Universal power not implies class-preserving

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., class-preserving)
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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 be a class-preserving automorphism: it need not preserve conjugacy classes of elements.

Related facts

Stronger facts

• Universal power not implies center-fixing
• Universal power not implies IA

Corollaries

• 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, the conjugacy classes are singleton, so we have a universal power automorphism that is not class-preserving.

The example generalizes to any Abelian group whose exponent is bigger than two.