A Condition for the Supersolvability of Finite Groups

Abstract

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 ≠ j. If the Sylow subgroups of all G _i are cyclic, then G is supersolvable.

Key Words:

• Permutable subgroups
• Saturated formations
• Solvable groups
• Supersolvable groups

2000 Mathematics Subject Classification:

• 20D10
• 20D20
• 20D34

Notes

Communicated by M. Dixon.