Free group
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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).
Note that the notion of free group is different from the notion free abelian group.
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History
In 1882 Walther Dyck studied the concept of a free group, without naming the concept, in his paper Gruppentheoretische Studien which was published in the Mathematische Annalen. The term free group was introduced by Jakob Nielsen in 1924.
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 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 2n − 1.
Nielsen-Schreier theorem: Any subgroup of a free group is free.
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.
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. Independently, a proof for both problems, and a proof of the first problem, have been announced (both in the affirmative). Neither has yet been judged correct and complete. For details, see the open problems at [1].