Editing Free group (section)

== Construction ==
The '''free group''' ''F<sub>S</sub>'' 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''<sup>&minus;1</sup>, in a set ''S''<sup>&minus;1</sup>. Let ''T''&nbsp;=&nbsp;''S''&nbsp;∪&nbsp;''S''<sup>&minus;1</sup>, and define a '''[[word (group theory)|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''&nbsp;=&nbsp;{''a'',&nbsp;''b'',&nbsp;''c''}, then ''T''&nbsp;=&nbsp;{''a'',&nbsp;''a''<sup>&minus;1</sup>,&nbsp;''b'',&nbsp;''b''<sup>&minus;1</sup>,&nbsp;''c'',&nbsp;''c''<sup>&minus;1</sup>}, and
:<math>a b^3 c^{-1} c a^{-1} c\,</math>
is a word in ''S''.

If an element of ''S'' lies immediately next to its inverse, the word may be simplified by omitting the c,&nbsp;c<sup>&minus;1</sup> pair:
:<math>a b^3 c^{-1} c a^{-1} c\;\;\longrightarrow\;\;a b^3 \, a^{-1} c.</math>
A word that cannot be simplified further is called '''reduced'''.

The free group ''F<sub>S</sub>'' is defined to be the group of all reduced words in ''S'', with [[concatenation]] of words (followed by reduction if necessary) as group operation.  The identity is the empty word.

A reduced word is called '''cyclically reduced''' if its first and last letter are not inverse to each other. Every word is [[Conjugacy class|conjugate]] to a cyclically reduced word, and a cyclically reduced conjugate of a cyclically reduced word is a cyclic permutation of the letters in the word.  For instance ''b''<sup>&minus;1</sup>''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''.