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单词 FreeGroup
释义

free group


\\PMlinkescapephrase

multiplicative groupMathworldPlanetmath

Definition

If F is a group with a subset A such that for every group Gevery function ψ:AG extends to a unique homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/GroupHomomorphism)ϕ:FG, then F is said to be a free groupMathworldPlanetmath of rank |A|,and we say that A freely generates F.

Examples

A group with only one element is a free group of rank 0, freely generated by the empty setMathworldPlanetmath.

The infinite cyclic group is a free group of rank 1, freely generated by either {1} or {-1}.

An example of a free group of rank 2 is the multiplicative group of 2×2 integer matrices generated by

(1201)

and

(1021).

If F is a free group freely generated by a set A, where |A|>1, then for distinct a,bA the set {anbn0<n} generates a free group of countably infiniteMathworldPlanetmath rank.

Properties

If a free group F is freely generated by A, then A is a minimalPlanetmathPlanetmath generating setPlanetmathPlanetmath for F, and no set of smaller cardinality than A can generate F.It follows that if F is freely generated by both A and B, then |A|=|B|. So the rank of a free group is a well-defined conceptMathworldPlanetmath, and free groups of different ranks are non-isomorphic.

For every cardinal numberMathworldPlanetmath κ there is, up to isomorphismMathworldPlanetmathPlanetmath, exactly one free group of rank κ.The abelianizationMathworldPlanetmath of a free group of rank κ is a free abelian group of rank κ.

Every group is a homomorphic image of some free group.More precisely, if G is a group generated by a set of cardinality κ, then G is a homomorphic image of every free group of rank κ or more.

The Nielsen-Schreier Theorem (http://planetmath.org/NielsenSchreierTheorem) states that every subgroupMathworldPlanetmathPlanetmath (http://planetmath.org/Subgroup) of a free group is itself free.

Construction

For any set A, the following construction gives a free group of rank |A|.

Let A be a set with elements ai for i in some index setMathworldPlanetmathPlanetmath I.We refer to A as an alphabet and the elements of A as letters.A syllable is a symbol of the form ain for n.It is customary to write a for a1. Define a word to be a finite sequencePlanetmathPlanetmath of syllables. For example,

a23a1a4-1a32a2-3

is a five-syllable word. Notice that there exists a unique empty wordPlanetmathPlanetmath, i.e., the word with no syllables, usually written simply as 1.Denote the set of all words formed from elements of A by 𝒲[A].

Define a binary operationMathworldPlanetmath, called the productMathworldPlanetmathPlanetmathPlanetmathPlanetmath, on 𝒲[A] by concatenationMathworldPlanetmath of words. To illustrate, if a23a1 and a1-1a34 are elements of 𝒲[A] then their product is simply a23a1a1-1a34.This gives 𝒲[A] the structureMathworldPlanetmath of a monoid.The empty word 1 acts as a right and left identityPlanetmathPlanetmath (http://planetmath.org/LeftIdentityAndRightIdentity) in 𝒲[A], and is the only element which has an inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. In order to give 𝒲[A] the structure of a group, two more ideas are needed.

If v=u1ai0u2 is a word where u1,u2 are also words and ai is some element of A, an elementary contraction of type I replaces the occurrence of a0 by 1. Thus, after this type of contraction we get another word w=u1u2. If v=u1aipaiqu2 is a word, an elementary contraction of type II replaces the occurrence of aipaiq by aip+q which results in w=u1aip+qu2. In either of these cases, we also say that w is obtained from v by an elementary contraction, or that v is obtained from w by an elementary expansion.

Call two words u,v equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (denoted uv) if one can be obtained from the other by a finite sequence of elementary contractions or expansions. This is an equivalence relation on 𝒲[A]. Let [A] be the set of equivalence classesMathworldPlanetmath of words in 𝒲[A]. Then [A] is group under the operationMathworldPlanetmath

[u][v]=[uv]

where [u][A]. The inverse [u]-1 of an element [u] is obtained by reversing the order of the syllables of [u] and changing the sign of each syllable. For example, if [u]=[a1a32], then [u]-1=[a3-2a1-1].

It can be shown that [A] is a free group freely generated by the set {[x]xA}. Moreover, a group is free if and only if it is isomorphic to [A] for some set A.

Titlefree group
Canonical nameFreeGroup
Date of creation2013-03-22 12:28:39
Last modified on2013-03-22 12:28:39
Owneryark (2760)
Last modified byyark (2760)
Numerical id43
Authoryark (2760)
Entry typeDefinition
Classificationmsc 20E05
Related topicFreeModule
Related topicSubgroup
Related topicGroup
Related topicFreeProduct
Definesfreely generates
Definesfreely generated
Definesrank
Definesempty word
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