Free product

Free product

In abstract algebra, the free product of groups constructs a group from two or more given ones. Given, for example, groups "G" and "H", the free product "G*H" can be constructed as follows: given presentations of "G" and of "H", take the generators of "G" and of "H", take the disjoint union of those, and adjoin the corresponding relations for "G" and for "H". This is a presentation of "G*H", the point being that there should be no interaction between "G" and "H" in the free product. If "G" and "H" are infinite cyclic groups, for example, "G*H" is a free group on two generators.

The free product applies to the theory of fundamental groups in algebraic topology. If connected spaces "X" and "Y" are joined at a single point (via the wedge sum), the fundamental group of the resulting space will be the free product of the fundamental groups of "X" and of "Y". This is a special case of van Kampen's theorem. The modular group is a free product of cyclic groups of orders 2 and 3, up to a problem with defining it to within index 2. Groups can be shown to have free product structure by means of group actions on trees.

The above definition may not look like an intrinsic one. The dependence on the choice of presentation can be eliminated by showing that the free product is the coproduct in the category of groups.


The more general construction of free product with amalgamation is correspondingly a pushout in the same category. Suppose "G" and "H" are given as before, along with group homomorphisms

:varphi : F ightarrow Gmbox{ and }psi : F ightarrow H.

where "F" is some arbitrary group. Start with the free product "G*H" and adjoin as relations


for every "f" in "F". In other words take the smallest normal subgroup "N" of "G*H" containing all of those elements on the left-hand side, 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 φ and ψ, is the quotient group

:(G * H)/N.,

The amalgamation has forced an identification between φ("F") in "G" with ψ("F") in "H", element by element. This is the construction needed to compute the fundamental group of two connected spaces joined along a connected subspace, with "F" taking the role of the fundamental group of the subspace. See: Seifert-van Kampen theorem.

Free products with amalgamation and a closely related notion of HNN extension 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 of random variables play the same role in defining "freeness" in the theory of free probability that Cartesian products play in defining statistical independence in classical probability theory.

ee also

*Free group
*HNN extension
*Graph of groups
*Bass–Serre theory
*Free object
*Direct product
*Universal property



Wikimedia Foundation. 2010.

Look at other dictionaries:

  • free product — noun A group formed from others by concatenating their presentations (when those presentations are written with disjoint sets of generators) …   Wiktionary

  • Free probability — is a mathematical theory which studies non commutative random variables. The freeness property is the analogue of the classical notion of independence, and it is connected with free products. This theory was initiated by Dan Voiculescu around… …   Wikipedia

  • free offer — ➔ offer2 * * * free offer UK US noun [C] MARKETING ► a free product or service that a company offers to its customers in order to get more business: »Register online for printable coupons, promotional codes and other free offers from your… …   Financial and business terms

  • Product of groups — In mathematics, a product of groups usually refers to a direct product of groups, but may also mean: *semidirect product *product of subgroups *wreath product *free product …   Wikipedia

  • free group — noun A group (set with a particular kind of binary operation) that has a presentation without relators; equivalently, a free product of some number of copies of ℤ …   Wiktionary

  • Free trade debate — Free trade is one of the most debated topics in economics of the 20th and 21st century Fact|date=December 2007. Arguments over free trade can be divided into economic, moral, and socio political arguments. The academic debate among economists is… …   Wikipedia

  • Product Red — Product Red, styled as (PRODUCT)RED, is a brand licensed to partner companies such as Nike, American Express (UK), Apple Inc., Starbucks, Converse, Bugaboo, Penguin Classics (UK International), Gap, Emporio Armani, Hallmark (US) and Dell. It was… …   Wikipedia

  • Free Grace theology — refers to a distinct view of Christian topics, such as faith, repentance, assurance of salvation, and perseverance, that is tied to dispensationalism. Two organizations that promote it are the Grace Evangelical Society (GES), and the Free Grace… …   Wikipedia

  • product liability — liability in tort or delict in respect of produce. Essentially an application of the law of tort although the term can be used to cover liability under sale and supply of goods. Although there is now a Europe wide regime of strict liability, the… …   Law dictionary

  • Free software — or software libre is software that can be used, studied, and modified without restriction, and which can be copied and redistributed in modified or unmodified form either without restriction, or with minimal restrictions only to ensure that… …   Wikipedia