Free group


Image:F2 Cayley Graph.png|right|thumb|Diagram showing the Cayley graph for the free group on two generators. Each vertex represents an element of the free group, and each edge represents multiplication by a or b.
In mathematics, the free group over a given set consists of all words that can be built from members of, considering two words to be different unless their equality follows from the group axioms. The members of are called generators of, and the number of generators is the rank of the free group.
An arbitrary group is called free if it is isomorphic to for some subset of, that is, if there is a subset of such that every element of can be written in exactly one way as a product of finitely many elements of and their inverses.
A related but different notion is a abelian group">abelian group">abelian group; both notions are particular instances of a free object from universal algebra. As such, free groups are defined by their universal property.

History

Free groups first arose in the study of hyperbolic geometry, as examples of Fuchsian groups. In an 1882 paper, Walther von Dyck pointed out that these groups have the simplest possible presentations. The algebraic study of free groups was initiated by Jakob Nielsen in 1924, who gave them their name and established many of their basic properties. Max Dehn realized the connection with topology, and obtained the first proof of the full Nielsen–Schreier theorem. Otto Schreier published an algebraic proof of this result in 1927, and Kurt Reidemeister included a comprehensive treatment of free groups in his 1932 book on combinatorial topology. Later on in the 1930s, Wilhelm Magnus discovered the connection between the lower central series of free groups and free Lie algebras.

Examples

The group of integers is free of rank 1; a generating set is. The integers are also a free abelian group, although all free groups of rank are non-abelian. A free group on a two-element set occurs in the proof of the Banach–Tarski paradox and is described there.
On the other hand, any nontrivial finite group cannot be free, since the elements of a free generating set of a free group have infinite order.
In algebraic topology, the fundamental group of a bouquet of k circles is the free group on a set of k elements.

Construction

The free group with free generating set can be constructed as follows. is a set of symbols, and we suppose for every in there is a corresponding "inverse" symbol,, in a set. Let, and define a word in to be any written product of elements of. That is, a word in is an element of the monoid generated by. The empty word is the word with no symbols at all. For example, if, then, and
is a word in.
If an element of lies immediately next to its inverse, the word may be simplified by omitting the pair:
A word that cannot be simplified further is called reduced.
The free group is defined to be the group of all reduced words in, with concatenation of words as group operation. The identity is the empty word.
A reduced word is called cyclically reduced if its first and last letter are not inverse to each other. Every word is conjugate to a cyclically reduced word, and a cyclically reduced conjugate of a cyclically reduced word is a cyclic permutation of the letters in the word. For instance is not cyclically reduced, but is conjugate to, which is cyclically reduced. The only cyclically reduced conjugates of are,, and.

Universal property

The free group is the universal group generated by the set. This can be formalized by the following universal property: given any function from to a group, there exists a unique homomorphism making the following diagram commute :
[Image:Free Group Universal.svg|center|100px]
That is, homomorphisms are in one-to-one correspondence with functions. For a non-free group, the presence of relations would restrict the possible images of the generators under a homomorphism.
To see how this relates to the constructive definition, think of the mapping from to as sending each symbol to a word consisting of that symbol. To construct for the given, first note that sends the empty word to the identity of and it has to agree with on the elements of. For the remaining words, can be uniquely extended, since it is a homomorphism, i.e.,.
The above property characterizes free groups up to isomorphism, and is sometimes used as an alternative definition. It is known as the universal property of free groups, and the generating set is called a basis for. The basis for a free group is not uniquely determined.
Being characterized by a universal property is the standard feature of free objects in universal algebra. In the language of category theory, the construction of the free group is a functor from the category of sets to the category of groups. This functor is left adjoint to the forgetful functor from groups to sets.

Facts and theorems

Some properties of free groups follow readily from the definition:
  1. Any group is the homomorphic image of some free group. Let be a set of generators of. The natural map : is an epimorphism, which proves the claim. Equivalently, is isomorphic to a quotient group of some free group. If can be chosen to be finite here, then is called finitely generated. The kernel is the set of all relations in the presentation of ; if can be generated by the conjugates of finitely many elements of, then is finitely presented.
  2. If has more than one element, then is not abelian, and in fact the center of is trivial.
  3. Two free groups and are isomorphic if and only if and have the same cardinality. This cardinality is called the rank of the free group. Thus for every cardinal number k, there is, up to isomorphism, exactly one free group of rank k.
  4. A free group of finite rank n > 1 has an exponential growth rate of order 2n − 1.
A few other related results are:
  1. The Nielsen–Schreier theorem: Every subgroup of a free group is free. Furthermore, if the free group has rank n and the subgroup has index e in, then is free of rank 1 + e.
  2. A free group of rank k clearly has subgroups of every rank less than k. Less obviously, a free group of rank at least 2 has subgroups of all countable ranks.
  3. The commutator subgroup of a free group of rank k > 1 has infinite rank; for example for, it is freely generated by the commutators for non-zero m and n.
  4. The free group in two elements is SQ universal; the above follows as any SQ universal group has subgroups of all countable ranks.
  5. Any group that acts on a tree, freely and preserving the orientation, is a free group of countable rank.
  6. The Cayley graph of a free group of finite rank, with respect to a free generating set, is a tree on which the group acts freely, preserving the orientation. As a topological space, this Cayley graph is contractible. For a finitely presented group, the natural homomorphism defined above,, defines a covering map of Cayley graphs, in fact a universal covering. Hence, the fundamental group of the Cayley graph is isomorphic to the kernel of, the normal subgroup of relations among the generators of. The extreme case is when, the trivial group, considered with as many generators as, all of them trivial; the Cayley graph is a bouquet of circles, and its fundamental group is itself.
  7. Any subgroup of a free group,, corresponds to a covering space of the bouquet of circles, namely to the Schreier coset graph of. This can be used to give a topological proof of the Nielsen-Schreier theorem above.
  8. The groupoid approach to these results, given in the work by P.J. Higgins below, is related to the use of covering spaces above. It allows more powerful results, for example on Grushko's theorem, and a normal form for the fundamental groupoid of a graph of groups. In this approach there is considerable use of free groupoids on a directed graph.
  9. Grushko's theorem has the consequence that if a subset of a free group on n elements generates and has n elements, then generates freely.

Free abelian group

The free abelian group on a set is defined via its universal property in the analogous way, with obvious modifications:
Consider a pair, where is an abelian group and is a function. is said to be the free abelian group on with respect to if for any abelian group and any function, there exists a unique homomorphism such that
The free abelian group on can be explicitly identified as the free group modulo the subgroup generated by its commutators,, i.e.
its abelianisation. In other words, the free abelian group on is the set of words that are distinguished only up to the order of letters. The rank of a free group can therefore also be defined as the rank of its abelianisation as a free abelian group.

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. answered the first question by showing that any two nonabelian free groups have the same first-order theory, and answered both questions, showing that this theory is decidable.
A similar unsolved question in free probability theory asks whether the von Neumann group algebras of any two non-abelian finitely generated free groups are isomorphic.