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 $M$:

$H_{p}(G_{1}\times G_{2};M)\cong \left(\sum _{i+j=p}H_{i}(G_{1};M)\otimes H_{j}(G_{2};M)\right)\oplus \left(\sum _{u+v=p-1}\operatorname {Tor} _{\mathbb {Z} }^{1}(H_{u}(G_{1};M),H_{v}(G_{2};M))\right)$

Related facts

• Kunneth formula for group cohomology
• Cohomology group for trivial group action commutes with direct product in second coordinate