Homology group

Definition
Let $$G$$ be a group acting on an abelian group $$A$$, via an action $$\varphi:G \to \operatorname{Aut}(A)$$. Equivalently, $$A$$ is a module over the (possibly non-commutative) unital group ring $$\mathbb{Z}G$$ of $$G$$ over the ring of integers.

The homology groups $$H_{n,\varphi}(G,A)$$ ($$n = 0,1,2,3,\dots$$) are abelian groups defined in the following equivalent ways.

When $$\varphi$$ is understood from context, the subscript $${}_\varphi$$ may be omitted in the notation for the homology group, as well as the notation for the groups of $$n$$-cycles and $$n$$-boundaries.

Equivalence of definitions
The equivalence of (1) and (2) follows from the fact that (1) is the special case of (2) that arises if we choose our projective resolution as the bar resolution. The equivalence between (2) and (3) is by the definition of $$\operatorname{Tor}$$. The equivalence between (3) and (4) follows from the fact that the coinvariants functor (defined in (4)) sends $$A$$ to $$\mathbb{Z} \otimes_{\mathbb{Z}G} A$$ where $$\mathbb{Z}$$ is given the structure of a trivial $$\mathbb{Z}G$$-module.