# Cocycle for trivial group action

## Contents

## Definition

Suppose is a group and is an abelian group.

### Definition in terms of cocycle for a group action

A -**cocycle for trivial group action** is a -cocycle for a group action of on , where the action is trivial.

### Explicit definition

A -cocycle for trivial group action of on is a function satisfying the following for all :

## Particular cases

Value of | Condition for being a -cocycle for trivial group action | Further information |
---|---|---|

1 | , or | It becomes a homomorphism of groups from to , and hence, from the abelianization of to |

2 | , or . | 2-cocycle for trivial group action |