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


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.

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