Criterions of supersolubility of some finite factorizable groups

Abstract

Let A, B be subgroups of a group G and ∅ 6= X ⊆ G. A subgroup A is said to be X-permutable with B if for some x ∈ X we have ABx = BxA [1]. We obtain some new criterions for supersolubility of a finite group G = AB, where A and B are supersoluble groups. In particular, we prove that a finite group G = AB is supersoluble provided A, B are supersolube subgroups of G such that every primary cyclic subgroup of A X-permutes with every Sylow subgroup of B and if in return every primary cyclic subgroup of B X-permutes with every Sylow subgroup of A where X = F(G) is the Fitting subgroup of G.