group theory

# Contents

## Idea

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

## Definition

###### Definition

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

$G/K\phantom{\rule{thinmathspace}{0ex}}×\phantom{\rule{thinmathspace}{0ex}}K/H\stackrel{\simeq }{\to }G/H$G/K \,\times \, K/H \stackrel{\simeq}{\to} G/H

and accordingly the indices satisfy

$\mid G:K\mid \stackrel{˙}{\mid K:H\mid }=\mid G:H\mid \phantom{\rule{thinmathspace}{0ex}}.${\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 $n\in ℕ$ with $n\ge 1$ and $ℤ\stackrel{\cdot n}{↪}ℤ$ the subgroup of the integers given by those that are multiples of $n$, the index is $n$.

Revised on September 21, 2012 18:46:37 by Toby Bartels (98.23.131.250)