Quotients and inclusions of finite quasiprimitive permutation groups

✍ Scribed by Cheryl E. Praeger

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 283 KB
• Volume: 269
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(03)00400-9

✦ Synopsis

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.