Definition:Transversal (Group Theory)

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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 $D_3$.

SymmetryGroupEqTriangle.png

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