# Potentially characteristic not implies normal-potentially characteristic

From Groupprops

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., potentially characteristic subgroup) neednotsatisfy the second subgroup property (i.e., normal-potentially characteristic subgroup)

View a complete list of subgroup property non-implications | View a complete list of subgroup property implications

Get more facts about potentially characteristic subgroup|Get more facts about normal-potentially characteristic subgroup

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property potentially characteristic subgroup but not normal-potentially characteristic subgroup|View examples of subgroups satisfying property potentially characteristic subgroup and normal-potentially characteristic subgroup

## Contents

## Statement

It is possible to have a potentially characteristic subgroup that is not a normal-potentially characteristic subgroup.

## Related facts

### Weaker facts

- Normal not implies normal-potentially characteristic
- Normal not implies characteristic-potentially characteristic
- Potentially characteristic not implies characteristic-potentially characteristic

## Facts used

- Potentially characteristic not implies normal-extensible automorphism-invariant, which in turn follows from Normal not implies normal-extensible automorphism-invariant in finite and finite normal implies potentially characteristic.
- Normal-potentially characteristic implies normal-extensible automorphism-invariant

## Proof

The proof follows from facts (1) and (2).

### Example of the dihedral group

`Further information: dihedral group:D8`

Let be the dihedral group of order eight, and be one of the Klein four-subgroups.

- is not a normal-potentially characteristic subgroup of : Using the fact that every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible, every automorphism of can be extended to an automorphism of for any group containing as a normal subgroup. But since there is an automorphism of not sending to itself, cannot be characteristic in .
- is potentially characteristic in : for instance, we can realize as the -Sylow subgroup of the symmetric group of degree four, in such a way that becomes characteristic.