Intermediately subnormal-to-normal subgroup
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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:
- Whenever it is subnormal in any intermediate subgroup, then it is also normal in that intermediate subgroup.
- Whenever it is 2-subnormal in any intermediate subgroup, then it is also normal in that intermediate subgroup.
- 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.
- It has a subnormalizer, and the subnormalizer is equal to the normalizer.
- Whenever it is ascendant in any intermediate subgroup, then it is also normal in that intermediate subgroup.
- 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
- Normal subgroup
- Abnormal subgroup
- Weakly abnormal subgroup
- Pronormal subgroup
- Weakly pronormal subgroup
- Paranormal subgroup
- Polynormal subgroup
- Intermediately normal-to-characteristic subgroup: For full proof, refer: Intermediately normal-to-characteristic implies intermediately subnormal-to-normal
- Self-normalizing subgroup: For full proof, refer: Self-normalizing implies intermediately subnormal-to-normal
- Image-closed intermediately subnormal-to-normal subgroup
Weaker properties
- Subgroup having a subnormalizer
- Subnormal-to-normal subgroup
- Subgroup with self-normalizing normalizer: For full proof, refer: Normalizer of intermediately subnormal-to-normal implies self-normalizing
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 is intermediately subnormal-to-normal in , it is also intermediately subnormal-to-normal in any intermediate subgroup .
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.