Hereditarily permutable subgroup

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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, 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 hereditarily permutable if every subgroup of that subgroup is a permutable subgroup in the whole group.

Definition with symbols

A subgroup $H$ of a group $G$ is termed hereditarily permutable in $G$ if whenever $K\leq H$, $K$ is a permutable subgroup of $G$.

Formalisms

In terms of the hereditarily operator

This property is obtained by applying the hereditarily operator to the property: permutable subgroup
View other properties obtained by applying the hereditarily operator

Relation with other properties

Stronger properties

• Hereditarily normal subgroup
• Central subgroup

Weaker properties

• Right-transitively permutable subgroup
• Intersection-transitively permutable subgroup
• Permutable subgroup

Facts

• The Baer norm of a group, defined as the intersection of normalizers of all its subgroups, is hereditarily permutable. For full proof, refer: Baer norm is hereditarily permutable