# Right transversal of a subgroup

## Contents

## Definition

### Definition with symbols

Let be a subgroup of a group . Then a subset of is termed a **right transversal** of in if intersects every right coset of at exactly one element.

is also termed a **system of right coset representatives** of and the elements of are termed coset representatives of .

Sometimes, the term **section** is also used for this notion.

## Dual notion

The dual notion is that of left transversal of a subgroup.

## Algebra loop structure to the right transversal

Consider a subgroup of a group and a right transversal of in . Then, we can endow with a binary operation as follows. For , we define as the left coset representative (with respect to ) of in . It is easy to see that this gives the structure of an algebra loop.

### When the transversal is a subgroup

If we choose the transversal such that it forms a subgroup, then the algebra loop structure is just the usual group multiplication, so the algebra loop is canonically isomorphic to the subgroup.

### When the original subgroup is normal

If the original subgroup is normal, then the algebra loop structure on *any* left transversal is a group, and this group is isomorphic to the quotient group for that normal subgroup.