A Remark on З-Permutability of Finite Groups

Let 3 be a complete set of Sylow subgroups of a finite group G,that is,3 contains exactly one and only one Sylow p-subgroup of G for each prime p.A subgroup of a finite group G is said to be 3-permutable if it permutes with every member of 3.Recently,using the Classification of Finite Simple Groups,Heliel,Li and Li proved the following result: If the cyclic subgroups of prime order or order 4 (if p=2) of every member of 3 are 3=permutable subgroups in G,then G is supersolvable. In this paper,we give an elementary proof of this theorem and generalize it in terms of formation.