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index of a subgroup

Contents

Idea

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

Definition

Definition

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

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

of the set G/HG/H of cosets.

Properties

Multiplicativity

If HH is a subgroup of GG, the coset projection H:GG/H-H: G\to G/H sends an element gg of GG to its orbit gHgH.

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

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

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

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

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

Returning to the case of ordinary groups, i.e. group objects internal to SetSet, 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 HKGH \hookrightarrow K \hookrightarrow G of subgroups

|G:K||K:H|˙=|G:H|. {\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 GG is a finite group, then the index of any subgroup is the quotient

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

of the order (cardinality = number of elements) of GG by that of HH.

=–

Examples

  • For nn \in \mathbb{N} with n1n \geq 1 and n\mathbb{Z} \stackrel{\cdot n}{\hookrightarrow} \mathbb{Z} the subgroup of the integers given by those that are multiples of nn, the index is nn.

Revised on April 9, 2014 21:07:18 by Urs Schreiber (82.169.114.243)