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]

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