Subgroups & cosets

A subgroup $H$ of a group $G$ is a subset that is itself a group under the same operation. Lagrange's theorem states that for any finite group $G$ and subgroup $H$, the order $|H|$ divides $|G|$. The index $[G:H]$ is the number of distinct left cosets $gH=\{gh:h\in H\}$, and $|G|=[G:H]|H|$. Cosets partition $G$ into equal-sized pieces, so every element belongs to exactly one coset of $H$.