# Normal not implies strongly 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., characteristic-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 characteristic-potentially characteristic subgroup

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property normal subgroup but not characteristic-potentially characteristic subgroup|View examples of subgroups satisfying property normal subgroup and characteristic-potentially characteristic subgroup

## Statement

### Verbal statement

A normal subgroup need not be characteristic-potentially characteristic.

### Statement with symbols

It is possible to have a group and a normal subgroup of such that there is no group containing in which both and are characteristic subgroups.

## Facts used

- Characteristic-potentially characteristic implies normal-potentially characteristic
- Normal not implies normal-potentially characteristic

## Proof

The proof follows directly from facts (1) and (2).