Cyclic quotients of transitive groups

Let A be a transitive subgroup of S-n. We show that the largest cyclic quot ient of A has order at most n. This can be interpreted as an equivalent res ult about extensions of constants in the Galois closure of a covering of cu rves over a finite field. We also prove that the point stabilizer in a fini te primitive permutation group always has a faithful orbit. (C) 2000 Academ ic Press.