# Normal 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., normal 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 normal subgroup|Get more facts about normal-potentially characteristic subgroup

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## Contents

## Statement

### Verbal statement

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

### Statement with symbols

We can have a group with a subgroup such that is normal in , but whenever is a group containing as a normal subgroup, is *not* a characteristic subgroup in .

## Related facts

### Stronger facts

### Weaker facts

## Facts used

- Normal not implies normal-extensible automorphism-invariant
- Normal-potentially characteristic implies normal-extensible automorphism-invariant

## Proof

The proof follows directly 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 normal in : This is obvious.