# Kunneth formula for group homology

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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 $A$:
$H^n(G_1 \times G_2; A) \cong \left(\sum_{i+j = n} H_i(G_1;A) \otimes H_j(G_2;A) \right) \oplus \left(\sum_{p + q = n - 1} \operatorname{Tor}^1_{\mathbb{Z}}(H_p(G_1;A),H_q(G_2;A)\right)$