Kunneth formula for group homology

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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 homology groups for trivial group action of G_1 \times G_2 on A in terms of the homology 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}^1_{\mathbb{Z}}(H_p(G_1;A),H_q(G_2;A)\right)