# Intermediately characteristic not implies isomorph-containing in abelian group

From Groupprops

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) neednotsatisfy the second subgroup property (i.e., isomorph-containing subgroup)

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## Definition

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

## Related facts

- Equivalence of definitions of intermediately characteristic subgroup of finite abelian group: This says that in a finite abelian group, any intermediately characteristic subgroup is an isomorph-containing subgroup.
- Characteristic equals fully invariant in odd-order abelian group
- Equivalence of definitions of image-closed characteristic subgroup of finite abelian group

## Proof

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