First cohomology set with coefficients in a non-abelian group

Definition
Suppose $$G$$ is a group and $$A$$ is a (possibly non-abelian) group. Suppose we are given a homomorphism of groups $$\varphi:G \to \operatorname{Aut}(A)$$, i.e., a group action of $$G$$ on $$A$$ by automorphisms.

The first cohomology set $$H^1_\varphi(G;A)$$ is defined as follows:


 * Call a function $$f:G \to A$$ to be a 1-cocycle if $$f(gh) = f(g)(\varphi(g)(f(h)))$$.
 * Declare two functions $$f_1,f_2: G \to A$$ to be equivalent if there exits $$a \in A $$ such that $$af_1(g) = f_2(g)(\varphi(g)(a))$$. This defines an equivalence relation on functions from $$G$$ to $$A$$.
 * $$H^1_\varphi(G;A)$$ is defined as the quotient of the set of all 1-cocycles by the equivalence relation.

Note that when the group $$A$$ is an abelian group, this just becomes the first cohomology group, because:


 * The 1-cocycles form a group, see 1-cocycle for a group action.
 * The equivalence relation is equivalent to differing by a 1-coboundary, and the 1-coboundaries form a subgroup.
 * Thus, the quotient set by the equivalence relation is the quotient group of the group of 1-cocycles by the subgroup of 1-coboundaries, which gives the first cohomology group.