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 $S$.

A left transversal is also known as a set of left coset representatives.

Right Transversal

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

A right transversal is also known as a set of right coset representatives.

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 $G$.

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 6.3$