On the residual of a factorized group with widely supersoluble factors

Abstract

Let $\mathbb{P}$ be the set of all primes. A subgroup $H$ of a group $G$ is called {\it $\mathbb{P}$-subnormal} in $G$, if either $H=G$, or there exists a chain of subgroups $H=H_0\le H_1\le \dots \le H_n=G,|H_i:H_{i-1}|\in \mathbb{P},\forall i.$ A group $G$ is called {\it widely supersoluble}, $\mathrm{w}$-supersoluble for short, if every Sylow subgroup of $G$ is $\mathbb{P}$-subnormal in $G$. A group $G=AB$ with $\mathbb{P}$-subnormal $\mathrm{w}$-supersoluble subgroups $A$ and $B$ is studied. The structure of its $\mathrm{w}$-supersoluble residual is obtained. In particular, it coincides with the nilpotent residual of the $\mathcal{A}$-residual of $G$. Here $\mathcal{A}$ is the formation of all groups with abelian Sylow subgroups. Besides, we obtain new sufficient conditions for the $\mathrm{w}$-supersolubility of such group $G$.