Proof of Lagrange's theorem

Proof

1. Let $G$ be a finite group and let $H$ be a subgroup of $G$.
2. For $a, b \in G$, let $a \sim b$ when $a^{-1}b \in H$. Since $H$ contains the identity, inverses and products, this relation is reflexive, symmetric and transitive.
3. The class of $a$ under $\sim$ is the left coset $aH$, because $a^{-1}b \in H$ exactly when $b \in aH$.
4. The equivalence classes form a partition of $G$. Thus $G$ is the disjoint union of $k$ distinct left cosets for some integer $k \ge 1$.
5. [[Each left coset has $|H|$ elements|0DKMBWTD7P]].
6. Thus $|G| = k\,|H|$, and $|H|$ divides $|G|$.