2-coboundary for a group action

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This article gives a basic definition in the following area: group cohomology
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This term is defined, and makes sense, in the context of a group action on an Abelian group. In particular, it thus makes sense for a linear representation of a group

Definition

Let G be a group acting on an Abelian group A. A 2-coboundary for the action of G on A, is a function f: G \times G \to A such that there exists a function \phi:G \to A such that:

f = (g,h) \mapsto g.\phi(h) - \phi(gh) + \phi(g)

Importance

Suppose we want to classify all groups E which arise as the group extension with normal subgroup A and quotient G. One approach to describing such a group E is to define a collection S of coset representatives for A in E. This can be viewed as a map from G to E. Call the coset representative for g b_g. Define f(g,h) as the element of A given by b_{gh}^{-1}b_gb_h.

It turns out that if we pick two different collections of coset representatives and let f_1 and f_2 be the functions corresponding to them, then f_1 - f_2 is a 2-coboundary.

It's also true that each of f_1 and f_2 needs to be a 2-cocycle, and thus the collection of possible extensions of G by A is classified by the second cohomology group for the action of G on A.

Further information: second cohomology group