Definition:Coset

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Definition

Let $G$ be a group, and let $H \le G$.


Left Coset

The left coset of $x$ modulo $H$, or left coset of $H$ by $x$, is:

$x H = \set {y \in G: \exists h \in H: y = x h}$

This is the equivalence class defined by left congruence modulo $H$.


That is, it is the subset product with singleton:

$x H = \set x H$


Right Coset

The right coset of $y$ modulo $H$, or right coset of $H$ by $y$, is:

$H y = \set {x \in G: \exists h \in H: x = h y}$

This is the equivalence class defined by right congruence modulo $H$.


That is, it is the subset product with singleton:

$H y = H \set y$


Examples

Symmetry Group of Equilateral Triangle: Cosets of Reflection Subgroup

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 $r$.


The left cosets of $H$ are:

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


The right cosets of $H$ are:

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


Dihedral Group $D_3$: Cosets of $\gen b$

Consider the dihedral group $D_3$.

$D_3 = \gen {a, b: a^3 = b^2 = e, a b = b a^{-1} }$


Let $H \subseteq D_3$ be defined as:

$H = \gen b$

where $\gen b$ denotes the subgroup generated by $b$.


As $b$ has order $2$, it follows that:

$\gen b = \set {e, b}$


Left Cosets

The left cosets of $H$ are:

\(\displaystyle e H\) \(=\) \(\displaystyle \set {e, b}\)
\(\displaystyle \) \(=\) \(\displaystyle b H\)
\(\displaystyle \) \(=\) \(\displaystyle H\)


\(\displaystyle a H\) \(=\) \(\displaystyle \set {a, a b}\)
\(\displaystyle \) \(=\) \(\displaystyle a b H\)


\(\displaystyle a^2 H\) \(=\) \(\displaystyle \set {a^2, a^2 b}\)
\(\displaystyle \) \(=\) \(\displaystyle a^2 b H\)


Right Cosets

The right cosets of $H$ are:

\(\displaystyle H e\) \(=\) \(\displaystyle \set {e, b}\)
\(\displaystyle \) \(=\) \(\displaystyle H b\)
\(\displaystyle \) \(=\) \(\displaystyle H\)


\(\displaystyle H a\) \(=\) \(\displaystyle \set {a, a^2 b}\)
\(\displaystyle \) \(=\) \(\displaystyle H a^2 b\)


\(\displaystyle H a^2\) \(=\) \(\displaystyle \set {a^2, a b}\)
\(\displaystyle \) \(=\) \(\displaystyle H a b\)


Subgroup of Infinite Cyclic Group

Let $G = \gen a$ be an infinite cyclic group.

Let $s \in \Z_{>0}$ be a (strictly) positive integer.

Let $H$ be the subgroup of $G$ defined as:

$H := \gen {a^s}$

Then a complete repetition-free list of the cosets of $H$ in $G$ is:

$S = \set {H, aH, a^2 H, \ldots, a^{s - 1} H}$


Also see

  • Results about cosets can be found here.


Sources