Quotients and inclusions of finite quasiprimitive permutation groups

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    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(Omega) with G, H quasiprimitive and imprimitive. Then for each non-trivial H-invariant partition B of Omega, we have an inclusion G(B) < H-B less than or equal to Sym(B) with H-B congruent to H and G(B) congruent to 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. (C) 2003 Elsevier Inc. All rights reserved.
    Original languageEnglish
    Pages (from-to)329-346
    JournalJournal of Algebra
    Volume269
    Issue number1
    DOIs
    Publication statusPublished - 2003

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