Free Product - Generalization: Free Product With Amalgamation

Generalization: Free Product With Amalgamation

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

where F is some arbitrary group. Start with the free product GH and adjoin as relations

for every f in F. In other words take the smallest normal subgroup N of GH containing all elements on the left-hand side of the above equation, which are tacitly being considered in GH 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

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.

Read more about this topic:  Free Product

Famous quotes containing the words free and/or product:

    Louise Bryant: I’m sorry if you don’t believe in mutual independence and free love and respect.
    Eugene O’Neill: Don’t give me a lot of parlor socialism that you learned in the village. If you were mine, I wouldn’t share you with anybody or anything. It would be just you and me. You’d be at the center of it all. You know it would feel a lot more like love than being left alone with your work.
    Warren Beatty (b. 1937)

    For man is not the creature and product of Mechanism; but, in a far truer sense, its creator and producer.
    Thomas Carlyle (1795–1881)