Lagrange's Theorem states that if 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 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.

Lagrange theorem on subgroup size