Table of contents |
2 Construction 3 Universal property 4 Facts and theorems |
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.
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 F1 and F2 are two free groups on the set S, then F1 and F2 are isomorphic, and furthermore there exists precisely one group isomorphism f : F1 -> F2 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.
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 (Nielsen-Schreier theorem).
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.
This fact can be used to prove the Nielsen-Schreier 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, and in fact the center of F(S) is trivial (that is, consists only of the identity element).Examples
Construction
Universal property
Facts and theorems