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

If $H$ is a subgroup of $G$, the coset projection $-H:G\to G/H$ sends an element $g$ of $G$ to its orbit $\mathrm{gH}$.

If $s:G/H\to G$ is a section of the coset projection $-H:G\to G/H$, then $G/H×H\to G$ given by $\left(gH,h\right)↦s\left(gH\right){h}^{-1}$ is a bijection. Its inverse is given by the set map $G\to G/H×H$ given by $g↦\left(gH,{g}^{-1}s\left(gH\right)\right)$. Note that the induced product projections $G\to G/H$ conincides with the coset projection.

This argument can be internalized to a group object $G$ and a subgroup object $G$ in a category $C$. In this case, the coset projection $-H:G\to G/H$ is the coequalizer of the action on $G$ by multiplication of $H$. The coset projection need not have a section. However, in case such sections exist, each section $s$ of the coset projection, the above argument internalized yields an isomorphism

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

Even more generally, if $H↪K↪G$ is a sequence of subgroup objects, then each section of the projection $G/H\to G/K$ yields an isomorphism

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

Returning to the case of ordinary groups, i.e. group objects internal to $\mathrm{Set}$, where the external axiom of choice is assumed to hold, the coset projection, being a coequalizer and hence an epimorphism, has a section. This gives the multiplicative property of the indices of a sequence $H↪K↪G$ of subgroups

$\mid G:K\mid \stackrel{˙}{\mid K:H\mid }=\mid G:H\mid \phantom{\rule{thinmathspace}{0ex}}.${\vert G : K\vert} \dot {\vert K : H\vert} = {\vert G : H\vert} \,.

Finite groups

The concept of index is meaningful especially for finite groups, i.e. groups internal to $\mathrm{FinSet}$. See, for example, its role in the classification of finite simple groups.

Multiplicativity of the index has the following corollary, which is known as 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 October 31, 2013 01:00:29 by Colin Tan (137.132.250.13)