Group having an abelian contranormal subgroup: Difference between revisions
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==Definition== | ==Definition== | ||
A [[group]] <math>G</math> is termed a '''group having an abelian contranormal subgroup''' if there exists a subgroup <math>H</math> of <math>G</math> that is [[defining ingredient::abelian group|abelian as a group]] and is a [[defining ingredient::contranormal subgroup]] of <math>G</math>, i.e., the [[defining ingredient::normal closure]] of <math>< | A [[group]] <math>G</math> is termed a '''group having an abelian contranormal subgroup''' if there exists a subgroup <math>H</math> of <math>G</math> that is [[defining ingredient::abelian group|abelian as a group]] and is a [[defining ingredient::contranormal subgroup]] of <math>G</math>, i.e., the [[defining ingredient::normal closure]] of <math>H</math> in <math>G</math> equals <math>G</math>. | ||
==Relation with other properties== | ==Relation with other properties== | ||
Latest revision as of 22:00, 26 February 2010
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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
Definition
A group is termed a group having an abelian contranormal subgroup if there exists a subgroup of that is abelian as a group and is a contranormal subgroup of , i.e., the normal closure of in equals .
Relation with other properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Abelian group | ||||
| Simple group | ||||
| Group that is the normal closure of a singleton subset | ||||
| Group with two conjugacy classes | ||||
| Group having a cyclic conjugate-dense subgroup | ||||
| Group having an abelian conjugate-dense subgroup |