Coset/Examples
Examples of Cosets
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:
- $S = \set {H, aH, a^2 H, \ldots, a^{s - 1} H}$