Let $(G,\ast)$ be a \reftext{def:group-2026a}{group} whose underlying set is \reftext{def:finite-set-2026a}{finite}, and let $H$ be a \reftext{def:subgroup-2026a}{subgroup} of $G$. Then the following hold.

1. The sets $H$ and $G/H$, where $G/H$ is the set of \reftext{def:left-coset-order-index-2026a}{left cosets} of $H$ in $G$, are finite and nonempty, and

where $|G|$ is the order of $G$, $|H|$ is the \reftext{def:number-of-elements-2026a}{number of elements} of $H$, and $[G:H]=|G/H|$ is the index of $H$ in $G$.

2. $|H|$ \reftext{def:divides-natural-numbers-2026a}{divides} $|G|$.