Coset/Examples/Symmetry Group of Equilateral Triangle/Cosets of Reflection Subgroup

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Examples of Cosets

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


Let $H \subseteq D_3$ be defined as:

$H = \set {e, r}$


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

The left cosets of $H$ are:

\(\ds H\) \(=\) \(\ds \set {e, r}\)
\(\ds \) \(=\) \(\ds e H\)
\(\ds \) \(=\) \(\ds r H\)
\(\ds s H\) \(=\) \(\ds \set {s e, s r}\)
\(\ds \) \(=\) \(\ds \set {s, q}\)
\(\ds \) \(=\) \(\ds q H\)
\(\ds t H\) \(=\) \(\ds \set {t e, t r}\)
\(\ds \) \(=\) \(\ds \set {t, p}\)
\(\ds \) \(=\) \(\ds p H\)

The right cosets of $H$ are:

\(\ds H\) \(=\) \(\ds \set {e, r}\)
\(\ds \) \(=\) \(\ds H e\)
\(\ds \) \(=\) \(\ds H r\)
\(\ds H s\) \(=\) \(\ds \set {e s, r s}\)
\(\ds \) \(=\) \(\ds \set {s, p}\)
\(\ds \) \(=\) \(\ds H p\)
\(\ds H t\) \(=\) \(\ds \set {e t, r t}\)
\(\ds \) \(=\) \(\ds \set {t, q}\)
\(\ds \) \(=\) \(\ds H q\)


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.