A polynomial bound for the number of maximal systems of imprimitivity of a finite transitive permutation group

Abstract

We show that, there exists a constant a such that, for every subgroup H of a finite group G, the number of maximal subgroups of G containing H is bounded above by a|G : H| 3/2 . In particular, a transitive permutation group of degree n has at most an 3/2 maximal systems of imprimitivity. When G is soluble, generalizing a classic result of Tim Wall, we prove a much stroger bound, that is, the number of maximal subgroups of G containing H is at most |G : H| − 1.2010 Mathematics Subject Classification. primary 20E28; secondary 20B15, 20F05.