Normal-potentially characteristic not implies characteristic

From Groupprops
Jump to: navigation, search
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

Statement

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

Facts used

  1. subgroup of finite abelian group implies abelian-potentially characteristic

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 G is a finite abelian group and H is a non-characteristic subgroup of G, there exists some abelian group K containing G in which H is characteristic. Note that G is normal in K, because K is abelian.

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