The **free group** *F _{S}* with

**free generating set**

*S*can be constructed as follows.

*S*is a set of symbols and we suppose for every

*s*in

*S*there is a corresponding "inverse" symbol,

*s*−1, in a set

*S*−1. Let

*T*=

*S*∪

*S*−1, and define a

**word**in

*S*to be any written product of elements of

*T*. That is, a word in

*S*is an element of the monoid generated by

*T*. The empty word is the word with no symbols at all. For example, if

*S*= {

*a*,

*b*,

*c*}, then

*T*= {

*a*,

*a*−1,

*b*,

*b*−1,

*c*,

*c*−1}, and

is a word in *S*. If an element of *S* lies immediately next to its inverse, the word may be simplified by omitting the *s*, *s*−1 pair:

A word that cannot be simplified further is called **reduced**. The free group *F _{S}* is defined to be the group of all reduced words in

*T*. The group operation in

*F*is concatenation of words (followed by reduction if necessary). The identity is the empty word. A word is called

_{S}**cyclically reduced**, if its first and last letter are not inverse to each other. Every word is conjugate to a cyclically reduced word, and the cyclically reduced conjugates of a cyclically reduced word are all cyclic permutations. For instance

*b*−1

*abcb*is not cyclically reduced, but is conjugate to

*abc*, which is cyclically reduced. The only cyclically reduced conjugates of

*abc*are

*abc*,

*bca*, and

*cab*.

Read more about Free Group: Universal Property, Facts and Theorems, Free Abelian Group, Tarski's Problems

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