Exercise 14, § 5

A group of permutations $\Gamma$ of a set E is called $r$-ply transitive if, for any two sequences $(a_1, a_2, \ldots, a_r), (b_1, b_2, \ldots, b_r)$ of $r$ distinct elements of E, there exists a permutation $\sigma \in \Gamma$ such that $\sigma(a_i) = b_i$ for $1 \leq i \leq r$, this property not holding for at least one ordered pair of sequences of $r + 1$ distinct elements of E.

(a) Show that an $r$-ply transitive group is primitive if $r > 1$.

(b) The order of a permutation group $\Gamma$ operating on $n$ objects which is $r$-ply transitive is of the form $n(n-1) \ldots (n-r+1)d$, where $d$ is a divisor of $(n-r)!$ (consider a subgroup of the permutations of $\Gamma$ leaving invariant $r$ elements and calculate its index).