# Holomorph of cyclic group of odd prime order is complete

From Groupprops

## Contents

## Statement

Suppose is an odd prime and is the group of order . Then, the holomorph of is a Complete group (?).

## Related facts

- Odd-order cyclic group is characteristic in holomorph
- Odd-order abelian group not is characteristic in holomorph

## Facts used

## Proof

### Proof that the holomorph is centerless

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### Proof that every automorphism is inner

The automorphism group of a cyclic group of order is a group of order . Thus, the holomorph is obtained as the semidirect product with a group of coprime order .Further, since this group is the whole automorphism group, it is tautologically a self-normalizing subgroup of the automorphism group. Thus, fact (1) tells us that every automorphism of the holomorph is inner.