A CRITERION FOR THE p-SUPERSOLUBILITY OF FINITE GROUPS

Our main result here is the following theorem: Let G = AT, where A is a Hall pi-subgroup of G and T is p-nilpotent for some prime p is not an element of pi, 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 p(k) such that 1 < p(k) < |P| and A permutes with every subgroup of P of order p(k) and with every cyclic subgroup of P of order 4 (if p(k) = 2 and P is non-abelian). Then G is p-supersoluble.