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

==Statement==

Suppose <math>G</math> is a [[group]] and <math>A</math> is an [[abelian group]]. Denote by <math>H_1(G,A)</math> the [[first homology group]] for the ''trivial'' action of <math>G</math> on <math>A</math>. We then have:

<math>H_1(G,A) \cong G^{\operatorname{ab}} \otimes_{\mathbb{Z}} A</math>

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

==Related facts==

* [[First cohomology group for trivial group action is naturally isomorphic to group of homomorphisms]]