Subgroup having a left transversal that is also a right transversal

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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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VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

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		|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		|FULL LIST, MORE INFO
Retract	has a normal complement	(via permutably complemented, via subset-conjugacy-closed)		|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			|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			|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).
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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