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

From Groupprops

## Statement

Suppose is a group and is an abelian group. Denote by the first homology group for the *trivial* action of on . We then have:

where is the abelianization of , i.e., the quotient group of by its derived subgroup, and denotes the tensor product of abelian groups.