Center not is 1-automorphism-invariant

This article gives the statement, and possibly proof, of the fact that for a group, the subgroup obtained by applying a given subgroup-defining function (i.e., center) does not always satisfy a particular subgroup property (i.e., 1-automorphism-invariant subgroup)
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Statement

The center of a group need not be a 1-automorphism-invariant subgroup.

Related facts

• Quasiautomorphism-invariant not implies 1-automorphism-invariant
• Center is quasiautomorphism-invariant

Proof

Let $p$ be an odd prime, let $G$ be the prime-cube order group:U(3,p), i.e., the unique non-abelian group of order $p^{3}$ and exponent $p$, and let $H$ be the center of $G$. $H$ is a cyclic subgroup of order $p$ in $G$.

There exist 1-automorphisms of $G$ that do not preserve $H$. In fact, we can achieve any permutation of the cyclic subgroups of order $p$ using a 1-automorphism.