Intermediately subnormal-to-normal subgroup

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 defining ingredient::subgroup with self-normalizing normalizer. In other words, its normalizer in any intermediate subgroup is a defining ingredient::self-normalizing subgroup.
 * 4) It has a defining ingredient::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.

Stronger properties

 * Weaker than::Normal subgroup
 * Weaker than::Abnormal subgroup
 * Weaker than::Weakly abnormal subgroup
 * Weaker than::Pronormal subgroup
 * Weaker than::Weakly pronormal subgroup
 * Weaker than::Paranormal subgroup
 * Weaker than::Polynormal subgroup
 * Weaker than::Intermediately normal-to-characteristic subgroup:
 * Weaker than::Self-normalizing subgroup:
 * Weaker than::Image-closed intermediately subnormal-to-normal subgroup

Weaker properties

 * Stronger than::Subgroup having a subnormalizer
 * Stronger than::Subnormal-to-normal subgroup
 * Stronger than::Subgroup with self-normalizing normalizer:

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