
The group (Z,+) of integers is free; we can take S = {1}. A free group on a twoelement subset S occurs in the proof of the BanachTarski paradox and is described there.
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^{1}s 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.
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 all universal constructions, they give rise to a pair of adjoint functors.
Any group G is a quotient group of some free group F(S). If S can be chosen to be finite here, then G is called finitely generated.
Any subgroup of a free group is free (NielsenSchreier theorem).
Any connected graph can be viewed as a pathconnected 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. This fact can be used to prove the NielsenSchreier theorem.
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.
If S has more than one element, then F(S) is not abelian.
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