Subgroup having a 1-closed transversal

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Definition

No.	Shorthand	A subgroup of a group is termed a subgroup having a 1-closed transversal if ...	A subgroup $H$ of a group $G$ is termed a subgroup having a 1-closed transversal if ...
1	left transversal that is 1-closed	it has a left transversal that is a 1-closed subset, i.e., is a union of subgroups of $G$	there exists a subset $S\subseteq G$ such that $S$ intersects every left coset of $H$ in exactly one element, and $S$ is a union of subgroups of $G$
2	right transversal that is 1-closed	it has a left transversal that is a 1-closed subset, i.e., is a union of subgroups of $G$	there exists a subset $S\subseteq G$ such that $S$ intersects every right coset of $H$ in exactly one element, and $S$ is a union of subgroups of $G$
3	left transversal + right transversal + 1-closed	it has a left transversal that is also a right transversal and is also 1-closed	there exists a subset $S\subseteq G$ such that $S$ intersects every left coset of $H$ in exactly one element, $S$ intersects every right coset of $H$ in exactly one element, and $S$ is a union of subgroups of $G$

Examples

• Any permutably complemented subgroup has a 1-closed transversal, because we can take the transversal as the complementary subgroup.
• In a group of prime exponent, any subgroup has a 1-closed transversal.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
permutably complemented subgroup	has a transversal that is a subgroup	(any subgroup is a union of subgroups)	1-closed transversal not implies permutably complemented	|FULL LIST, MORE INFO
retract	has a transversal that is a normal subgroup	(via permutably complemented)	(via permutably complemented)	|FULL LIST, MORE INFO
direct factor	normal subgroup with a normal complement	(via retract)	(via retract)	|FULL LIST, MORE INFO
subgroup having a transversal comprising involutions	subgroup with a transversal all of whose non-identity elements are of order two			|FULL LIST, MORE INFO
subgroup having a twisted subgroup as transversal	subgroup with a transversal that is a twisted subgroup			|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	has a left transversal that is also a right transversal			|FULL LIST, MORE INFO
Subgroup having a symmetric transversal	has a left transversal that is a symmetric subset	follows because any 1-closed subset is symmetric		|FULL LIST, MORE INFO