Left Coset Space forms Partition

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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

$\blacksquare$


Also see


Sources