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:
“Without this, as well as with it, we could have declared our independence of Great Britain; but without it, we could not, I think, have secured our free government, and consequent prosperity. No oppressed, people will fight, and endure, as our fathers did, without the promise of something better, than a mere change of masters.”
—Abraham Lincoln (1809–1865)
“[As teenager], the trauma of near-misses and almost- consequences usually brings us to our senses. We finally come down someplace between our parents’ safety advice, which underestimates our ability, and our own unreasonable disregard for safety, which is our childlike wish for invulnerability. Our definition of acceptable risk becomes a product of our own experience.”
—Roger Gould (20th century)