# (n-1)th power map is endomorphism taking values in the center implies nth power map is endomorphism

From Groupprops

## Statement

Suppose is a group and is an integer such that the power map is an endomorphism of and is in the center of for all in . Then, the map is also an endomorphism of .

## Related facts

### Applications

### Converse

The precise converse is not true, but a partial converse is: nth power map is surjective endomorphism implies (n-1)th power map is endomorphism taking values in the center

## Proof

**Given**: A group , an integer such that the map is an endomorphism taking values in the center.

**To prove**: The map is an endomorphism of , i.e., for all .

**Proof**: We have the following for all .

Step no. | Assertion/construction | Given data used | Previous steps used |
---|---|---|---|

1 | |||

2 | is an endomorphism | ||

3 | Step (2) plugged into Step (1) | ||

4 | is in the center of , so commutes with | Step (3) | |

5 | Step (4) (simplified) |