Free Group

The free group FS 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 FS is defined to be the group of all reduced words in T. The group operation in FS is concatenation of words (followed by reduction if necessary). The identity is the empty word. A word is called 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−1abcb 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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