Universal power not implies IA

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., IA-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 be an IA-automorphism: it need not preserve the cosets of the commutator subgroup (i.e., it need not be identity on the Abelianization).

Related facts

• Universal power not implies center-fixing

Corollaries

• Universal power not implies class-preserving
• Subgroup-conjugating not implies IA
• Subgroup-conjugating not implies class-preserving
• Normal not implies IA
• 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 Abelianization, so we have a universal power automorphism that does not fix every element of the Abelianization.