2-hypernormalized subgroup

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 2-hypernormalized if its normalizer is a normal subgroup.

Relation with other properties

Stronger properties

• Normal subgroup

Weaker properties

• Finitarily hypernormalized subgroup
  • Intermediately finitarily hypernormalized subgroup
  • Hypernormalized subgroup
• 2-subnormal subgroup: Also related:
  • Subnormal subgroup
  • Conjugate-permutable subgroup

Metaproperties

Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
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ABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

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.
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ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

Transfer condition

This subgroup property does not satisfy the transfer condition