# 2-coboundary for a group action

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## Contents

This article gives a basic definition in the following area: group cohomology
View other basic definitions in group cohomology |View terms related to group cohomology |View facts related to group cohomology

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