# Subgroup having a left transversal that is also a right transversal

From Groupprops

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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]

## Statement

Suppose is a subgroup of a group . We say that is a **subgroup having a left transversal that is also a right transversal** if there exists a subset of such that is a left transversal of and is also a right transversal of .

## Examples

### Non-examples

See subgroup need not have a left transversal that is also a right transversal for an example of a subgroup that does *not* satisfy this property. In the example, the whole group is isomorphic to free group:F2 and the subgroup is free on a countable generating set.

## Relation with other properties

### Stronger properties

## Metaproperties

### 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

### 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 conditionABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition