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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-1ut-1''). NB. Note that the notion of free group is different from the notion free abelian group. == 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-1s'' by the empty string. This generates an equivalence relation on the set of strings; its quotient set is defined to be F(''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 most universal constructions, they give rise to a pair of adjoint functors. ==Facts and theorems== Any group ''G'' is isomorphic to a quotient group of some free group F(''S''). If ''S'' can be chosen to be finite here, then ''G'' is called ''finitely generated''. 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''. For every cardinal number ''k'', there is, up to isomorphism, exactly one free group of rank ''k''. 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). A free group of finite rank ''n'' > 1 has an exponential growth rate of order 2''n'' − 1. Any subgroup of free group is free. This statement easily follows from the following: 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 and any free group is isomorphic to a fundamental group of a graph. A free group of rank ''k'' clearly has subgroups of every rank less than ''k''. Less obviously, a free group of rank greater than 1 has subgroups of all countable ranks. ==See also== * Generating set of a group de:Freie Gruppe
