Abstract
Let G be a finite group. We say that 3 is a complete set of Sylow subgroups of G if for each prime p dividing the order of G, 3 contains exactly one Sylow p-subgroup of G, G(p), say. A subgroup of G is said to be 3-permutable in G if it permutes with every member of 3. A subgroup H of G is said to be weakly 3-permutable in G if there exists a subnormal subgroup K of G such that G = H K and H boolean AND K <= H-3, where H3 is the subgroup of H generated by all those subgroups of H which are 3-permutable in G. In this paper, we prove that G is supersolvable if the maximal subgroups of G(p) boolean AND F* (G) are weakly 3-permutable in G, for every G(p) is an element of 3, where F* (G) is the generalized Fitting subgroup of G. Also, we prove that if F is a saturated formation containing the class of all supersolvable groups, then G is an element of F a if and only if there is a normal subgroup H in G such that G/H is an element of F and the maximal subgroups of G(p) boolean AND F*(11.) are weakly 3-permutable in G, for every G(p) is an element of 3.