# Subgroup having a twisted subgroup as transversal

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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

Suppose is a subgroup of a group . We say that is a **subgroup having a twisted subgroup as transversal** if there exists a twisted subgroup of that is also a left transversal for in , i.e., it intersects every left coset of in at exactly one point. Note that will automatically also be a right transversal for in .

## Relation with other properties

### Stronger properties

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

permutably complemented subgroup | |FULL LIST, MORE INFO | |||

direct factor | (via permutably complemented) | (via permutably complemented) | |FULL LIST, MORE INFO | |

retract | (via permutably complemented) | (via permutably complemented) | |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 | |FULL LIST, MORE INFO | |||

subgroup having a symmetric transversal | |FULL LIST, MORE INFO | |||

subgroup having a left transversal that is also a right transversal | Subgroup having a 1-closed transversal|FULL LIST, MORE INFO |