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

Let σ=σi|i∈I be some partition of the set of all primes P, G a finite group and σ (G) =σi|σi∩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 σi∈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.