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:
 * $\left[{G : H}\right] = 1 \iff G = H$

where $\left[{G : H}\right]$ denotes the index of $H$ in $G$.

Proof
From Lagrange's Theorem:
 * $\left[{G : H}\right] = \dfrac {\left\vert{G}\right\vert} {\left\vert{H}\right\vert}$

But then:
 * $\dfrac {\left\vert{G}\right\vert} {\left\vert{H}\right\vert} = 1 \iff \left\vert{G}\right\vert = \left\vert{H}\right\vert$

Hence the result.