RO-NMO 2007 Problem 124

Let $n\geq 3$ be an integer and $S_{n}$ the permutation group. $G$ is a subgroup of $S_{n}$ , generated by $n-2$ transpositions. For all $k\in\{1,2,\ldots,n\}$ , denote by $S(k)$ the set $\{\sigma(k) \ : \ \sigma\in G\}$ .

Show that for any $k$ , $|S(k)|\leq n-1$ .