# 3-cocycle for a group action

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 be a group acting (on the left) on an abelian group via a homomorphism of groups where is the automorphism group of .

### Explicit definition

A 3-cocycle for the action is a function satisfying the following for all (here, the s are allowed to be equal):

If we suppress and use for the action, we can rewrite this as:

or equivalently:

Note that a function (without any conditions) is sometimes termed a 3-cochain for the group action.

### Definition as part of the general definition of cocycle

A 3-cocycle for a group action is a special case of a cocycle for a group action, namely . This, in turn, is the notion of cocycle corresponding to the Hom complex from the bar resolution of to as -modules.

### Group structure

The set of 3-cocycles for the action of on forms a group under pointwise addition.

### As a group of homomorphisms

For any group , we can construct a -module such that for any abelian group , the group of 3-cocycles can be identified with the group of -module maps from to .