# Universal power not implies IA

From Groupprops

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) neednotsatisfy 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 power for some integer ) 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

### 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 , where 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.