# Subgroup having a transversal comprising involutions

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 transversal comprising involutions if ... A subgroup $H$ of a group $G$ is termed a subgroup having a transversal comprising involutions if ...
1 left transversal comprising involutions it has a left transversal all of whose non-identity elements are involutions there is a subset $S$ of $G$ such that $S$ intersects each left coset of $H$ at exactly one point and such that all the non-identity elements of $S$ are involutions
2 right transversal comprising involutions it has a left transversal all of whose non-identity elements are involutions there is a subset $S$ of $G$ such that $S$ intersects each right coset of $H$ at exactly one point and such that all the non-identity elements of $S$ are involutions
3 left transversal, also right transversal, comprising involutions it has a left transversal that is also a right transversal, such that all the non-identity elements are involutions there is a subset $S$ of $G$ such that $S$ intersects each left coset of $H$ at exactly one point, $S$ intersects each right coset of $H$ at exactly one point and such that all the non-identity elements of $S$ are involutions.

## Relation with other properties

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Subgroup having a 1-closed transversal left transversal (equivalently, right transversal) that is a 1-closed subset (any set of involutions, along with the identity, is 1-closed) |FULL LIST, MORE INFO
Subgroup having a symmetric transversal left transversal (equivalently, right transversal) that is a symmetric subset Subgroup having a 1-closed transversal|FULL LIST, MORE INFO
Subgroup having a left transversal that is also a right transversal Subgroup having a 1-closed transversal, Subgroup having a symmetric transversal|FULL LIST, MORE INFO