# Definition:Transversal (Group Theory)

## Contents

## 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 $G$** if and only if every left coset of $H$ contains **exactly one** element of $S$.

### Right Transversal

$S$ is a **right transversal for $H$ in $G$** if and only if 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 $\index G H$ elements, where $\index G H$ denotes the index of $H$ in $G$.

## Examples

### Reflection Subgroup in Equilateral Triangle

Consider the symmetry group of the equilateral triangle $D_3$.

Let $H \subseteq D_3$ be defined as:

- $H = \set {e, r}$

where:

- $e$ denotes the identity mapping
- $r$ denotes reflection in the line $r$.

Some of the left transversals of $H$ are given by:

- $\set {e, s, t}$
- $\set {e, q, p}$
- $\set {r, s, p}$

and so on.

### Integer Multiples in Integers

Let $\struct {\Z, +}$ denote the additive group of integers.

Let $\struct {n \Z, +}$ denote the additive group of integer multiples.

Then a transversal for $\struct {n \Z, +}$ in $\struct {\Z, +}$ is:

- $\set {0, 1, \ldots, n - 1}$

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 6.3$. Index. Transversals