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 $|G|$.

Idea

The left cosets of the subgroup cut the group into blocks of equal size. The order of the group is the number of blocks times the size of one block.