Right Coset Space forms Partition

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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 $G$:

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


Follows directly from:

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


Also see