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 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 elements on the left-hand side of the above equation, 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
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)
“The product of the artist has become less important than the fact of the artist. We wish to absorb this person. We wish to devour someone who has experienced the tragic. In our society this person is much more important than anything he might create.”
—David Mamet (b. 1947)