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 $$G$$