Finite Groups with Given Weakly σ-Permutable Subgroups

Let G be a finite group and let σ = {σ i | i ∈ I} be a partition of the set of all primes P. A set ℋ of subgroups of G is said to be a complete Hall σ-set of G if each nonidentity member of ℋ is a Hall σ i -subgroup of G and ℋ has exactly one Hall σ i -subgroup of G for every σ i ∈ σ(G). A subgroup H of G is said to be σ-permutable in G if G possesses a complete Hall σ-set ℋ such that HA x = A x H for all A ∈ ℋ and all x ∈ G. A subgroup H of G is said to be weakly σ-permutable in G if there exists a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ H σG , where H σG is the subgroup of H generated by all those subgroups of H which are σ-permutable in G. We study the structure of G under the condition that some given subgroups of G are weakly σ-permutable in G. In particular, we give the conditions under which a normal subgroup of G is hypercyclically embedded. Some available results are generalized.