nLab index of a subgroup

Idea

For $H \hookrightarrow G$ a subgroup inclusion, its index is the number ${\vert G: H\vert}$ of $H$-cosets in $G$, hence roughly is the number of copies of $H$ that appear in $G$.

Definition

Definition

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

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

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 $gH$.

If $s : G/H\to G$ is a section of the coset projection $-H: G\to G/H$, then $G/H \times H \to G$ given by $( g H , h )\mapsto s( g H ) h^{-1}$ is a bijection. Its inverse is given by the set map $G\to G/H \times H$ given by $g\mapsto ( g H , g^{-1} s( g H ) )$. 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

$$G/H \times H \overset{\simeq}\rightarrow G \, .$$

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

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

Returning to the case of ordinary groups, i.e. group objects internal to $Set$, 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 $H \hookrightarrow K \hookrightarrow G$ of subgroups

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

Finite groups

The concept of index is meaningful especially for finite groups, i.e. groups internal to 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

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

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

Examples

• For $n \in \mathbb{N}$ with $n \geq 1$ 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 $n$.