Universal power not implies center-fixing

This article gives the statement and possibly, proof, of a non-implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., universal power automorphism) need not satisfy the second automorphism property (i.e., center-fixing automorphism)
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Statement

A universal power automorphism of a group (i.e., an automorphism obtained by taking the $n^{th}$ power for some integer $n$) need not fix every element of the center of the group.

Related facts

• Universal power not implies IA

Corollaries

• Subgroup-conjugating not implies center-fixing
• Subgroup-conjugating not implies class-preserving
• Normal not implies class-preserving

Proof

Example of an Abelian group

Consider the cyclic group of order $p$, where $p$ is an odd prime. The inverse map is a universal power automorphism of this group, and is not the identity map. Since the group is Abelian, it equals its center, so we have a universal power automorphism that does not fix every element of the center.

This example can be generalized to any Abelian group whose exponent is not two: the inverse map is a universal power automorphism that is not center-fixing.