Notes:Topology - Munkres/Section 68

Section 68: Free Products of Groups

• Part overview: Part II - Algebraic Topology
• Chapter overview: Chapter 11 - The Seifert-van Kampen Theorem
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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]