Definition:Transversal (Group Theory)

Definition
Let $G$ be a group.

Let $H$ be a subgroup of $G$.

Let $S \subseteq G$ be a subset of $G$.

Left Transversal
$S$ is a left transversal for $H$ in $G$ iff every left coset of $H$ contains exactly one element of $S$.

Right Transversal
$S$ is a right transversal for $H$ in $G$ iff every right coset of $H$ contains exactly one element of $S$.

Transversal
A transversal for $H$ in $G$ is either a left transversal or a right transversal.

Clearly if $S$ is a transversal for $H$ it contains $\left[{G : H}\right]$ elements, where $\left[{G : H}\right]$ denotes the index of $H$ in $G$