# Right transversal of a subgroup

## Definition

### Definition with symbols

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

$S$ is also termed a system of right coset representatives of $H$ and the elements of $S$ are termed coset representatives of $H$.

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 $H$ of a group $G$ and a right transversal $S$ of $H$ in $G$. Then, we can endow $S$ with a binary operation as follows. For $x,y \in S$, we define $x \circ y$ as the left coset representative (with respect to $H$) of $xy$ in $S$. It is easy to see that this gives $S$ 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.