# Normal not implies normal-potentially relatively 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 relatively characteristic subgroup)

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

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## Statement

A normal subgroup of a group need not be a normal-potentially relatively characteristic subgroup.

## Facts used

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

## 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, is not invariant under this automorphism, which extends to an automorphism of .
- is normal in : This is obvious.