# Difference between revisions of "Intermediately characteristic not implies isomorph-containing in abelian group"

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− | Let <math>G</math> be the [[group of integers]] under addition, and <math>H</math> be the subgroup of even integers. Then, <math>H</math> is a maximal subgroup of <math>G</math>, and is characteristic in <math>G</math> (because any automorphism sends even integers to even integers). Hence, <math>H</math> is intermediately characteristic in <math>G</math>. On the other hand, <math>H</math> is ''not'' an isomorph-containing subgroup of <math>G</math>. For instance, <math>H</math> is isomorphic to <math>G</math> itself, which is not contained in <math>H</math>. | + | Let <math>G</math> be the [[particular example::group of integers]] under addition, and <math>H</math> be the subgroup of even integers. Then, <math>H</math> is a maximal subgroup of <math>G</math>, and is characteristic in <math>G</math> (because any automorphism sends even integers to even integers). Hence, <math>H</math> is intermediately characteristic in <math>G</math>. On the other hand, <math>H</math> is ''not'' an isomorph-containing subgroup of <math>G</math>. For instance, <math>H</math> is isomorphic to <math>G</math> itself, which is not contained in <math>H</math>. |

## Latest revision as of 18:21, 30 October 2009

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)

View all subgroup property non-implications | View all subgroup property implications

## 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 .