# Characteristic not implies isomorph-free 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-free subgroup)

View all subgroup property non-implications | View all subgroup property implications

## Contents

## Statement

### Statement with symbols

There exists a finite group and a characteristic subgroup of such that is *not* an isomorph-free subgroup of . In other words, there exists another subgroup of that is isomorphic to .

## Related facts

- Characteristic not implies isomorph-containing
- Characteristic not implies sub-isomorph-free in finite
- Characteristic not implies isomorph-normal in finite
- Characteristic not implies sub-(isomorph-normal characteristic) in finite
- Characteristic not implies injective endomorphism-invariant

## Proof

### Example of the dihedral group

`Further information: dihedral group:D8, subgroup structure of dihedral group:D8, center of dihedral group:D8`

Let be the dihedral group of order eight, given as follows, where denotes the identity element of :

.

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

- is characteristic.
- is not isomorph-free: The subgroup of is isomorphic to .