# Universal power not implies center-fixing

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., center-fixing 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 fix every element of the center of the group.

## Related facts

### 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 , 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 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.