# Center not is 1-automorphism-invariant

From Groupprops

This article gives the statement, and possibly proof, of the fact that for a group, the subgroup obtained by applying a given subgroup-defining function (i.e., center) doesnotalways satisfy a particular subgroup property (i.e., 1-automorphism-invariant subgroup)

View subgroup property satisfactions for subgroup-defining functions View subgroup property dissatisfactions for subgroup-defining functions

## Statement

The center of a group need not be a 1-automorphism-invariant subgroup.

## Related facts

- Quasiautomorphism-invariant not implies 1-automorphism-invariant
- Center is quasiautomorphism-invariant

## Proof

Let be an odd prime, let be the prime-cube order group:U(3,p), i.e., the unique non-abelian group of order and exponent , and let be the center of . is a cyclic subgroup of order in .

There exist 1-automorphisms of that do not preserve . In fact, we can achieve any permutation of the cyclic subgroups of order using a 1-automorphism.