Exercise 15, § 5

Let $\Gamma$ be an $r$-ply transitive group of permutations of a set E with $n$ elements; for a permutation $\sigma \in \Gamma$ distinct from the identity permutation, let $n - s$ be the number of elements of E invariant under $\sigma$. If $s > r$, show that there exists a permutation $\tau \in \Gamma$ such that $\sigma^{-1} \tau \sigma \tau^{-1}$ is distinct from the identity permutation and the number of elements of E it leaves invariant is $\geq n - 2(s - r + 1)$ (use the decomposition of $\sigma$ into its component cycles). If $s = r$, show similarly that there exists $\tau \in \Gamma$ such that $\sigma^{-1} \tau \sigma \tau^{-1}$ is a cycle of order 3. Deduce that, if $r \geq 3$ and $\Gamma$ does not contain the alternating group $\mathfrak{A}_n$, then $s \geq 2r - 2$ for every permutation in $\Gamma$ (use Exercise 10). Conclude finally that, if $\Gamma$ is not identical with $\mathfrak{A}_n$ or $\mathfrak{S}_n$, then $r \leq n/3 + 1$.