Coset. If $H \subset G$ is a Subgroup, then it defines an Equivalence relation, and its equivalence classes are called cosets (Left coset), denoted $a H$ (for some element $a \in G$, like notation $[a]$).

$a \equiv b ~ \text{mod}H$ if $\exists h \in H$ s.t. $a=bh$

All cosets are bijective, so they all have the same cardinality, and they all have the same cadinality as $H$ (because of the coset $1H = H$).

Quotient group

Index of a subgroup