First cohomology set with coefficients in a non-abelian group

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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.