First homology group for trivial group action equals tensor product with abelianization

Statement

Suppose $G$ is a group and $A$ is an abelian group. Denote by $H_1(G,A)$ the first homology group for the trivial action of $G$ on $A$. We then have:

$H_1(G,A) \cong G^{\operatorname{ab}} \otimes_{\mathbb{Z}} A$

where $G^{\operatorname{ab}}$ is the abelianization of $G$, i.e., the quotient group of $G$ by its derived subgroup, and $\otimes_{\mathbb{Z}}$ denotes the tensor product of abelian groups.

Related facts

• First cohomology group for trivial group action is naturally isomorphic to group of homomorphisms