Normal-potentially characteristic not implies characteristic
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) need not satisfy 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
Get more facts about normal-potentially characteristic subgroup|Get more facts about characteristic subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property normal-potentially characteristic subgroup but not characteristic subgroup|View examples of subgroups satisfying property normal-potentially characteristic subgroup and characteristic subgroup
- is not a characteristic subgroup of .
- There exists a group containing as a normal subgroup and as a characteristic subgroup.
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.