Holomorph of cyclic group of odd prime order is complete

Statement
Suppose $$p$$ is an odd prime and $$P$$ is the group of order $$p$$. Then, the holomorph of $$P$$ is a fact about::complete group.

Related facts

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

Facts used

 * 1) uses::Semidirect product with self-normalizing subgroup of automorphism group of coprime order implies every automorphism is inner

Proof that every automorphism is inner
The automorphism group of a cyclic group of order $$p$$ is a group of order $$p - 1$$. 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.