Quotients and inclusions of finite quasiprimitive permutation groups

Abstract

A permutation group is said to be quasiprimitive if each non-trivial normal subgroup is transitive. Finite quasiprimitive permutation groups may be classified into eight types, in a similar fashion to the case division of finite primitive permutation groups provided by the O'Nan–Scott Theorem. The action induced by an imprimitive quasiprimitive permutation group on a non-trivial block system is faithful and quasiprimitive, but may have a different quasiprimitive type from that of the original permutation action. All possibilities for such differences are determined. Suppose that G<H< Sym(Ω) with G, H quasiprimitive and imprimitive. Then for each non-trivial H-invariant partition B of Ω, we have an inclusion G B <H B ⩽ Sym( B) with H B ≅H and G B ≅G , and H B is primitive if B is maximal. The inclusions (G B ,H B ) in the case where H B is primitive have been described in work of Baddeley and the author, but it turns out that many of them have no proper liftings to imprimitive quasiprimitive inclusions ( G, H). We show that either G and H have the same socle and the same quasiprimitive type, or the inclusion ( G, H) is associated in a well defined way with a proper factorisation S= AT where S and T are both non-abelian simple groups.