Intermediately characteristic not implies isomorph-containing in abelian group
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)
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- 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
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 .