A CRITERION FOR THE p-SUPERSOLUBILITY OF FINITE GROUPS

Abstract

Our main result here is the following theorem: Let G = AT, where A is a Hall π-subgroup of G and T is p-nilpotent for some prime p ∉ π, let P denote a Sylow p-subgroup of T and assume that A permutes with every Sylow subgroup of T. Suppose that there is a number pk such that 1 < pk < |P| and A permutes with every subgroup of P of order pk and with every cyclic subgroup of P of order 4 (if pk = 2 and P is non-abelian). Then G is p-supersoluble.