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 could be constructed in this way: given presentations of G and of H, take the generators of G and of H, take the disjoint union of those, add the corresponding relations for G and for H. That 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, G*H is then 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 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 given G and H as before, but with group homomorphisms
- <math>\varphi : F \rightarrow G\mbox{ and }\psi : F \rightarrow H.<math>
Start with the free product G*H and add as relations
- <math>\varphi(f)\psi(f)^{-1}=e<math>
for every f in F. In other words take the smallest normal subgroup N of G*H containing all of those elements on the LHS, 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
- <math>(G*H)/N.<math>
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 fundamental group of the subspace. See: Seifert-van Kampen theorem.
One may similarly define free products of other algebraic structures than groups, including algebras over a field. Free products of algebras of random variables are 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.