nLab
index of a subgroup

Idea

For $H\hookrightarrow G$ a subgroup, its index is the number $\mid G:H\mid$ of $H$-cosets in $G$.

Definition

Definition

For $H\hookrightarrow G$ a subgroup, its index is the cardinality

$$\mid G:H\mid ≔\mid G/H\mid$${\vert G : H\vert} \coloneqq {\vert G/H\vert}

of the set $G/H$ of cosets.

Properties

Multiplicativity

Proposition

If $H\hookrightarrow K\hookrightarrow G$ is a sequence of subgroups, then there is a (non-canonical) bijection of (products of) cosets

$$G/K\times K/H\stackrel{\simeq }{\to }G/H$$G/K \,\times \, K/H \stackrel{\simeq}{\to} G/H

and accordingly the indices satisfy

$$\mid G:K\mid \dot{\mid K:H\mid }=\mid G:H\mid .$${\vert G : K\vert} \dot {\vert K : H\vert} = {\vert G : H\vert} \,.

Finite groups

Theorem

(Lagrange’s theorem)

If$G$ is a finite group, then the index of any subgroup is the quotient

$$\mid G:H\mid =\frac{\mid G\mid }{\mid H\mid }$${\vert G : H\vert} = \frac{{\vert G\vert}}{\vert H\vert}

of the order (cardinality = number of elements) of $G$ by that of $H$.

Examples

• For $n\in \mathbb{N}$ with $n\ge 1$ and $\mathbb{Z}\stackrel{\cdot n}{\hookrightarrow }\mathbb{Z}$ the subgroup of the integers given by those that are multiples of $n$, the index is $n$.