- 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 thedisjoint 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" areinfinite cyclic group s, for example, "G*H" is afree group on two generators.The free product applies to the theory of

fundamental group s inalgebraic topology . Ifconnected space s "X" and "Y" are joined at a single point (via thewedge 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 ofvan Kampen's theorem . Themodular group is a free product ofcyclic group s 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 ofgroup action s 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 thecategory of groups .**Generalization**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 withgroup homomorphism s:$varphi\; :\; F\; ightarrow\; Gmbox\{\; and\; \}psi\; :\; F\; ightarrow\; H.$

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

:$varphi(f)psi(f)^\{-1\}=e$

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 inBass–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 variable s play the same role in defining "freeness" in the theory offree probability thatCartesian product s play in definingstatistical independence in classicalprobability theory .**ee also***

Free group

*HNN extension

*Graph of groups

*Bass–Serre theory

*Free object

*Direct product

*Universal property **References***

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