# Difference between revisions of "Generating set of a group"

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===Cayley graph=== | ===Cayley graph=== | ||

− | {further|[[Cayley graph of a group]]}} | + | {{further|[[Cayley graph of a group]]}} |

Given a generating set of a group, we can construct the [[Cayley graph of a group|Cayley graph]] of the group with respect to that generating set. This is a graph whose vertex set is the set of elements of the group and where there is an edge between two vertices whenever one can be taken to the other by left multiplying by a generator. | Given a generating set of a group, we can construct the [[Cayley graph of a group|Cayley graph]] of the group with respect to that generating set. This is a graph whose vertex set is the set of elements of the group and where there is an edge between two vertices whenever one can be taken to the other by left multiplying by a generator. | ||

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The generators of a group may not in general be independent. That is, there may be words in the generators that simplify to the identity in the given group. Such a word is termed a [[relation]] between the generators. Relations can be viewed as elements of the free group on symbols corresponding to the generators. Thus relations form a [[normal subgroup]] of the free group. Specifying a set of relations whose normal closure is this normal subgroup is what we call a [[presentation of a group]]. | The generators of a group may not in general be independent. That is, there may be words in the generators that simplify to the identity in the given group. Such a word is termed a [[relation]] between the generators. Relations can be viewed as elements of the free group on symbols corresponding to the generators. Thus relations form a [[normal subgroup]] of the free group. Specifying a set of relations whose normal closure is this normal subgroup is what we call a [[presentation of a group]]. | ||

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+ | ==Study of this notion== | ||

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+ | {{msc class|20F05}} | ||

[[Category: Properties of subsets of groups]] | [[Category: Properties of subsets of groups]] | ||

[[Category: Terminology related to combinatorial group theory]] | [[Category: Terminology related to combinatorial group theory]] |

## Revision as of 04:37, 25 August 2007

## Contents

## Definition

### Symbol-free definition

A subset of a group is termed a **generating set** if it satisfies the following equivalent conditions:

- Every element of the group can be expressed in terms of the elements of this subset by means of the group operations of multiplication and inversion.
- There is no proper subgroup of the group containing this subset
- There is a surjective map from a free group on that many generators to the given group, that sends the generators of the free group to the elements of this
*generating set*.

The elements of the generating set are termed generators.

### Definition with symbols

**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE]

## Constructs

### Cayley graph

`Further information: Cayley graph of a group`
Given a generating set of a group, we can construct the Cayley graph of the group with respect to that generating set. This is a graph whose vertex set is the set of elements of the group and where there is an edge between two vertices whenever one can be taken to the other by left multiplying by a generator.

### Presentation

The generators of a group may not in general be independent. That is, there may be words in the generators that simplify to the identity in the given group. Such a word is termed a relation between the generators. Relations can be viewed as elements of the free group on symbols corresponding to the generators. Thus relations form a normal subgroup of the free group. Specifying a set of relations whose normal closure is this normal subgroup is what we call a presentation of a group.

## Study of this notion

### Mathematical subject classification

Under the Mathematical subject classification, the study of this notion comes under the class: 20F05