# Subgroup having a symmetric transversal

From Groupprops

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]

## Contents

## Definition

A subgroup of a group is termed a **subgroup having a symmetric transversal** if there exists a symmetric subset of such that is a left transversal of . 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 | Subgroup having a 1-closed 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 twisted subgroup as transversal|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 | Subgroup having a 1-closed transversal|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 |