# Cocycle for a group action

From Groupprops

## Contents

## Definition

Suppose is a group and is an abelian group, with an action of on . In other words, is a homomorphism of groups from to , the automorphism group of .

### Definition in terms of bar resolution

A -cocycle is an element in the cocycle group for the Hom complex from the bar resolution of to , in the sense of -modules.

### Explicit definition

For a nonnegative integer, a -cocycle for the action of on is a function such that, for all :

If we suppress the symbol and denote the action by , this becomes:

In particular, when the action is trivial, this is equivalent to saying that:

## Particular cases

Condition for being a -cocycle | Further information | |
---|---|---|

1 | For all , we have , equivalently | 1-cocycle for a group action |

2 | For all , we have , equivalently | 2-cocycle for a group action |

3 | For all , we have , or equivalently, | 3-cocycle for a group action |