# free group

## Definition

If $F$ is a group with a subset $A$ such that for every group $G$
every function $\psi :A\to G$ extends to a unique homomorphism^{} (http://planetmath.org/GroupHomomorphism)
$\varphi :F\to G$, then $F$ is said to be a *free group ^{}* of

*rank*$|A|$, and we say that $A$

*freely generates*$F$.

## Examples

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

$$\left(\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)$$ |

and

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

If $F$ is a free group freely generated by a set $A$, where $|A|>1$, then for distinct $a,b\in A$ the set $$ 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 |