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 subgroupsA 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.