Intermediately subnormal-to-normal subgroup: Difference between revisions

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A [[subgroup]] of a [[group]] is termed '''intermediately subnormal-to-normal''' if it satisfies the following equivalent conditions:
A [[subgroup]] of a [[group]] is termed '''intermediately subnormal-to-normal''' if it satisfies the following equivalent conditions:


* Whenever it is [[defining ingredient::subnormal subgroup|subnormal]] in any intermediate subgroup, then it is also [[defining ingredient::normal subgroup|normal]] in that intermediate subgroup.
# Whenever it is [[defining ingredient::subnormal subgroup|subnormal]] in any intermediate subgroup, then it is also [[defining ingredient::normal subgroup|normal]] in that intermediate subgroup.
* Whenever it is [[defining ingredient::2-subnormal subgroup|2-subnormal]] in any intermediate subgroup, then it is also [[defining ingredient::normal subgroup|normal]] in that intermediate subgroup.
# Whenever it is [[defining ingredient::2-subnormal subgroup|2-subnormal]] in any intermediate subgroup, then it is also [[defining ingredient::normal subgroup|normal]] in that intermediate subgroup.
# In any intermediate subgroup, it is a [[defining ingredient::subgroup with self-normalizing normalizer]]. In other words, its [[defining ingredient::normalizer of a subgroup|normalizer]] in any intermediate subgroup is a [[defining ingredient::self-normalizing subgroup]].
# It has a [[defining ingredient::subnormalizer]], and the subnormalizer is equal to the normalizer.
# Whenever it is [[defining ingredient::ascendant subgroup|ascendant]] in any intermediate subgroup, then it is also [[defining ingredient::normal subgroup|normal]] in that intermediate subgroup.
# Whenever it is [[defining ingredient::hypernormalized subgroup|hypernormalized]] in any intermediate subgroup, then it is also [[defining ingredient::normal subgroup|normal]] in that intermediate subgroup.
 
===Equivalence of definitions===
 
{{further|[[Equivalence of definitions of intermediately subnormal-to-normal subgroup]]}}


==Formalisms==
==Formalisms==


{{obtainedbyapplyingthe|intermediately operator|subnormal-to-normal subgroup}}
{{obtainedbyapplyingthe|intermediately operator|subnormal-to-normal subgroup}}
{{obtainedbyapplyingthe|intermediately operator|subgroup with self-normalizing normalizer}}


==Relation with other properties==
==Relation with other properties==
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* [[Weaker than::Paranormal subgroup]]
* [[Weaker than::Paranormal subgroup]]
* [[Weaker than::Polynormal subgroup]]
* [[Weaker than::Polynormal subgroup]]
* [[Weaker than::Intermediately contranormal subgroup]]
* [[Weaker than::Intermediately normal-to-characteristic subgroup]]: {{proofat|[[Intermediately normal-to-characteristic implies intermediately subnormal-to-normal]]}}
* [[Weaker than::Intermediately normal-to-characteristic subgroup]]: {{proofat|[[Intermediately normal-to-characteristic implies intermediately subnormal-to-normal]]}}
* [[Weaker than::Self-normalizing subgroup]]: {{proofat|[[Self-normalizing implies intermediately subnormal-to-normal]]}}
* [[Weaker than::Image-closed intermediately subnormal-to-normal subgroup]]


===Weaker properties===
===Weaker properties===


* [[Stronger than::Subgroup having a subnormalizer]]
* [[Stronger than::Subnormal-to-normal subgroup]]
* [[Stronger than::Subnormal-to-normal subgroup]]
* [[Stronger than::Subgroup with self-normalizing normalizer]]: {{proofat|[[Normalizer of intermediately subnormal-to-normal implies self-normalizing]]}}


==Metaproperties==
==Metaproperties==
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If <math>H</math> is intermediately subnormal-to-normal in <math>G</math>, it is also intermediately subnormal-to-normal in any intermediate subgroup <math>K</math>.
If <math>H</math> is intermediately subnormal-to-normal in <math>G</math>, it is also intermediately subnormal-to-normal in any intermediate subgroup <math>K</math>.
==References==
* {{paperlink|KurdachenkoSubbotin}}: In this page, the term ''transitively normal'' is used for this concept. Note that [[transitively normal subgroup]] means something different on this wiki.

Latest revision as of 20:54, 28 February 2009

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]


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Definition

Symbol-free definition

A subgroup of a group is termed intermediately subnormal-to-normal if it satisfies the following equivalent conditions:

  1. Whenever it is subnormal in any intermediate subgroup, then it is also normal in that intermediate subgroup.
  2. Whenever it is 2-subnormal in any intermediate subgroup, then it is also normal in that intermediate subgroup.
  3. In any intermediate subgroup, it is a subgroup with self-normalizing normalizer. In other words, its normalizer in any intermediate subgroup is a self-normalizing subgroup.
  4. It has a subnormalizer, and the subnormalizer is equal to the normalizer.
  5. Whenever it is ascendant in any intermediate subgroup, then it is also normal in that intermediate subgroup.
  6. Whenever it is hypernormalized in any intermediate subgroup, then it is also normal in that intermediate subgroup.

Equivalence of definitions

Further information: Equivalence of definitions of intermediately subnormal-to-normal subgroup

Formalisms

In terms of the intermediately operator

This property is obtained by applying the intermediately operator to the property: subnormal-to-normal subgroup
View other properties obtained by applying the intermediately operator

In terms of the intermediately operator

This property is obtained by applying the intermediately operator to the property: subgroup with self-normalizing normalizer
View other properties obtained by applying the intermediately operator

Relation with other properties

Stronger properties

Weaker properties

Metaproperties

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If H is intermediately subnormal-to-normal in G, it is also intermediately subnormal-to-normal in any intermediate subgroup K.

References

  • Transitivity of normality and pronormal subgroups by L. A. Kurdachenko and I. Ya. Subbotin, Combinatorial group theory, discrete groups, and number theory, Volume 421, Page 201 - 210(Year 2006): More info: In this page, the term transitively normal is used for this concept. Note that transitively normal subgroup means something different on this wiki.
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