# Characteristic not implies normal-isomorph-free

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., normal-isomorph-free subgroup)

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## Statement

It is possible to have a finite group and subgroups of such that is a characteristic subgroup of , is a normal subgroup of , and are isomorphic groups.

## Facts used

- Characteristic not implies characteristic-isomorph-free in finite (combined with normal-isomorph-free implies characteristic-isomorph-free)
- Characteristic-isomorph-free not implies normal-isomorph-free in finite (combined with characteristic-isomorph-free implies characteristic)
- Characteristic not implies series-isomorph-free (combined with normal-isomorph-free implies series-isomorph-free)

## Proof

The proof could be obtained using an example for *any* of the stronger facts (1)-(3).