On the residual of a factorized group with widely supersoluble factors

Abstract

Let P be the set of all primes. A subgroup H of a group G is called P-subnormal in G, if either H = G, or there exists a chain of subgroups H = H0 ≤ H1 ≤ . . . ≤ Hn = G, |Hi : Hi−1| ∈ P, ∀i. A group G is called widely supersoluble, w-supersoluble for short, if every Sylow subgroup of G is P-subnormal in G. A group G = AB with P-subnormal w-supersoluble subgroups A and B is studied. The structure of its w-supersoluble residual is obtained. In particular, it coincides with the nilpotent residual of the A-residual of G. Here A is the formation of all groups with abelian Sylow subgroups. Besides, we obtain new sufficient conditions for the w-supersolubility of such group G.