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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.
View a complete list of subgroup properties|Get subgroup property lookup help |Get exploration suggestions[SHOW MORE]

VIEW RELATED: Subgroup metaproperty satisfactions |
VIEW FACTS THAT: use, or start with, a subgroup satisfying the property| prove, or show that, a subgroup satisfies the property
RANDOM SUBGROUP PROPERTY: 2-subnormal subgroup: A normal subgroup of a normal subgroup. Need not be normal, because normality is not transitive.

Contents 1 Definition 1.1 Symbol-free definition 2 Relation with other properties 2.1 Stronger properties 2.2 Weaker properties 3 Metaproperties 3.1 Transitivity 3.2 Intermediate subgroup condition 3.3 Transfer condition

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
ABOUT THIS PROPERTY: |
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.
ABOUT THIS PROPERTY: |
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