free group

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Definition

If $F$ is a group with a subset $A$ such that for every group $G$ every function $\psi\colon A\to G$ extends to a unique homomorphism (http://planetmath.org/GroupHomomorphism) $\phi\colon F\to G$ , then $F$ is said to be a free group 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 set.

The infinite cyclic group $\mathbb{Z}$ 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 $2\times 2$ integer matrices generated by

$\left(\begin{array}[]{cc}1&2\\ 0&1\end{array}\right)\phantom{.}$

and

$\left(\begin{array}[]{cc}1&0\\ 2&1\end{array}\right).$

If $F$ is a free group freely generated by a set $A$ , where $|A|>1$ , then for distinct $a,b\in A$ the set $\{a^{n}b^{n}\mid 0<n\in\mathbb{Z}\}$ generates a free group of countably infinite rank.

Properties

If a free group $F$ is freely generated by $A$ , then $A$ is a minimal generating set 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 concept, and free groups of different ranks are non-isomorphic.

For every cardinal number $\kappa$ there is, up to isomorphism, exactly one free group of rank $\kappa$ . The abelianization of a free group of rank $\kappa$ is a free abelian group of rank $\kappa$ .

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

The Nielsen-Schreier Theorem (http://planetmath.org/NielsenSchreierTheorem) states that every subgroup (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 $a_{i}$ for $i$ in some index set $I$ . We refer to $A$ as an alphabet and the elements of $A$ as letters. A syllable is a symbol of the form $a_{i}^{n}$ for $n\in\mathbb{Z}$ . It is customary to write $a$ for $a^{1}$ . Define a word to be a finite sequence of syllables. For example,

$a_{2}^{3}a_{1}a_{4}^{-1}a_{3}^{2}a_{2}^{-3}$

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

Define a binary operation, called the product, on $\mathcal{W}[A]$ by concatenation of words. To illustrate, if $a_{2}^{3}a_{1}$ and $a_{1}^{-1}a_{3}^{4}$ are elements of $\mathcal{W}[A]$ then their product is simply $a_{2}^{3}a_{1}a_{1}^{-1}a_{3}^{4}$ . This gives $\mathcal{W}[A]$ the structure of a monoid. The empty word $1$ acts as a right and left identity (http://planetmath.org/LeftIdentityAndRightIdentity) in $\mathcal{W}[A]$ , and is the only element which has an inverse. In order to give $\mathcal{W}[A]$ the structure of a group, two more ideas are needed.

If $v=u_{1}a_{i}^{0}u_{2}$ is a word where $u_{1},u_{2}$ are also words and $a_{i}$ is some element of $A$ , an elementary contraction of type I replaces the occurrence of $a^{0}$ by $1$ . Thus, after this type of contraction we get another word $w=u_{1}u_{2}$ . If $v=u_{1}a_{i}^{p}a_{i}^{q}u_{2}$ is a word, an elementary contraction of type II replaces the occurrence of $a_{i}^{p}a_{i}^{q}$ by $a_{i}^{p+q}$ which results in $w=u_{1}a_{i}^{p+q}u_{2}$ . 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 $u, v$ equivalent (denoted $u\sim v$ ) if one can be obtained from the other by a finite sequence of elementary contractions or expansions. This is an equivalence relation on $\mathcal{W}[A]$ . Let $\mathcal{F}[A]$ be the set of equivalence classes of words in $\mathcal{W}[A]$ . Then $\mathcal{F}[A]$ is group under the operation

$[u][v]=[uv]$

where $[u]\in\mathcal{F}[A]$ . 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 $[u]=[a_{1}a_{3}^{2}]$ , then $[u]^{-1}=[a_{3}^{-2}a_{1}^{-1}]$ .

It can be shown that $\mathcal{F}[A]$ is a free group freely generated by the set $\{[x]\mid x\in A\}$ . Moreover, a group is free if and only if it is isomorphic to $\mathcal{F}[A]$ 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