# Definition:Coset

## Definition

Let $\struct {S, \circ}$ be an algebraic structure.

Let $\struct {H, \circ}$ be a subgroup of $\struct {S, \circ}$.

### Left Coset

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

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

That is, it is the subset product with singleton:

$x \circ H = \set x \circ H$

### Right Coset

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

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

That is, it is the subset product with singleton:

$H \circ y = H \circ \set y$

## Also defined as

It is usual for the algebraic structure $S$ in fact to be a group.

However, this does not prevent the more general definition to be applied when necessary.

## 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:

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

The right cosets of $H$ are:

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

### Symmetric Group on 3 Letters: Cosets of Alternating Subgroup

Consider the symmetric group on 3 Letters.

Let $S_3$ denote the set of permutations on $3$ letters.

The symmetric group on $3$ letters is the algebraic structure:

$\struct {S_3, \circ}$

where $\circ$ denotes composition of mappings.

Let $H \subseteq S_3$ be defined as:

$H = \set {e, \tuple {1 2 3}, \tuple {1 3 2} }$

The cosets of $H$ are:

 $\ds e H$ $=$ $\ds \set {e, \tuple {1 2 3}, \tuple {1 3 2} }$ $\ds$ $=$ $\ds \tuple {1 2 3} H$ $\ds$ $=$ $\ds \tuple {1 3 2} H$ $\ds$ $=$ $\ds H$
 $\ds \tuple {1 2} H$ $=$ $\ds \set {\tuple {1 2}, \tuple {1 2} \tuple {1 2 3}, \tuple {1 2} \tuple {1 3 2} }$ $\ds$ $=$ $\ds \set {\tuple {1 2}, \tuple {2 3}, \tuple {1 3} }$

### 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:

 $\ds e H$ $=$ $\ds \set {e, b}$ $\ds$ $=$ $\ds b H$ $\ds$ $=$ $\ds H$

 $\ds a H$ $=$ $\ds \set {a, a b}$ $\ds$ $=$ $\ds a b H$

 $\ds a^2 H$ $=$ $\ds \set {a^2, a^2 b}$ $\ds$ $=$ $\ds a^2 b H$

### Right Cosets

The right cosets of $H$ are:

 $\ds H e$ $=$ $\ds \set {e, b}$ $\ds$ $=$ $\ds H b$ $\ds$ $=$ $\ds H$

 $\ds H a$ $=$ $\ds \set {a, a^2 b}$ $\ds$ $=$ $\ds H a^2 b$

 $\ds H a^2$ $=$ $\ds \set {a^2, a b}$ $\ds$ $=$ $\ds H a b$

### 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.