# Group of cyclicity-preserving 2-cocycles for trivial group action is naturally identified with group of homomorphisms from a particular abelian group

From Groupprops

## Contents

## Statement

Suppose is a group. Then, there exists an abelian group such that, for any abelian group , the group of cyclicity-preserving 2-cocycles for the trivial group action can be identified with the group of homomorphisms under pointwise addition.

## Examples

### Extreme examples

If is a cyclic group or more generally a locally cyclic group, the corresponding group is a trivial group.

### Examples of 2-groups

When is a 2-group, is also a 2-group.

### Examples of other p-groups

We consider the case .

Group | Order | Prime-base logarithm of order | Group that acts as source of homomorphisms | Order | Prime-base logarithm of order |
---|---|---|---|---|---|

elementary abelian group:E9 | 9 | 2 | elementary abelian group:E27 | 27 | 3 |

direct product of Z9 and Z3 | 27 | 3 | elementary abelian group:E243 | 243 | 5 |