# Normal-potentially characteristic not implies 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-potentially characteristic subgroup) neednotsatisfy the second subgroup property (i.e., characteristic subgroup)

View a complete list of subgroup property non-implications | View a complete list of subgroup property implications

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## Statement

It is possible to have a group and a subgroup such that both the following hold:

- is
*not*a characteristic subgroup of . - There exists a group containing as a normal subgroup and as a characteristic subgroup.

## Facts used

## Proof

By fact (1), any subgroup of a finite abelian group can be realized as a characteristic subgroup in some bigger abelian group. Thus, if is a finite abelian group and is a non-characteristic subgroup of , there exists some abelian group containing in which is characteristic. Note that is normal in , because is abelian.

For instance, if we take to be the Klein four-group, and as one of the subgroups of order two, then we can take as a direct product of Z4 and Z2, containing as one of its cyclic subgroups.