In

_{''i''} is either an element of ''G'' or an element of ''H''. Such a word may be reduced using the following operations:
* Remove an instance of the identity element (of either ''G'' or ''H'').
* Replace a pair of the form ''g''_{1}''g''_{2} by its product in ''G'', or a pair ''h''_{1}''h''_{2} by its product in ''H''.
Every reduced word is an alternating product of elements of ''G'' and elements of ''H'', e.g.
:$g\_1\; h\_1\; g\_2\; h\_2\; \backslash cdots\; g\_k\; h\_k.$
The free product ''G'' ∗ ''H'' is the group whose elements are the reduced words in ''G'' and ''H'', under the operation of concatenation followed by reduction.
For example, if ''G'' is the infinite cyclic group $\backslash langle\; x\backslash rangle$, and ''H'' is the infinite cyclic group $\backslash langle\; y\backslash rangle$, then every element of ''G'' ∗ ''H'' is an alternating product of powers of ''x'' with powers of ''y''. In this case, ''G'' ∗ ''H'' is isomorphic to the free group generated by ''x'' and ''y''.

_{''G''} is a set of generators and ''R''_{''G''} is a set of relations), and suppose that
:$H\; =\; \backslash langle\; S\_H\; \backslash mid\; R\_H\; \backslash rangle$
is a presentation for ''H''. Then
:$G\; *\; H\; =\; \backslash langle\; S\_G\; \backslash cup\; S\_H\; \backslash mid\; R\_G\; \backslash cup\; R\_H\; \backslash rangle.$
That is, ''G'' ∗ ''H'' is generated by the generators for ''G'' together with the generators for ''H'', with relations consisting of the relations from ''G'' together with the relations from ''H'' (assume here no notational clashes so that these are in fact

_{''n''} denotes the free group on ''n'' generators.
Another example is the

The subgroups of a free product of two groups with an amalgamated subgroup

Transactions of the American Mathematical Society 150: 227–255. For example, the homomorphisms from $G$ and $H$ to the quotient group $(G\; *\; H)/N$ that are induced by $\backslash varphi$ and $\backslash psi$ are both injective, as is the induced homomorphism from $F$. Free products with amalgamation and a closely related notion of

Categories and Groupoids, by Philip Higgins

Group products Algebraic topology Free algebraic structures

mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, specifically group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...

, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

s, 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 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 (the universal group with a given set of generators).
The free product is the coproduct
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The copr ...

in the category of groups. That is, the free product plays the same role in group theory that disjoint union
In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A (t ...

plays in set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...

, or that the direct sum plays in module theory. Even if the groups are commutative, their free product is not, unless one of the two groups is the trivial group. Therefore, the free product is not the coproduct in the category of abelian groups In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab.
Properties
The zero object of Ab ...

.
The free product is important in algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify u ...

because of van Kampen's theorem, which states that the fundamental group of the union of two path-connected topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...

s whose intersection is also path-connected is always an amalgamated free product of the fundamental groups of the spaces. In particular, the fundamental group of the wedge sum
In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if ''X'' and ''Y'' are pointed spaces (i.e. topological spaces with distinguished basepoints x_0 and y_0) the wedge sum of ''X'' and ''Y'' is the qu ...

of two spaces (i.e. the space obtained by joining two spaces together at a single point) is simply the free product of the fundamental groups of the spaces.
Free products are also important in Bass–Serre theory 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 ...

, the study of groups acting
Acting is an activity in which a story is told by means of its enactment by an actor or actress who adopts a character—in theatre, television, film, radio, or any other medium that makes use of the mimetic mode.
Acting involves a broad ...

by automorphisms on trees. Specifically, any group acting with finite vertex stabilizers on a tree may be constructed from finite groups using amalgamated free products and HNN extension In mathematics, the HNN extension is an important construction of combinatorial group theory.
Introduced in a 1949 paper ''Embedding Theorems for Groups'' by Graham Higman, Bernhard Neumann, and Hanna Neumann, it embeds a given group ''G'' into ...

s. Using the action of the modular group
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractiona ...

on a certain tessellation
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...

of the hyperbolic plane, it follows from this theory that the modular group is isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

to the free product of cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binary ...

s of orders 4 and 6 amalgamated over a cyclic group of order 2.
Construction

If ''G'' and ''H'' are groups, aword
A word is a basic element of language
Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communic ...

