Intermediately characteristic not implies isomorph-containing in abelian group

From Groupprops
Jump to: navigation, search
This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a abelian group. That is, it states that in a abelian group, every subgroup satisfying the first subgroup property (i.e., intermediately characteristic subgroup) need not satisfy the second subgroup property (i.e., isomorph-containing subgroup)
View all subgroup property non-implications | View all subgroup property implications


It is possible to have an abelian group G and an intermediately characteristic subgroup H of G that is not an isomorph-containing subgroup of G.

Related facts


Let G be the group of integers under addition, and H be the subgroup of even integers. Then, H is a maximal subgroup of G, and is characteristic in G (because any automorphism sends even integers to even integers). Hence, H is intermediately characteristic in G. On the other hand, H is not an isomorph-containing subgroup of G. For instance, H is isomorphic to G itself, which is not contained in H.