# Subgroup of finite index has a left transversal that is also a right transversal

From Groupprops

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., subgroup of finite index) must also satisfy the second subgroup property (i.e., subgroup having a left transversal that is also a right transversal)

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## Contents

## Statement

Suppose is a group and is a subgroup of finite index in , i.e., the index of in is finite. Then, there exists a subset of such that is both a left transversal and a Right transversal (?) of in . In other words, intersects every left coset of in exactly one element, and it also intersects every right coset of in exactly one element.

## Related facts

### Other conditions that guarantee the existence of a left transversal that is also a right transversal

Property | Meaning | Proof of implication |
---|---|---|

normal subgroup | each left coset is a right coset | |

permutably complemented subgroup | there is a permutable complement | |

subgroup of finite group | the whole group is finite | subgroup of finite group has a left transversal that is also a right transversal |

subset-conjugacy-closed subgroup | subset-conjugacy-closed implies left transversal that is also a right transversal | |

retract | has a normal complement | (via subset-conjugacy-closed, via permutably complemented) |

### Other facts about left and right cosets

- Left cosets partition a group
- Left cosets are in bijection via left multiplication
- Left and right coset spaces are naturally isomorphic

## Facts used

## Proof

### Proof outline

We use fact (2) to *quotient out* by the normal core, and we use fact (1) to prove the result in the quotient and then *pull back* to the original group.