Transitive subgroups of primitive permutation groups

We investigate the finite primitive permutation groups G which have a trans itive subgroup containing no nontrivial subnormal subgroup of G. The conclu sion is that such primitive groups are rather rare, and that their existenc e is intimately connected with factorisations of almost simple groups. A co rollary is obtained on primitive groups which contain a regular subgroup. H eavily involved in our proofs are some new results on subgroups of simple g roups which have orders divisible by various primes. For example, another c orollary implies that for every simple group T apart from L-3(3), U-3(3), a nd L-2(p) with p a Mersenne prime, there is a collection IT consisting of t wo or three odd prime divisors of \T\, such that if M is a subgroup of T of order divisible by every prime in Pi, then \M\ is divisible by all the pri me divisors of \T\, and we obtain a classification of such subgroups M. (C) 2000 Academic Press.