# Finite direct power-closed characteristic not implies 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., finite direct power-closed characteristic subgroup) neednotsatisfy the second subgroup property (i.e., fully invariant subgroup)

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## Statement

### Verbal statement

It is possible to have a finite direct power-closed characteristic subgroup of a group that is *not* a fully invariant subgroup.

## Facts used

## Proof

`Further information: direct product of S3 and Z2`

The proof follows by piecing together facts (1) and (2).

An explicit example of (2), and hence of this result as well, is when the whole group is the direct product of S3 and Z2 and is the center of .