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

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{{termrelatedto|combinatorial group theory}} | {{termrelatedto|combinatorial group theory}} | ||

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==Definition== | ==Definition== | ||

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* 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. (note that if the subset is a [[symmetric subset]], i.e., it is closed under taking inverses, then every element of the group must be a product of elements in the subset. Symmetric subsets arise, for instance, when we take a union of subgroups). | * 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. (note that if the subset is a [[symmetric subset]], i.e., it is closed under taking inverses, then every element of the group must be a product of elements in the subset. Symmetric subsets arise, for instance, when we take a union of subgroups). | ||

− | * There is no proper subgroup of the group containing this subset | + | * There is no proper subgroup of the group containing this subset<section end=beginner/> |

* 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''. | * 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''. | ||

− | + | <section begin=beginner/> | |

The elements of the generating set are termed generators. | The elements of the generating set are termed generators. | ||

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where for each <math>a_i</math>, either <math>a_i \in S</math> or <math>a_i^{-1} \in S</math> (here, the <math>a_i</math>s are not necessarily distinct). In the situation where <math>S</math> is a [[symmetric subset]] (i.e. <math>a_i \in S \implies a_i^{-1} \in S</math>) we do not need to throw in inverses. This happens, for instance, when <math>S</math> is a union of subgroups of <math>G</math>. | where for each <math>a_i</math>, either <math>a_i \in S</math> or <math>a_i^{-1} \in S</math> (here, the <math>a_i</math>s are not necessarily distinct). In the situation where <math>S</math> is a [[symmetric subset]] (i.e. <math>a_i \in S \implies a_i^{-1} \in S</math>) we do not need to throw in inverses. This happens, for instance, when <math>S</math> is a union of subgroups of <math>G</math>. | ||

− | * If <math>H</math> is a [[proper subgroup]] of <math>G</math> (i.e. <math>H</math> is a [[subgroup]] of <math>G</math> that is not equal to the whole of <math>G</math>), then <math>H</math> cannot contain <math>S</math>. | + | * If <math>H</math> is a [[proper subgroup]] of <math>G</math> (i.e. <math>H</math> is a [[subgroup]] of <math>G</math> that is not equal to the whole of <math>G</math>), then <math>H</math> cannot contain <math>S</math>.<section end=beginner/> |

* Consider the natural map from the free group on as many generators as elements of <math>S</math>, to the group <math>G</math>, which maps the freely generating set to the elements of <math>S</math>. This gives a surjective homomorphism from the free group, to <math>G</math>. | * Consider the natural map from the free group on as many generators as elements of <math>S</math>, to the group <math>G</math>, which maps the freely generating set to the elements of <math>S</math>. This gives a surjective homomorphism from the free group, to <math>G</math>. | ||

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==Constructs== | ==Constructs== |

## Revision as of 11:08, 4 July 2008

This article defines a property of subsets of groups

View other properties of subsets of groups|View properties of subsets of abelian groups|View subgroup properties

This term is related to: combinatorial group theory

View other terms related to combinatorial group theory | View facts related to combinatorial group theory

## 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. (note that if the subset is a symmetric subset, i.e., it is closed under taking inverses, then every element of the group must be a product of elements in the subset. Symmetric subsets arise, for instance, when we take a union of subgroups).
- 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

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

- For any element , we can write:

where for each , either or (here, the s are not necessarily distinct). In the situation where is a symmetric subset (i.e. ) we do not need to throw in inverses. This happens, for instance, when is a union of subgroups of .

- If is a proper subgroup of (i.e. is a subgroup of that is not equal to the whole of ), then cannot contain .
- Consider the natural map from the free group on as many generators as elements of , to the group , which maps the freely generating set to the elements of . This gives a surjective homomorphism from the free group, to .

## 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