_{2};A)\right)\oplus \left(\sum _{p+q=n+1}\operatorname {Tor} _{\mathbb {Z} }^{1}(H^{p}(G_{1};A),H^{q}(G_{2};A)\right)$

For nontrivial group action

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Related facts

• Second cohomology group for trivial group action commutes with direct product in second coordinate