# Characteristic not implies potentially fully invariant

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., characteristic subgroup) neednotsatisfy the second subgroup property (i.e., potentially fully invariant subgroup)

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## Statement

It is possible to have a subgroup of a group such that is a characteristic subgroup of but is *not* a potentially fully invariant subgroup of , i.e., there is *no* group containing such that is a fully invariant subgroup of .

## Facts used

- Normal not implies potentially fully invariant: There exists an example of a group and a normal subgroup of such that is not fully invariant in any group containing .
- NPC theorem: This states that if is a normal subgroup of , there exists a group containing such that is a characteristic subgroup of .

## Proof

Let and be a group-subgroup pair as given by fact (1). By fact (2), there exists a group containing such that is a characteristic subgroup of .

We want to show that there is no group containing such that is fully invariant in . The reason for this is: if such a group exists, it would also contain , so we'd have a group containing in which is fully invariant. This contradicts the choice of and as being examples for fact (1).