Kunneth formula for group cohomology

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Statement

For trivial group action

Suppose G_1,G_2 are groups and M is an abelian group. We have the following formula for the cohomology groups for trivial group action of G_1 \times G_2 on M in terms of the cohomology groups for trivial group action of G_1 and G_2 respectively on M:

H^p(G_1 \times G_2; M) \cong \left(\sum_{i+j = p} H^i(G_1;M) \otimes H^j(G_2;M) \right) \oplus \left(\sum_{u + v = p + 1} \operatorname{Tor}^1_{\mathbb{Z}}(H^u(G_1;M),H^v(G_2;M))\right)

For nontrivial group action

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