# Potentially characteristic not implies characteristic-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., potentially characteristic 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 potentially characteristic subgroup|Get more facts about characteristic-potentially characteristic subgroup

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## Statement

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

## Facts used

- Potentially characteristic not implies normal-potentially characteristic, which in turn uses potentially characteristic not implies normal-extensible automorphism-invariant, which in turn uses normal not implies normal-extensible automorphism-invariant in finite

## Proof

The proof follows directly from fact (1), and the observation that any characteristic-potentially characteristic subgroup is also normal-potentially characteristic.