Characteristically simple subgroups of quasiprimitive permutation groups

Abstract

We say that G (..) Sym(..) is transitive if G has just one orbit on , namely (..). If G is transitive on (..) and the only partitions of (..) preserved by G are (..) and (..), then we say that G is primitive. The O'Nan-Scott Theorem [25] classifies the finite primitive permutation groups by dividing them into classes, according to the structure of their minimal normal subgroups. An important result in this classification is that every permutation group admits at most two distinct transitive minimal normal subgroups [8, Lemma 5.1]. A permutation group is quasiprimitive if all its nontrivial normal subgroups are transitive. For example, all primitive permutation groups are quasiprimitive. Finite quasiprimitive groups were characterized by Cheryl Praeger [28], who showed that they can be classified similarly to the O'Nan-Scott classification of finite primitive permutation groups. The inclusion problem for a permutation group H asks to determine the possible (primitive or quasiprimitive) subgroups of the symmetric group that contain H. In other words, given a permutation group H (..) Sym(..), we are asking about its overgroups. For instance, it is a common situation in algebraic combinatorics that we know a part of the group of automorphisms of a combinatorial structure (for example, a Cayley graph) and we wish to determine a larger automorphism group which may be primitive or quasiprimitive. In this work we describe all inclusions H (..) G such that H is a transitive nonabelian characteristically simple group and G is a finite primitive or quasiprimitive permutation group with nonabelian socle. The study of these inclusions is possible since we have detailed information concerning factorizations of finitenonabelian simple groups. For this reason, many of the results presented here rely on the classification of finite simple groups, specially chapters 4 and 7.