On Products of Finite w-Supersoluble Group

Abstract

Let G be a finite group. A subgroup H of G is ℙ -subnormal in G whenever either H = G or there exists a chain of subgroups H = H₀ ≤ H₁ ≤ · · · ≤ Hn = G, such that |Hᵢ : Hᵢ₋₁ | is a prime for every i = 1, . . . , n; G is said to be w -supersoluble if every Sylow subgroup of G is ℙ -subnormal in G. We study conditions under which the group G = AB, where A and B are ℙ -subnormal subgroups of G, belongs to a subgroup-closed saturated formation containing all finite supersoluble groups and contained in the class of all w-supersoluble groups.