Lagrange's theorem

Cosets. For $g\in G$, the left coset $gH=\{gh:h\in H\}$ has cardinality $|H|$ (the map $h\mapsto gh$ is a bijection).

Partition. The left cosets form a partition of $G$: they cover $G$ (each $g\in gH$) and are disjoint (if $g_1H\cap g_2H\neq\emptyset$, then $g_1H=g_2H$).

Count. If there are $k$ distinct cosets, then $|G|=k|H|$, so $|H|$ divides $|G|$.

Conclusion. Lagrange's theorem holds; the integer $k=|G|/|H|$ is the index of $H$ in $G$.