Exercise 18, § 7

Let S be a symmetric subset of a group G. Two elements $g_1, g_2$ of G are called S-neighbours if $g_2^{-1}g_1$ belongs to S. A sequence $(g_1, \ldots, g_n)$ of elements of G is called an S-chain if $g_i$ and $g_{i+1}$ are S-neighbours for $1 \leq i \leq n-1$; the elements $g_1$ and $g_n$ are called respectively the beginning and the end of the chain.

(a) Let Y be a subset of G and $R_Y \{a, b\}$ the relation "there exists an S-chain in Y beginning at a and ending at b". Show that $R_Y$ is an equivalence relation on Y (cf. Set Theory, II, § 6, Exercise 10); the equivalence classes under this relation are called the connected S-components of Y. Y is called S-connected if it has at most one connected component.

(b) Show that G is S-connected if and only if S generates G.

(c) Suppose that no element $s \in S$ satisfies the relation $s^2 = e$; choose a subset X of S such that S is the disjoint union of X and $X^{-1}$. Show the equivalence of the following conditions: (i) The family X is free. (ii) There exists no S-chain $(g_1, \ldots, g_n)$ in G consisting of distinct elements $n \geq 3$ in number such that $g_1$ and $g_n$ are S-neighbours. (Use Exercise 13.)