# Normal subgroup having a 1-closed transversal

From Groupprops

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and subgroup having a 1-closed transversal

View other subgroup property conjunctions | view all subgroup properties

## Contents

## Definition

A **normal subgroup having a 1-closed transversal** is a normal subgroup that is also a subgroup having a 1-closed transversal: it has a left transversal that is also a 1-closed subset of the group (i.e., a union of subgroups of the group).

## Relation with other properties

### Stronger properties

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

Complemented normal subgroup | normal, has a permutable complement | |FULL LIST, MORE INFO | ||

Direct factor | normal, has a normal complement | Complemented normal subgroup|FULL LIST, MORE INFO |

### Weaker properties

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

Subgroup having a 1-closed transversal | ||||

Normal subgroup | ||||

Subgroup having a left transversal that is also a right transversal | |FULL LIST, MORE INFO |