Holomorph of cyclic group of odd prime order is complete

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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

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.