On primitive overgroups of quasiprimitive permutation groups
A permutation group is called quasiprimitive if all its non-trivial normal subgroups are transitive. It is elementary that any primitive group is quasiprimitive. The authors investigate the pairs \((G,H)\) of permutation groups of finite degree such that \(G\leq H\) with \(G\) quasiprimitive and \(H\) primitive. Since the second author already studied the case where \(G\) is primitive [in \textit{C. E. Praeger}, Proc. Lond. Math. Soc., III. Ser. 60, No. 1, 68--88 (1990; Zbl 0653.20005)], they assume in this paper \(G\) imprimitive. They partition the class of quasiprimitive groups into 8 families, 3 of which containing only primitive groups, corresponding to 8 families of primitive groups. Hence there are 40 possible types of embeddings of \(G\) imprimitive quasiprimitive into \(H\) primitive. A lot of these cases are impossible or exist only if \(G\) is primitive. The existing embeddings are described, except in the cases where \(H\) is either almost simple or the blow-up of an almost simple group. The results rely on the classification of the finite simple groups.
scientific article; zbMATH DE number 1930294
scientific article