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

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(Proof)
 
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==Proof==
 
==Proof==
  
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>.
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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) need not satisfy 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 G and an intermediately characteristic subgroup H of G that is not an isomorph-containing subgroup of G.

Related facts

Proof

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.