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 Complete group (?).

Related facts

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

Facts used

1. Semidirect product with self-normalizing subgroup of automorphism group of coprime order implies every automorphism is inner

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 $p$ 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.