---
tag: 04QEBF8S49
kind: definition
name: Index of a subgroup
msc: [20A05]
defines: 0DTXBQPA2S
uses: [05FD317V2W]
sources:
  - url: https://en.wikipedia.org/wiki/Index_of_a_subgroup
    locator: lead section
---

## Definition

Let $G$ be a [[group|G5S1M9]] and let $H$ be a [[subgroup|0X1JXT47GA]] of $G$.
The index of $H$ in $G$, written $[G : H]$ or $|G : H|$, is the cardinality of the set of [[left cosets|03DEH43F4H]] 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 $2$.