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 $H_{\varphi }^{1}(G;A)$ 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 $A$.
• $H_{\varphi }^{1}(G;A)$ 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.