Holomorph of cyclic group of odd prime order is complete
- Odd-order cyclic group is characteristic in holomorph
- Odd-order abelian group not is characteristic in holomorph
- Semidirect product with self-normalizing subgroup of automorphism group of coprime order implies every automorphism is inner
Proof that the holomorph is centerlessPLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
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.