Transversal (Group Theory)/Examples

Examples of Transversals in the Context of Group Theory

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:

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