Lagrange's Theorem

Let $(G,\ast)$ be a group whose underlying set is finite, and let $H$ be a subgroup of $G$. Then the following hold.

1. The sets $H$ and $G/H$, where $G/H$ is the set of left cosets of $H$ in $G$, are finite and nonempty, and

$$|G|=|H|\,[G:H],$$

where $|G|$ is the order of $G$, $|H|$ is the number of elements of $H$, and $[G:H]=|G/H|$ is the index of $H$ in $G$.

2. $|H|$ divides $|G|$.