Finite groups with some σ-primary subgroups weakly m-ℋ-permutable

Abstract

Let σ = { σ i | i ∈ I } be some partition of the set of all primes P , G a finite group and σ ( G ) = { σ i | σ i ∩ π ( G ) ≠ ∅ } . A set H of subgroups of G is said to be a complete Hall σ-set of G if every nonidentity member of H is a Hall σi -subgroup of G for some i and H contains exactly one Hall σi -subgroup of G for every σ i ∈ σ ( G ) . Let H be a complete Hall σ-set of G. A subgroup H of G is said to be H -permutable if HA = AH for every member A ∈ H ; m- H -permutable if H = 〈 A , B 〉 for some modular subgroup A and H -permutable subgroup B of G; weakly m- H -permutable in G if there exists a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ S ≤ H for some m- H -permutable subgroup S of G. In this paper, we study the structure of finite groups G by assuming that some σ-primary subgroups are weakly m- H -permutable in G. Some recent results are generalized and unified.