# Cyclicity-preserving 2-cocycle for trivial group action

From Groupprops

## Contents

## Definition

Suppose is a group and is an abelian group. A function is termed a **cyclicity-preserving 2-cocycle for trivial group action** if it satisfies the following conditions:

Condition name | Expression for condition |
---|---|

2-cocycle for a group action (particularly, 2-cocycle for trivial group action) | for all |

cyclicity-preserving | whenever is cyclic |

The group of cyclicity-preserving 2-cocycles is denoted .

## Facts

### Existence of a source group

For any group , there exists an abelian group such that for any abelian group , the group of cyclicity-preserving 2-cocycles can be identified with the group .

`Further information: group of cyclicity-preserving 2-cocycles for trivial group action is naturally identified with group of homomorphisms from a particular abelian group`

## Examples

### Extreme examples

- If is a cyclic group or a locally cyclic group, the group of cyclicity-preserving 2-cocycles is a trivial group.
- If is a trivial group, the group of cyclicity-preserving 2-cocycles is a trivial group.

## Relation with other properties

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

IIP 2-cocycle for trivial group action | 2-cocycle such that for all | |FULL LIST, MORE INFO | ||

normalized 2-cocycle for trivial group action | 2-cocycle such that for all | |FULL LIST, MORE INFO | ||

2-cocycle for trivial group action | IIP 2-cocycle for trivial group action|FULL LIST, MORE INFO |