# Left-transitively permutable subgroup

From Groupprops

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

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]

## 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 of a group is termed **left-transitively permutable** if for any embedding of in some group , such that is permutable inside , is also permutable inside .

## 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 transitiveABOUT 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