On weakly ????-permutable subgroups of finite groups

Abstract Let σ = {σi ∣i ∈ I} be some partition of the set of all primes ℙ, G be a finite group and σ(G) = {σi ∣σi ∩ π(G) ≠ ∅}. G is said to be σ-primary if ∣σ(G)∣ ≤ 1. A subgroup H of G is said to be σ-subnormal in G if there exists a subgroup chain H = H 0 ≤ H 1 ≤ … ≤ Ht = G such that either H i−1 is normal in Hi or Hi /(H i−1) Hi is σ-primary for all i = 1, …, t. A set ???? of subgroups of G is said to be a complete Hall σ-set of G if every non-identity member of ???? is a Hall σi -subgroup of G for some i and ???? contains exactly one Hall σi -subgroup of G for every σi ∈ σ(G). Let ???? be a complete Hall σ-set of G. A subgroup H of G is said to be ????-permutable if HA = AH for all A ∈ ????. We say that a subgroup H of G is weakly ????-permutable in G if there exists a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ H ????, where H ???? is the subgroup of H generated by all those subgroups of H which are ????-permutable. By using the weakly ????-permutable subgroups, we establish some new criteria for a group G to be σ-soluble and supersoluble, and we also give the conditions under which a normal subgroup of G is hypercyclically embedded.