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In mathematics, the **free group***F*_{S} over a given set *S* consists of all words that can be built from members of *S*, considering two words to be different unless their equality follows from the group axioms (e.g. *st* = *suu*^{−1}*t*, but *s* ≠ *t*^{−1} for *s*,*t*,*u* ∈ *S*). The members of *S* are called **generators** of *F*_{S}, and the number of generators is the **rank** of the free group. An arbitrary group *G* is called **free** if it is isomorphic to *F*_{S} for some subset *S* of *G*, that is, if there is a subset *S* of *G* such that every element of *G* can be written in exactly one way as a product of finitely many elements of *S* and their inverses (disregarding trivial variations such as *st* = *suu*^{−1}*t*).

- History
- Examples
- Construction
- Universal property
- Facts and theorems
- Free abelian group
- Tarski's problems
- See also
- Notes
- References

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.

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.^{ [1] } 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.^{ [2] }^{ [3] }^{ [4] } Max Dehn realized the connection with topology, and obtained the first proof of the full Nielsen–Schreier theorem.^{ [5] } Otto Schreier published an algebraic proof of this result in 1927,^{ [6] } and Kurt Reidemeister included a comprehensive treatment of free groups in his 1932 book on combinatorial topology.^{ [7] } Later on in the 1930s, Wilhelm Magnus discovered the connection between the lower central series of free groups and free Lie algebras.

The group (**Z**,+) of integers is free of rank 1; a generating set is *S* = {1}. 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 *S* 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.

The **free group***F _{S}* with

is a word in *S*.

If an element of *S* lies immediately next to its inverse, the word may be simplified by omitting the c, c^{−1} pair:

A word that cannot be simplified further is called **reduced**.

The free group *F _{S}* is defined to be the group of all reduced words in

A 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 *b*^{−1}*abcb* is not cyclically reduced, but is conjugate to *abc*, which is cyclically reduced. The only cyclically reduced conjugates of *abc* are *abc*, *bca*, and *cab*.

The free group *F _{S}* is the universal group generated by the set

That is, homomorphisms *F _{S}* →

To see how this relates to the constructive definition, think of the mapping from *S* to *F _{S}* as sending each symbol to a word consisting of that symbol. To construct

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 *S* is called a **basis** for *F _{S}*. 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.

Some properties of free groups follow readily from the definition:

- Any group
*G*is the homomorphic image of some free group F(*S*). Let*S*be a set of*generators*of*G*. The natural map*f*: F(*S*) →*G*is an epimorphism, which proves the claim. Equivalently,*G*is isomorphic to a quotient group of some free group F(*S*). The kernel of*φ*is a set of*relations*in the presentation of*G*. If*S*can be chosen to be finite here, then*G*is called**finitely generated**. - 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). - Two free groups F(
*S*) and F(*T*) are isomorphic if and only if*S*and*T*have the same cardinality. This cardinality is called the**rank**of the free group*F*. Thus for every cardinal number*k*, there is, up to isomorphism, exactly one free group of rank*k*. - A free group of finite rank
*n*> 1 has an exponential growth rate of order 2*n*− 1.

A few other related results are:

- The Nielsen–Schreier theorem: Every 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 (*nonabelian!*) free group of rank at least 2 has subgroups of all countable ranks. - The commutator subgroup of a free group of rank
*k*> 1 has infinite rank; for example for F(*a*,*b*), it is freely generated by the commutators [*a*^{m},*b*^{n}] for non-zero*m*and*n*. - The free group in two elements is SQ universal; the above follows as any SQ universal group has subgroups of all countable ranks.
- 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).
- 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.
- The groupoid approach to these results, given in the work by P.J. Higgins below, is kind of extracted from an approach using covering spaces. 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.
- Grushko's theorem has the consequence that if a subset
*B*of a free group*F*on*n*elements generates*F*and has*n*elements, then*B*generates*F*freely.

