# Difference between revisions of "Tour:Index of a subgroup"

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{{quotation|'''WHAT'S MORE''': Some further facts about index of a subgroup, some of which you'll see soon in the tour, and others, that'll be seen much later. Ignore those that use unfamiliar terminology.}} | {{quotation|'''WHAT'S MORE''': Some further facts about index of a subgroup, some of which you'll see soon in the tour, and others, that'll be seen much later. Ignore those that use unfamiliar terminology.}} | ||

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{{guided tour-bottom|beginners|Introduction three|Lagrange's theorem|Left and right coset spaces are naturally isomorphic}} | {{guided tour-bottom|beginners|Introduction three|Lagrange's theorem|Left and right coset spaces are naturally isomorphic}} |

## Revision as of 22:13, 3 July 2008

**This article adapts material from the main article:** index of a subgroup

This page is part of the Groupprops Guided tour for beginners (Jump to beginning of tour)PREVIOUS: Left and right coset spaces are naturally isomorphic |UP: Introduction three (beginners) |NEXT: Lagrange's theorem

WHAT YOU NEED TO DO:

- Read the two equivalent definitions of index (as number of left cosets, and number of right cosets)
- Convince yourself of why the two definitions are equivalent.

## Contents

## Definition

### Symbol-free definition

The index of a subgroup in a group is the following equivalent things:

- The number of left cosets of the subgroup
- The number of right cosets of the subgroup

The collection of left cosets is sometimes termed the coset space, so in this language, the index of a subgroup is the cardinality of its coset space.

### Definition with symbols

Given a subgroup of a group , the index of in , denoted , is defined in the following ways:

- It is the number of left cosets of in , i.e. the number of sets of the form .
- It is the number of right cosets of in , i.e. the number of sets of the form .

The collection of left cosets of in is sometimes termed the coset space, and is denoted . With this notation, the index of in , is the cardinality .

### Equivalence of definitions

The equivalence of definitions follows from the fact that there is a natural bijection between the collection of left cosets of a subgroup, and the collection of its right cosets, given by the map

`Further information: Left and right coset spaces are naturally isomorphic`

WHAT'S MORE: Some further facts about index of a subgroup, some of which you'll see soon in the tour, and others, that'll be seen much later. Ignore those that use unfamiliar terminology.

### Further note for finite groups

When the group is finite, then by Lagrange's theorem, the index of a subgroup is the ratio of the order of the group to the order of the subgroup.

## Facts

### Multiplicativity of the index

`Further information: Index is multiplicative`
If , then we have:

In other words, the number of cosets of in equals the number of cosets of in , times the number of cosets of in .

In fact, more is true. We can set up a bijection as follows:

However, this bijection is not a *natural* one, and, in order to define it, we first need to choose a system of coset representatives of .

### Effect of intersection on the index

`Further information: Conjugate-intersection index theorem`
If and are two subgroups of , then the index of is bounded above by the product of the indices of and of .

This follows as a consequence of the product formula. Note that equality holds if and only if .

Note that in case and are conjugate subgroups of index , the index of is bounded above by .

This page is part of the Groupprops Guided tour for beginners (Jump to beginning of tour). If you found anything difficult or unclear, make a note of it; it is likely to be resolved by the end of the tour.PREVIOUS: Left and right coset spaces are naturally isomorphic |UP: Introduction three (beginners) |NEXT: Lagrange's theorem