proof of Lagrange’s theorem

We know that the cosets $Hg$ form a partition of $G$ (see the coset entry for proof of this.) Since $G$ is finite, we know it can be completely decomposed into a finite number of cosets. Call this number $n$. We denote the $i$th coset by $H{a}_i$ and write $G$ as

$$G=H{a}_1\cup H{a}_2\cup \mathrm{\cdots }\cup H{a}_n$$

since each coset has $|H|$ elements, we have

$$|G|=|H|\cdot n$$

and so $|H|$ divides $|G|$, which proves Lagrange’s theorem. $\mathrm{\square }$