A Condition for the Supersolvability of Finite Groups.

Let G be a finite group and H, K two nontrivial subgroups of G. We say that G is a mutually m-permutable product of H and K if G = HK and every maximal subgroup of H permutes with K and every maximal subgroup of K permutes with H. We prove the following result: Let G = G1G2...Gr be a finite group such that G1, G2,...Gr are pairwise permutable subgroups of G and GiGj is a mutually m-permutable product for all i and j with i ≠ j. If the Sylow subgroups of all Gi are cyclic, then G is supersolvable. [ABSTRACT FROM AUTHOR]