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 $(G,\times)$ 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 $x$