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 = G(1)G(2)...G(r) be a finite group such that G(1), G(2),...G(r) are pairwise permutable subgroups of G and G(i)G(j) is a mutually m-permutable product for all i and j with i not equal j. If the Sylow subgroups of all G(i) are cyclic, then G is supersolvable.