in ''G'' and ''H'' is a product of the form
:$s\_1\; s\_2\; \backslash cdots\; s\_n,$
where each ''s''Presentation

Suppose that :$G\; =\; \backslash langle\; S\_G\; \backslash mid\; R\_G\; \backslash rangle$ is apresentation
A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...

for ''G'' (where ''S''disjoint union
In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A (t ...

s).
Examples

For example, suppose that ''G'' is a cyclic group of order 4, :$G\; =\; \backslash langle\; x\; \backslash mid\; x^4\; =\; 1\; \backslash rangle,$ and ''H'' is a cyclic group of order 5 :$H\; =\; \backslash langle\; y\; \backslash mid\; y^5\; =\; 1\; \backslash rangle.$ Then ''G'' ∗ ''H'' is the infinite group :$G\; *\; H\; =\; \backslash langle\; x,\; y\; \backslash mid\; x^4\; =\; y^5\; =\; 1\; \backslash rangle.$ Because there are no relations in a free group, the free product of free groups is always a free group. In particular, :$F\_m\; *\; F\_n\; \backslash cong\; F\_,$ where ''F''modular group
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractiona ...

$PSL\_2(\backslash mathbf\; Z)$. It is isomorphic to the free product of two cyclic groups
:$PSL\_2(\backslash mathbf\; Z)\; =\; (\backslash mathbf\; Z\; /\; 2\; \backslash mathbf\; Z)\; \backslash ast\; (\backslash mathbf\; Z\; /\; 3\; \backslash mathbf\; Z).$
Generalization: Free product with amalgamation

The more general construction of free product with amalgamation is correspondingly a special kind of pushout in the samecategory
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
...

. Suppose $G$ and $H$ are given as before, along with monomorphisms (i.e. injective group homomorphisms):
:$\backslash varphi\; :\; F\; \backslash rightarrow\; G\; \backslash \; \backslash ,$ and $\backslash \; \backslash ,\; \backslash psi\; :\; F\; \backslash rightarrow\; H,$
where $F$ is some arbitrary group. Start with the free product $G\; *\; H$ and adjoin as relations
:$\backslash varphi(f)\backslash psi(f)^=1$
for every $f$ in $F$. In other words, take the smallest normal subgroup $N$ of $G\; *\; H$ containing all elements on the left-hand side of the above equation, which are tacitly being considered in $G\; *\; H$ by means of the inclusions of $G$ and $H$ in their free product. The free product with amalgamation of $G$ and $H$, with respect to $\backslash varphi$ and $\backslash psi$, is the quotient group
:$(G\; *\; H)/N.\backslash ,$
The amalgamation has forced an identification between $\backslash varphi(F)$ in $G$ with $\backslash psi(F)$ in $H$, element by element. This is the construction needed to compute the fundamental group of two connected spaces joined along a path-connected subspace, with $F$ taking the role of the fundamental group of the subspace. See: Seifert–van Kampen theorem.
Karrass and Solitar have given a description of the subgroups of a free product with amalgamation.A. Karrass and D. Solitar (1970The subgroups of a free product of two groups with an amalgamated subgroup

Transactions of the American Mathematical Society 150: 227–255. For example, the homomorphisms from $G$ and $H$ to the quotient group $(G\; *\; H)/N$ that are induced by $\backslash varphi$ and $\backslash psi$ are both injective, as is the induced homomorphism from $F$. Free products with amalgamation and a closely related notion of

HNN extension In mathematics, the HNN extension is an important construction of combinatorial group theory.
Introduced in a 1949 paper ''Embedding Theorems for Groups'' by Graham Higman, Bernhard Neumann, and Hanna Neumann, it embeds a given group ''G'' into ...

are basic building blocks in Bass–Serre theory of groups acting on trees.
In other branches

One may similarly define free products of other algebraic structures than groups, including algebras over a field. Free products of algebras ofrandom variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...

s play the same role in defining " freeness" in the theory of free probability that Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\tim ...

s play in defining statistical independence
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of o ...

in classical probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

.
See also

* Direct product of groups *Coproduct
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The copr ...

* Graph of groups
* Kurosh subgroup theorem
* Normal form for free groups and free product of groups
* Universal property
Notes

References

* * {{planetmath reference, urlname=FreeProductWithAmalgamatedSubgroup, title=Free product with amalgamated subgroupCategories and Groupoids, by Philip Higgins

Group products Algebraic topology Free algebraic structures