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

## Contents

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]

## Statement

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

## 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

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Subgroup of finite group the whole group is finite subgroup of finite group has a left transversal that is also a right transversal |FULL LIST, MORE INFO
Subgroup of finite index the index of the subgroup in the whole group is finite subgroup of finite index has a left transversal that is also a right transversal |FULL LIST, MORE INFO
Normal subgroup the left cosets are the same as the right cosets every left transversal is a right transversal. Note that existence of transversals requires the axiom of choice Subgroup having a symmetric transversal|FULL LIST, MORE INFO
Permutably complemented subgroup has a permutable complement we can take the permutable complement as the left transversal that is also a right transversal Subgroup having a 1-closed transversal, Subgroup having a symmetric transversal, Subgroup having a twisted subgroup as transversal|FULL LIST, MORE INFO
Retract has a normal complement (via permutably complemented, via subset-conjugacy-closed) Subgroup having a 1-closed transversal, Subgroup having a symmetric transversal, Subgroup having a twisted subgroup as transversal|FULL LIST, MORE INFO
Subgroup having a symmetric transversal has a left transversal (equivalently, right transversal) that is a symmetric subset |FULL LIST, MORE INFO
Subgroup having a 1-closed transversal has a left transversal (equivalently, right transversal) that is a 1-closed subset, i.e., a union of subgroups Subgroup having a symmetric transversal|FULL LIST, MORE INFO
Subgroup having a transversal comprising involutions has a left transversal (equivalently, right transversal) all of whose non-identity elements are involutions Subgroup having a 1-closed transversal, Subgroup having a symmetric transversal|FULL LIST, MORE INFO

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