# Characteristic not implies isomorph-normal in finite group

From Groupprops

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

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## Statement

### Statement with symbols

It is possible to have a group and a subgroup of such that is a characteristic subgroup of but is not isomorph-normal in : there exists a subgroup of isomorphic to that is not normal in .

## Proof

### Example of the dihedral group

`Further information: dihedral group:D8`

Let be the dihedral group of order eight, given by:

.

Let be the center of . is a subgroup of order two generated by .

- is characteristic.
- is not isomorph-normal: The subgroup of is isomorphic to , but is not normal in , because conjugation by sends it to the subgroup .