Kunneth formula for group homology

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 $A$ :

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

Statement

For trivial group action

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