# 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 = SS−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.

### Other articles related to "groups, group, free group, free groups, free":

Xenogears - Plot - Story
... to defend the village against both groups of gears ... The group follows Stone to Zeboim, an excavation site ... Stone takes the girl while the group fights Id, the mysterious red gear's pilot, who wants the girl, but is stopped by Wiseman ...
Free Group - Tarski's Problems
... Around 1945, Alfred Tarski asked whether the free groups on two or more generators have the same first order theory, and whether this theory is decidable ... question by showing that any two nonabelian free groups have the same first order theory, and Kharlampovich Myasnikov (2006) answered both questions, showing that this theory is decidable ... A similar unsolved (in 2011) question in free probability theory asks whether the von Neumann group algebras of any two non-abelian finitely generated free groups are isomorphic ...
Boundedly Generated Group - Free Groups Are Not Boundedly Generated
... literature that it is obvious that finitely generated free groups are not boundedly generated ... rather than algebraic, so can be applied to a wider class of groups, for example Gromov-hyperbolic groups ... Since for any n ≥ 2, the free group on 2 generators F2 contains the free group on n generators Fn as a subgroup of finite index (in fact n – 1), once ...
Free Probability
... Free probability is a mathematical theory that studies non-commutative random variables ... The "freeness" or free independence property is the analogue of the classical notion of independence, and it is connected with free products ... This theory was initiated by Dan Voiculescu around 1986 in order to attack the free group factors isomorphism problem, an important unsolved problem in the theory of ...
Train Track Map - History
... Train track maps for free group automorphisms were introduced in a 1992 paper of Bestvina and Handel ... train tracks on surfaces, but the free group case is substantially different and more complicated ... to solve the Scott conjecture which says that for every automorphism α of a finitely generated free group Fn the fixed subgroup of α is free of rank at most n ...

### Famous quotes containing the words group and/or free:

It is not God that is worshipped but the group or authority that claims to speak in His name. Sin becomes disobedience to authority not violation of integrity.