Subgroup having a symmetric transversal

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Definition

A subgroup $H$ of a group $G$ is termed a subgroup having a symmetric transversal if there exists a symmetric subset $S$ of $G$ such that $S$ is a left transversal of $G$. Note that for a symmetric subset, being a left transversal is equivalent to being a right transversal, because left and right coset spaces are naturally isomorphic via the bijection induced by the inverse map.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
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 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

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Subgroup having a left transversal that is also a right transversal