# Subgroup having a 1-closed transversal

Jump to: navigation, search
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]

## 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 $G$

## 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 Subgroup having a twisted subgroup as transversal|FULL LIST, MORE INFO
retract has a transversal that is a normal subgroup (via permutably complemented) (via permutably complemented) Subgroup having a twisted subgroup as transversal|FULL LIST, MORE INFO
direct factor normal subgroup with a normal complement (via retract) (via retract) Normal subgroup having a 1-closed transversal, Subgroup having a twisted subgroup as transversal|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 Subgroup having a symmetric 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