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# free group

# 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
$\phi\colon F\to G$, 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 states that every 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 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 $A$.

## Mathematics Subject Classification

20E05*no label found*

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