In mathematics, a group

*G*is called

**free**if there is a subset

*S*of

*G*such that any element of

*G*can be written in one and only one way as a product of finitely many elements of

*S*and their inverses (disregarding trivial variations such as

*st*=

^{-1}*su*). Note that the notion of free group is different from the notion free abelian group: in this case the order in the product matters.

^{-1}ut^{-1}

Table of contents |

2 Construction 3 Universal property 4 Facts and theorems |

### Examples

The group (**Z**,+) of integers is free; we can take *S* = {1}. A free group on a two-element subset *S* occurs in the proof of the Banach-Tarski paradox and is described there.

### Construction

If *S* is any set, there always exists a free group on *S*. This free group on *S* is essentially unique in the following sense: if *F*_{1} and *F*_{2} are two free groups on the set *S*, then *F*_{1} and *F*_{2} are isomorphic, and furthermore there exists precisely one group isomorphism *f* : *F*_{1} `->` *F*_{2} such that *f*(*s*) = *s* for all *s* in *S*.

This free group on *S* is denoted by F(*S*) and can be constructed as follows. For every *s* in *S*, we introduce a new symbol *s*^{-1}. We then form the set of all finite strings consisting of symbols of *S* and their inverses. Two such strings are considered equivalent if one arises from the other by replacing two adjacent symbols *ss ^{-1}* or

*s*by the empty string. This generates an equivalence relation on the set of strings; its quotient set is defined to be F(

^{-1}s*S*). Because the equivalence relation is compatible with string concatenation, F(

*S*) becomes a group with string concatenation as operation.

If *S* is the empty set, then F(*S*) is the trivial group consisting only of its identity element.

### Universal property

The free group on *S* is characterized by the following universal property: if *G* is any group and *f* : *S* `->` *G* is any function, then there exists a unique group homomorphism *T* : F(*S*) `->` *G* such that *T*(*s*) = *f*(*s*) for all *s* in *S*.

Free groups are thus instances of the more general concept of free objects in category theory. Like all universal constructions, they give rise to a pair of adjoint functors.

### Facts and theorems

Any group *G* is a quotient group of some free group F(*S*). If *S* can be chosen to be finite here, then *G* is called *finitely generated*.

Any subgroup of a free group is free (*Nielsen-Schreier theorem*).

Any connected graph can be viewed as a path-connected topological space by treating an edge between two vertices as a continuous path between those vertices. With this understanding, the fundamental group of every connected graph is free. This fact can be used to prove the Nielsen-Schreier theorem.

If *F* is a free group on *S* and also on *T*, then *S* and *T* have the same cardinality. This cardinality is called the **rank** of the free group *F*.

If *S* has more than one element, then F(*S*) is not abelian, and in fact the center of F(*S*) is trivial (that is, consists only of the identity element).