Free group

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). 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 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.

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 all universal constructions, they give rise to a pair of adjoint functors.

Facts and theorems

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.