Why does $|G/H|=|G|/|H|$?

Question

This is used in my textbook without any real explanation. Is this something that follows from Lagrange?

Lagrange: If $H \subseteq G$ is a subgroup of a finite group $G$, then $$|G|=|G/H||H|$$

Other than this it is used several times without reference to anything.

Thanks!

Answer 1

The reason is that:

1. All cosets are in bijection with $H$, hence they all have the same number of elements, $|H|$;

2. They make up a partition of $G$,

   so you can apply the shepherd's principle.