Notes:Topology - Munkres/Section 68
From Maths
Contents
[hide]Section 68: Free Products of Groups
- Part overview: Part II - Algebraic Topology
- Chapter overview: Chapter 11 - The Seifert-van Kampen Theorem
- Previous section: Section 67 - Direct Sums of Abelian Groups
- Next section:
Content
Description: Words, reduction and reduced words
(page 412)
Definition: Free product
Let (G,×) be a group, let \{(G_\alpha,\times)\}_{\alpha\in I} be an arbitrary family of subgroups of G that generate G. Suppose that:
- \forall\alpha,\beta\in I[\alpha\ne\beta\implies G_\alpha\cap G_\beta=\{e\}] where e denotes the identity element of G
We say that G is the free product of \{G_\alpha\}_{\alpha\in I} if:
- for all x\in G there exists only one reduced word that represents x