Index of a subgroup

Remarks

The index is also the number of right cosets, because $gH \mapsto Hg^{-1}$ is a bijection between the two sets of cosets. In $(\mathbb{Z}, +)$, the subgroup $n\mathbb{Z}$ has index $n$ for each positive integer $n$. A subgroup of a finite group has index $|G| / |H|$, but an infinite group can have a subgroup of finite index, for example $2\mathbb{Z}$ in $\mathbb{Z}$.