FINITE GROUPS WHOSE ALL PROPER SUBGROUPS ARE GPST-GROUPS

Abstract

A set W = {W₁, . . . , W_t} of nilpotent Hall subgroups of G is a complete Wielandt set if (|Wi|,|Wj) = 1 for all i,j.  A finite group G is called a GPST-group if G has a complete Wielandt set \(\mathcal{W}\) such that every member in \(\mathcal{W}\) permutes all maximal subgroups of any non-cyclic subgroup S in \(\mathcal{W}\).  In this paper, we give a complete classification of those groups which are not GPST-groups but all of whose proper subgroups are GPST-groups, i.e., they are precisely minimal non-PST-groups.