# Left Coset Space forms Partition

## Theorem

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

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

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

## Proof

Follows directly from:

Left Congruence Modulo Subgroup is Equivalence Relation
Relation Partitions Set iff Equivalence

$\blacksquare$