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Wikipedia / Free group

In mathematics, the free group consists of all words, considering two words to be different unless their equality follows from the group axioms are called generators, and the number of generators is the rank of the free group. An arbitrary group is called free if it is isomorphic for some subset).

A related but different notion is a free 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 (discrete groups acting by isometries on the hyperbolic plane). 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

of integers. The integers are also a free abelian group 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 (a set of k loops having only one point in common) is the free group on a set of k elements.

Construction

The free group with free generating set, and define a word is an element of the monoid.

A word that cannot be simplified further is called reduced.

, with concatenation of words (followed by reduction if necessary) 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.

Universal property

is the universal. This can be formalized by the following universal property, there exists a unique homomorphism making the following diagram commute (where the unnamed mapping denotes the inclusion): . For a non-free group, the presence of relations would restrict the possible images of the generators under a homomorphism.

.

The above property characterizes free groups up to isomorphism, and is sometimes used as an alternative definition. It is known as the universal property is called a basis. 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 (similar to most constructions of free objects) 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. be a set of generators is an epimorphism is isomorphic to a quotient group is called finitely generated is the set of all relations in the presentation is finitely presented.

  2. is not abelian, and in fact the center is trivial (that is, consists only of the identity element).

  3. have the same cardinality. This cardinality is called the rank. 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 has rank n has index e is free of rank 1 + e(n–1).

  2. A free group of rank k clearly has subgroups of every rank less than k. Less obviously, a (nonabelian!) free group of rank at least 2 has subgroups of all countable ranks.

  3. The commutator subgroup of a free group of rank k, 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 (given by 1 plus the Euler characteristic of the quotient graph).

  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 (a one-dimensional simplicial complex is contractible, defines a covering map, in fact a universal covering. Hence, the fundamental group itself.

  7. , corresponds to a covering space of the bouquet of circles, namely to the Schreier coset graph. 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 on n and has n freely.

Free abelian group

is said to be the

, i.e. its abelianisation 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 (as of 2011) question in free probability theory asks whether the von Neumann group algebras of any two non-abelian finitely generated free groups are isomorphic.

See also

Notes

References