On primitive overgroups of quasiprimitive permutation groups

A permutation group is said to be quasiprimitive if all its non-trivial normal subgroups are transitive. We investigate pairs (G, H) of permutation groups of degree n such that G less than or equal to H less than or equal to S-n with G quasiprimitive and H primitive. An explicit classification of such pairs is obtained except in the cases where the primitive group H is either almost simple or the blow-up of an almost simple group. The theory in these remaining cases is investigated in separate papers. The results depend on the finite simple group classification. (C) 2003 Elsevier Science (USA). All rights reserved.