2-coboundary for a group action

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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 $ϕ : G \to A$ such that:

$f = (g, h) \mapsto g . ϕ (h) - ϕ (g h) + ϕ (g)$

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 $b_{g h}^{- 1} b_{g} b_{h}$ .

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