Lagrange's Theorem states that if $~$G$~$ is a finite Group and $~$H$~$ a subgroup, then the order $~$|H|$~$ of $~$H$~$ divides the order $~$|G|$~$ of $~$G$~$. It generalises to infinite groups: the statement then becomes that the left cosets form a [set_partition partition], and for any pair of cosets, there is a bijection between them.

# Proof

In full generality, the cosets form a partition and are all in bijection.

To specialise this to the finite case, we have divided the $~$|G|$~$ elements of $~$G$~$ into buckets of size $~$|H|$~$ (namely, the cosets), so $~$|G|/|H|$~$ must in particular be an integer.