Weakly abnormal 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]

|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

Definition

Definition with symbols

A subgroup $H$ of a group $G$ is termed weakly abnormal if, given any $g\in G$, $g\in H^{<g>}$. Here $H^{<g>}$ is the smallest subgroup of $G$ containing $H$, which is closed under conjugation by $g$.

Relation with other properties

Stronger properties

• Abnormal subgroup

Weaker properties

• Weakly pronormal subgroup
• Polynormal subgroup

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 a subgroup is weakly abnormal in the whole group, it is also weakly abnormal in every intermediate subgroup.