# 2-coboundary for a group action

## 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 be a group acting on an Abelian group . A **2-coboundary** for the action of on , is a function such that there exists a function such that:

## Importance

Suppose we want to classify all groups which arise as the group extension with normal subgroup and quotient . One approach to describing such a group is to define a collection of coset representatives for in . This can be viewed as a map from to . Call the coset representative for . Define as the element of given by .

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

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

`Further information: second cohomology group`