On Generalized σ-Subnormal Subgroups of Finite Groups

Let sigma= {sigma(i) | i is an element of I} be some partition of the set of all primes P, G a finite group and sigma(G) = {sigma(i) | sigma(i) boolean AND pi(G) not equal empty set}. A subgroup A of G is said to be generalized sigma - subnormal in G if A = < L, T >, where L is a modular subgroup and T is a sigma-subnormal subgroup of G. We study the structure of G being based on the assumptions that if all members of H and every maximal subgroup of any non-cyclic H-i is an element of H are generalized sigma-subnormal in G, where H is a complete Hall sigma-set of G.