Left-transitively 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 left-transitively permutable if whenever the group is embedded as a permutable subgroup of some bigger group, the subgroup is also permutable inside the bigger group.

Definition with symbols

A subgroup $H$ of a group $G$ is termed left-transitively permutable if for any embedding of $G$ in some group $K$, such that $G$ is permutable inside $K$, $H$ is also permutable inside $K$.

Relation with other properties

Weaker properties

• Characteristic subgroup: For full proof, refer: Left-transitively permutable implies characteristic
• Normal subgroup
• Permutable subgroup

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties