Cohomology 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 cohomology 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 cohomology group, as well as the notation for the groups of $$n$$-cocycles and $$n$$-coboundaries.

Definition in terms of twisted cohomology
See here: Math Overflow question on the subject.

Equivalence of definitions
The equivalence of (5) or (5') and (1) follows from the fact that (5) is the special case of (1) that arises if we choose our projective resolution as the bar resolution.

(1) and (2) are both explicit formulations of (3), based on the definition of $$\operatorname{Ext}$$.

The equivalence of definitions (3) and (4) follows from the fact that $$A^G \cong \operatorname{Hom}_{\mathbb{Z}G}(\mathbb{Z},A)$$ where $$A^G$$ is the set of fixed points of $$A$$ under $$G$$ and $$\mathbb{Z}$$ is treated as a trivial $$\mathbb{Z}G$$-module in the $$\operatorname{Hom}$$-set expression on the right side.