Definition:Coset

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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 $D_3$.

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:

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


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.


Sources

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