Kunneth formula for group homology

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 homology groups for trivial group action of $G_{1} \times G_{2}$ on $M$ in terms of the homology groups for trivial group action of $G_{1}$ and $G_{2}$ respectively on $M$ :

$H_{p} (G_{1} \times G_{2}; M) ≅ (\sum_{i + j = p} H_{i} (G_{1}; M) \otimes H_{j} (G_{2}; M)) \oplus (\sum_{u + v = p - 1} T_{o}^{r} (H_{u} (G_{1}; M), H_{v} (G_{2}; M))$

Statement

For trivial group action

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