Exercise 13, § 7

Let $G$ be a group and $X$ a subset of $G$ disjoint from $X^{-1}$. Let $S = X \cup X^{-1}$. Prove the equivalence of the following two properties: (i) $X$ is a free family in $G$. (ii) No product $s_1 \ldots s_n$, with $n \geq 1$, $s_i \in S$ and $s_i s_{i+1} \neq e$ for $1 \leq i \leq n - 1$, is equal to $e$.