Center not is 1-automorphism-invariant

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 particular example::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.