# Homomorphism in each coordinate implies cocycle for trivial group action

From Groupprops

## Statement

Suppose is a group and is an abelian group. Suppose is a function with the property that, for all , if we fix the entries in all coordinates but the coordinate, the induced function from to is a homomorphism of groups.

Then, is a -cocycle for trivial group action.

## Particular cases

What being a homomorphism in each coordinate means | What being a -cocycle means | Link to -cocycle page | |
---|---|---|---|

1 | being a homomorphism of groups from to | being a homomorphism of groups from to | -- |

2 | satisfies, for all , both of these: and | satisfies, for all , the following: | 2-cocycle for trivial group action |

3 | satisfies, for all , all of these: | satisfies, for all , the following: | 3-cocycle for trivial group action |