The free abelian group on a set *S* is defined via its universal property in the analogous way, with obvious modifications: Consider a pair (*F*, *φ*), where *F* is an abelian group and *φ*: *S* → *F* is a function. *F* is said to be the **free abelian group on S with respect to φ** if for any abelian group

*f*(*φ*(*s*)) =*ψ*(*s*), for all*s*in*S*.

The free abelian group on *S* can be explicitly identified as the free group F(*S*) modulo the subgroup generated by its commutators, [F(*S*), F(*S*)], i.e. its abelianisation. In other words, the free abelian group on *S* 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.

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. Sela (2006) answered the first question by showing that any two nonabelian free groups have the same first-order theory, and Kharlampovich & Myasnikov (2006) 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.

- ↑ von Dyck, Walther (1882). "Gruppentheoretische Studien (Group-theoretical Studies)".
*Mathematische Annalen*.**20**(1): 1–44. doi:10.1007/BF01443322. S2CID 179178038. - ↑ Nielsen, Jakob (1917). "Die Isomorphismen der allgemeinen unendlichen Gruppe mit zwei Erzeugenden".
*Mathematische Annalen*.**78**(1): 385–397. doi:10.1007/BF01457113. JFM 46.0175.01. MR 1511907. S2CID 119726936. - ↑ Nielsen, Jakob (1921). "On calculation with noncommutative factors and its application to group theory. (Translated from Danish)".
*The Mathematical Scientist*. 6 (1981) (2): 73–85. - ↑ Nielsen, Jakob (1924). "Die Isomorphismengruppe der freien Gruppen".
*Mathematische Annalen*.**91**(3): 169–209. doi:10.1007/BF01556078. S2CID 122577302. - ↑ See Magnus, Wilhelm; Moufang, Ruth (1954). "Max Dehn zum Gedächtnis".
*Mathematische Annalen*.**127**(1): 215–227. doi:10.1007/BF01361121. S2CID 119917209. - ↑ Schreier, Otto (1928). "Die Untergruppen der freien Gruppen".
*Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg*.**5**: 161–183. doi:10.1007/BF02952517. S2CID 121888949. - ↑ Reidemeister, Kurt (1972 (1932 original)).
*Einführung in die kombinatorische Topologie*. Darmstadt: Wissenschaftliche Buchgesellschaft.Check date values in:`|date=`

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In mathematics, an **abelian group**, also called a **commutative group**, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.

In mathematics, a **group** is a set equipped with a binary operation that combines any two elements to form a third element in such a way that three conditions called group axioms are satisfied, namely associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, and help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study.

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In group theory, a branch of abstract algebra, a **cyclic group** or **monogenous group** is a group that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element *g* such that every other element of the group may be obtained by repeatedly applying the group operation to *g* or its inverse. Each element can be written as a power of *g* in multiplicative notation, or as a multiple of *g* in additive notation. This element *g* is called a *generator* of the group.

In mathematics, a **presentation** is one method of specifying a group. A presentation of a group *G* comprises a set *S* of **generators**—so that every element of the group can be written as a product of powers of some of these generators—and a set *R* of **relations** among those generators. We then say *G* has presentation

In the theory of abelian groups, the **torsion subgroup***A _{T}* of an abelian group

In mathematics, a **free abelian group** or **free Z-module** is an abelian group with a basis, or, equivalently, a free module over the integers. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as a linear combination of basis elements with integer coefficients. For instance, the integers with addition form a free abelian group with basis {1}. Free abelian groups have properties which make them similar to vector spaces. They have applications in algebraic topology, where they are used to define chain groups, and in algebraic geometry, where they are used to define divisors. Integer lattices also form examples of free abelian groups, and lattice theory studies free abelian subgroups of real vector spaces.

A group is a set together with an associative operation which admits an identity element and such that every element has an inverse.

