Primitive permutation groups containing a cycle of prime-power length

group of degree n containing a pr-cycle is alternating or symmetric if 1 < pr < n—2 and p is a prime number congruent to 2 modulo 3. The context is a generalisation of Jordan's Theorem that a primitive group containing a cycle of prime length with more than 2 fixed points is alternating or symmetric, and the hope that the same might be true without any arithmetic restrictions whatever on the length of the cycle. My purpose in this note is to prove the result of Rowlinson and Williamson without their restriction on the prime p. THEOREM. / / G is a primitive permutation group of degree n containing a cycle of length pr, where p is a prime number and p # 3, and ifn>pr + 2, then G is alternating or symmetric. The proof depends on the following LEMMA. Suppose that F is a set with pr elements, that N is a transitive permutation group on T, and that Op(N), the maximal normal p-subgroup ofN, contains a transitive cyclic subgroup R. Then N ' < Op(N). Proof. Let g be a p'-element of the derived group N': I shall show that g — 1. The proof is by induction on r — the conclusion certainly holds if r is 0 (or 1). Let P: = OP(N) and let Z be the centre of P. Since R is transitive it is regular on F and therefore self-centralising (even in Sym(r)), and so Z ^ R. Therefore Z must be cyclic, its automorphism group must be abelian, and so N ' ^ CN(Z). In particular g centralises Z. Now let p be the equivalence relation on F such that a = /? (mod p) if and only if a, /MieinthesameZ-orbit. Since Z<3 N this relation is N-invariant and consequently, if F*: = T/p then N induces a group N*: = Nr * of permutations of F*. Clearly (with an obvious notation) P * < Op(N*), so R * is a cyclic subgroup of Op(N*) transitive on F*, and g * is a jp'-element of the derived group of N*. Since Z is not trivial (assuming that r> 0) we have |F* | = pr * with r * < r, and so our inductive hypothesis gives that g * = 1. This means that g fixes every Z-orbit as a set. Con-sequently, since g centralises Z and is a p'-element, it follows that g = 1 as promised