Intermediately subnormal-to-normal subgroup

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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

• 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 $H$ 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.