Check the details of Example 1 of §68.
Solution
Goal: Verify the foundational details of the construction of the free product of groups \(G = *_{\alpha \in J} G_\alpha\) from §68, including word reduction, the group axioms, and the universal mapping property.
1. Words and reduced words:Proof:
1.1. Let \(\{G_\alpha\}_{\alpha \in J}\) be a family of groups. Assume without loss of generality that the sets \(G_\alpha \setminus \{1_\alpha\}\) are pairwise disjoint.
1.2. A word is a finite sequence \(w = (x_1, x_2, \dots, x_m)\) where each \(x_i \in \bigcup_{\alpha \in J} (G_\alpha \setminus \{1_\alpha\})\). The empty sequence \(\varnothing\) is the empty word of length \(0\).
1.3. A word is reduced if no two adjacent elements \(x_i, x_{i+1}\) belong to the same group \(G_\alpha\).
1.4. Let \(G\) denote the set of all reduced words.
2. Group multiplication and associativity (van der Waerden representation):Proof:
2.1. To avoid tedious case analysis on word concatenation and reduction, let \(\operatorname{Sym}(G)\) be the group of permutations of the set of reduced words \(G\).
2.2. For each \(\alpha \in J\) and \(g \in G_\alpha\), define the action \(L_g: G \to G\) on a reduced word \(w = (x_1, \dots, x_m)\) by:\[L_g(w) = \begin{cases} (g, x_1, \dots, x_m) & \text{if } x_1 \notin G_\alpha, \\ (g x_1, x_2, \dots, x_m) & \text{if } x_1 \in G_\alpha \text{ and } g x_1 \neq 1_\alpha, \\ (x_2, \dots, x_m) & \text{if } x_1 \in G_\alpha \text{ and } g x_1 = 1_\alpha. \end{cases}\]
2.3. Direct verification shows \(L_{1_\alpha} = \operatorname{id}_G\) and \(L_{g h} = L_g \circ L_h\) for all \(g, h \in G_\alpha\).
2.4. Thus each \(L_g\) is a permutation with inverse \(L_{g^{-1}}\), defining an injective homomorphism \(\phi_\alpha: G_\alpha \to \operatorname{Sym}(G)\).
2.5. The map \(\Psi: G \to \operatorname{Sym}(G)\) given by \(\Psi(x_1, \dots, x_m) = L_{x_1} \circ \dots \circ L_{x_m}\) evaluates on the empty word as \(\Psi(w)(\varnothing) = w\), hence \(\Psi\) is injective.
2.6. Pulling back the group structure from \(\operatorname{Sym}(G)\) via \(\Psi\) equips \(G\) with an associative group multiplication whose identity is \(\varnothing\) and where \((x_1, \dots, x_m)^{-1} = (x_m^{-1}, \dots, x_1^{-1})\).
3. Canonical inclusions and Universal Mapping Property:Proof:
3.1. The inclusion maps \(i_\alpha: G_\alpha \hookrightarrow G\) sending \(g \mapsto (g)\) are injective group homomorphisms.
3.2. Let \(H\) be any group and let \(\{h_\alpha: G_\alpha \to H\}_{\alpha \in J}\) be a family of homomorphisms.
3.3. Define \(h: G \to H\) by \(h(\varnothing) = 1_H\) and \(h(x_1, \dots, x_m) = h_{\alpha_1}(x_1) \cdots h_{\alpha_m}(x_m)\) where \(x_j \in G_{\alpha_j}\).
3.4. The homomorphism property \(h(w \cdot w') = h(w) h(w')\) follows from the definition of reduction in \(G_\alpha\).
3.5. \(h\) is the unique homomorphism satisfying \(h \circ i_\alpha = h_\alpha\) for all \(\alpha \in J\).
4. Conclusion:The construction rigorously yields the free product \(*_{\alpha \in J} G_\alpha\) satisfying all group axioms and universal properties. Q.E.D.