Abstract
Let 3 be a complete set of Sylow subgroups of a finite group G, that is, for each prime p dividing the order of G, 3 contains exactly one and only one Sylow p-subgroup of G. A subgroup H of G is said to be 3-permutable of G if H permutes with every member of 3. A subgroup H of G is said to be a weakly 3-permutable subgroup of G if there exists a subnormal subgroup K of G such that G = HK and H boolean AND K <= H-3, where H-3 is the subgroup of H generated by all those subgroups of H which are 3-permutable subgroups of G. In this paper, we prove that if p is the smallest prime dividing the order of G and the maximal subgroups of G(p) is an element of 3 are weakly 3-permutable subgroups of G, then G is p-nilpotent. Moreover, we prove that if F is a saturated formation containing the class of all supersolvable groups, then G is an element of(sic) iff there is a solvable normal subgroup H in G such that G/H is an element of(sic) and the maximal subgroups of the Sylow subgroups of the Fitting subgroup F(H) are weakly 3-permutable subgroups of G. These two results generalize and unify several results in the literature.