Exercise 19, § 7

Let X be a set, G = F(X) the free group constructed on X, and S = X ∪ X⁻¹. a) If g ∈ G, a reduced decomposition of g is called any sequence (s₁, ..., sₙ) of elements of S such that g = s₁ ... sₙ and sᵢsᵢ₊₁ ≠ e for 1 ≤ i < n. Prove that every element of G admits one and only one reduced decomposition; the integer n is the length of g (cf. I, p. 147, exerc. 26). b) Let Y be a subset of G containing e. Prove the equivalence of the following conditions: b₁) Y is S-connected (cf. exerc. 18). b₂) For every g ∈ Y, if (s₁, ..., sₙ) is the reduced decomposition of g, one has s₁ ... sᵢ ∈ Y for 1 ≤ i ≤ n. c) Let Y₁ and Y₂ be two non-empty S-connected subsets of G, such that Y₁ ∩ Y₂ = ∅. Prove that there exists at most one pair (y₁, y₂) ∈ Y₁ × Y₂ such that y₁ and y₂ are S-neighbours. There exists one if and only if Y₁ ∪ Y₂ is S-connected; in this case one says that Y₁ and Y₂ are neighbours. d) Let (Y₁, ..., Yₙ) be a sequence of non-empty, disjoint, S-connected subsets of G. Suppose that Yᵢ and Yᵢ₊₁ are neighbours for 1 ≤ i < n. Prove that, if n ≥ 3, Y₁ and Yₙ are not neighbours.