free group


multiplicative groupMathworldPlanetmath


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


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




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.


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 ( states that every subgroupMathworldPlanetmathPlanetmath ( of a free group is itself free.


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,


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 ( 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


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.

Title free group
Canonical name FreeGroup
Date of creation 2013-03-22 12:28:39
Last modified on 2013-03-22 12:28:39
Owner yark (2760)
Last modified by yark (2760)
Numerical id 43
Author yark (2760)
Entry type Definition
Classification msc 20E05
Related topic FreeModule
Related topic Subgroup
Related topic Group
Related topic FreeProduct
Defines freely generates
Defines freely generated
Defines rank
Defines empty word