# Coset Space forms Partition

## Theorem

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

### Left Coset Space forms Partition

The left coset space of $H$ forms a partition of its group $G$, and hence:

 $\displaystyle x \equiv^l y \pmod H$ $\iff$ $\displaystyle x H = y H$ $\displaystyle \neg \paren {x \equiv^l y} \pmod H$ $\iff$ $\displaystyle x H \cap y H = \O$

### Right Coset Space forms Partition

The right coset space of $H$ forms a partition of its group $G$:

 $\displaystyle x \equiv^r y \pmod H$ $\iff$ $\displaystyle H x = H y$ $\displaystyle \neg \paren {x \equiv^r y} \pmod H$ $\iff$ $\displaystyle H x \cap H y = \O$

## Examples

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

<onlyinclude> Consider the dihedral group $D_3$.

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

From Dihedral Group $D_3$: Cosets of $\gen b$, the left cosets of of the subgroup $\gen b$ generated by $b$ are:

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

It follows from Coset Space forms Partition that these are consequences of:

 $\displaystyle b^{-1} e$ $=$ $\displaystyle b^{-1} = b \in H$ $\displaystyle \paren {a b}^{-1} a = b^{-1} a^{-1} a$ $=$ $\displaystyle b^{-1} = b \in H$ $\displaystyle \paren{a^2 b}^{-1} a^2 = b^{-1} a^{-2} a^2$ $=$ $\displaystyle b^{-1} = b \in H$