# Right Coset Space forms Partition

## Theorem

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

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

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

## Proof

Follows directly from:

Right Congruence Modulo Subgroup is Equivalence Relation
Relation Partitions Set iff Equivalence.

$\blacksquare$