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

From Groupprops

## Contents

## Statement

The following statements are equivalent for a group and an integer . Suppose the power map is a surjective endomorphism (such as an automorphism) of .

Then, the power map is an endomorphism of and is in the center of for all .

## Related facts

### Converse

A precise converse does not hold, but the following does: (n-1)th power map is endomorphism taking values in the center implies nth power map is endomorphism. We cannot guarantee surjectivity in general.

### Similar facts

- nth power map is endomorphism iff abelian (if order is relatively prime to n(n-1))
- nth power map is endomorphism implies every nth power and (n-1)th power commute
- Inverse map is automorphism iff abelian
- Square map is endomorphism iff abelian
- Cube map is endomorphism iff abelian (if order is not a multiple of 3)
- Cube map is automorphism implies abelian
- Frattini-in-center odd-order p-group implies p-power map is endomorphism
- Frattini-in-center odd-order p-group implies (p plus 1)-power map is automorphism

### Analogues in other algebraic structures

- Multiplication by n map is a derivation iff derived subring has exponent dividing n
- Multiplication by n map is an endomorphism iff derived subring has exponent dividing n(n-1)

## Facts used

## Proof

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**Given**: A group and an integer such that is a surjective endomorphism of .

**To prove**: is an endomorphism of and for all .

**Proof**

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

1 | for all . | Fact (1) | power map is endomorphism |
Given+Fact direct | |

2 | for all | power map is surjective |
Step (1) | [SHOW MORE] | |

3 | For any , , so the power map is an endomorphism | power map is endomorphism | Step (2) | [SHOW MORE] |

Steps (2) and (3) complete the proof.