Index of a subgroup

Definition

Let $G$ be a group and let $H$ be a subgroup of $G$. The index of $H$ in $G$, written $[G : H]$ or $|G : H|$, is the cardinality of the set of left cosets of $H$ in $G$.

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 $\mathbb{Z}$.

Explanations

L1 human

The index counts how many disjoint copies of the subgroup you need to cover the whole group. For the even integers inside the integers, you need two copies: the even and the odd integers. Thus the index of the even integers is $2$.