In mathematics, a **module** is one of the fundamental algebraic structures used in abstract algebra. A **module over a ring** is a generalization of the notion of vector space over a field, wherein the corresponding scalars are the elements of an arbitrary given ring and a multiplication is defined between elements of the ring and elements of the module. A module taking its scalars from a ring *R* is called an *R*-module.

In mathematics, specifically group theory, the **free product** is an operation that takes two groups *G* and *H* and constructs a new group *G* ∗ *H*. The result contains both *G* and *H* as subgroups, is generated by the elements of these subgroups, and is the “universal” group having these properties, in the sense that any two homomorphisms from *G* and *H* into a group *K* factor uniquely through a homomorphism from *G* ∗ *H* to *K*. Unless one of the groups *G* and *H* is trivial, the free product is always infinite. The construction of a free product is similar in spirit to the construction of a free group.

In group theory, a branch of mathematics, the **order of a group** is its cardinality, that is, the number of elements in its set. If the group is seen multiplicatively, the **order of an element***a* of a group, sometimes also called the **period length** or **period** of *a*, is the smallest positive integer *m* such that *a*^{m} = *e*, where *e* denotes the identity element of the group, and *a*^{m} denotes the product of *m* copies of *a*. If no such *m* exists, *a* is said to have infinite order.

In algebra, a **finitely generated group** is a group *G* that has some finite generating set *S* so that every element of *G* can be written as the combination of finitely many elements of the finite set *S* and of inverses of such elements.

In universal algebra, a **variety of algebras** or **equational class** is the class of all algebraic structures of a given signature satisfying a given set of identities. For example, the groups form a variety of algebras, as do the abelian groups, the rings, the monoids etc. According to Birkhoff's theorem, a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphic images, subalgebras and (direct) products. In the context of category theory, a variety of algebras, together with its homomorphisms, forms a category; these are usually called *finitary algebraic categories*.

In mathematics, the **Grothendieck group** construction constructs an abelian group from a commutative monoid *M* in the most universal way, in the sense that any abelian group containing a homomorphic image of *M* will also contain a homomorphic image of the Grothendieck group of *M*. The Grothendieck group construction takes its name from a specific case in category theory, introduced by Alexander Grothendieck in his proof of the Grothendieck–Riemann–Roch theorem, which resulted in the development of K-theory. This specific case is the monoid of isomorphism classes of objects of an abelian category, with the direct sum as its operation.

In mathematics, specifically in group theory, the **direct product** is an operation that takes two groups *G* and *H* and constructs a new group, usually denoted *G* × *H*. This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics.

**Bass–Serre theory** is a part of the mathematical subject of group theory that deals with analyzing the algebraic structure of groups acting by automorphisms on simplicial trees. The theory relates group actions on trees with decomposing groups as iterated applications of the operations of free product with amalgamation and HNN extension, via the notion of the **fundamental group of a graph of groups.** Bass–Serre theory can be regarded as one-dimensional version of the orbifold theory.

In the mathematical subject of group theory, the **rank of a group***G*, denoted rank(*G*), can refer to the smallest cardinality of a generating set for *G*, that is

In the mathematical subject of group theory, a **co-Hopfian group** is a group that is not isomorphic to any of its proper subgroups. The notion is dual to that of a Hopfian group, named after Heinz Hopf.

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*Journal of Algebra*.**302**(2): 451–552. doi:10.1016/j.jalgebra.2006.03.033. MR 2293770. Archived from the original (PDF) on 2016-10-21. Retrieved 2015-09-04. - W. Magnus, A. Karrass and D. Solitar, "Combinatorial Group Theory", Dover (1976).
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*Geom. Funct. Anal*.**16**(3): 707–730. doi:10.1007/s00039-006-0565-8. MR 2238945. S2CID 123197664. - Serre, Jean-Pierre,
*Trees*, Springer (2003) (English translation of "arbres, amalgames, SL_{2}", 3rd edition,*astérisque***46**(1983)) - P.J. Higgins,
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