First 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.

Definition in cohomology terms
The first cohomology group $$H^1_\varphi(G,A)$$ is an abelian group 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 1-cocycles and 1-coboundaries.

All these definitions have natural analogues for the $$n^{th}$$ cohomology group $$H^n_\varphi(G,A)$$ for all $$n \ge 0$$. For more, see cohomology group.

Definition in terms of stability automorphisms of extensions
Suppose $$E$$ is a group that has an abelian normal subgroup identified with $$A$$ and such that the quotient group $$E/A$$ is identified with $$G$$ (we abuse notation here and treat $$A$$ and $$G$$ as the actual subgroup and quotient group respectively). Further, assume that the induced action of the quotient on the subgroup is the same as the group action $$\varphi$$.

Then, $$H^1_\varphi(G,A)$$ is a group quotient:

(Group of stability automorphisms of the chain $$1 \underline{\triangleleft} A \underline{\triangleleft} E$$)/(Subgroup comprising the stability automorphisms of the chain that are induced from conjugation by elements of $$A$$)

More precisely, the group of all stability automorphisms can be naturally identified with the 1-cocycle group $$Z^1_\varphi(G,A)$$ and the group of stability automorphisms arising via conjugation by an element of $$A$$ can be naturally identified with the 1-coboundary group $$B^1_\varphi(G,A)$$.

Particular cases
If the action of $$G$$ on $$A$$ is the trivial group action (i.e., every element of $$G$$ fixes every element of $$A$$), then the first cohomology group $$H^1(G,A)$$ can be naturally identified with the set $$\operatorname{Hom}(G,A)$$ endowed with a group structure under pointwise addition. Specifically, the group of 1-cocycles is identified with the group of homomorphisms under pointwise addition and the group of 1-coboundaries is the trivial group.

Generalizations

 * First cohomology set with coefficients in a non-abelian group