# Definition:Coset

## 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 $D_3$. 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.