Notes:Topology - Munkres/Section 68
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Contents
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
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Content
Description: Words, reduction and reduced words
(page 412)
Definition: Free product
Let [ilmath](G,\times)[/ilmath] be a group, let [ilmath]\{(G_\alpha,\times)\}_{\alpha\in I} [/ilmath] be an arbitrary family of subgroups of [ilmath]G[/ilmath] that generate [ilmath]G[/ilmath]. Suppose that:
- [ilmath]\forall\alpha,\beta\in I[\alpha\ne\beta\implies G_\alpha\cap G_\beta=\{e\}][/ilmath] where [ilmath]e[/ilmath] denotes the identity element of [ilmath]G[/ilmath]
We say that [ilmath]G[/ilmath] is the free product of [ilmath]\{G_\alpha\}_{\alpha\in I} [/ilmath] if:
- for all [ilmath]x\in G[/ilmath] there exists only one reduced word that represents [ilmath]x[/ilmath]
