On weakly σ-semipermutable subgroups of finite groups

In 2015, A.N.Skiba in [1] introduce definition: A subgroup H of G is said to be sigma-subnormal in G if there is a subgroup chain H = H-0 <= H-1 <= center dot center dot center dot <= H-t = G such that either Hi - 1 (sic) H-i or H-i/(Hi - 1)(Hi) is sigma-primary for all i = 1, . . . , t. Later, Wenbin Guo and A.N.Skiba in [2] introduce the definition of sigma-semipermutable: A subgroup H of G is said to be sigma-semipermutable in G if G possesses a complete Hall sigma-set H such that H A(x) = A(x) H for all A is an element of H and all x is an element of G such that sigma (A) boolean AND sigma (H) = empty set. In this paper, we present a new generalized supplemented definition: A subgroup H of G is said to be: weakly sigma-semipermutable in G if there exists a sigma-subnormal subgroup T of G such that G = HT and H boolean AND T <= H-sigma G, where H-sigma G is the subgroup of H generated by all those subgroups of H which are sigma-semipermutable in G. Also, the structure of a finite group with some weakly sigma-semipermutable subgroups is investigated.