- Free product
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 unionof 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 groupon 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 groupis a free product of cyclic groups of orders 2 and 3, up toa 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
coproductin the category of groups.
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
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 extensionare basic building blocks in Bass–Serre theoryof 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 probabilitythat Cartesian products play in defining statistical independencein classical probability theory.
Graph of groups
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