Index is One iff Subgroup equals Group

Theorem
Let $G$ be a group whose identity element is $e$.

Let $H$ be a subgroup of $G$.

Then:
 * $\index G H = 1 \iff G = H$

where $\index G H$ denotes the index of $H$ in $G$.

Proof
From Lagrange's Theorem:
 * $\index G H = \dfrac {\order G} {\order H}$

But then:
 * $\dfrac {\order G} {\order H} = 1 \iff \order G = \order H$

Hence the result.