LAUSR

Let G be a finite group. A subgroup H of G is $${\mathbb {P}}$$ -subnormal in G whenever either $$H=G$$ or there exists a chain of subgroups $${H=H_0\le H_1\le \cdots \le H_n=G}$$ , such that $$|H_{i}:H_{i-1}|$$ is a prime for every $$i=1, \ldots , n$$ ; G is said to be $$\mathrm {w}$$ -supersoluble if every Sylow subgroup of G is $${\mathbb {P}}$$ -subnormal in G. We study conditions under which the group $$G=AB$$ , where A and B are $$\mathbb P$$ -subnormal subgroups of G, belongs to a subgroup-closed saturated formation containing all finite supersoluble groups and contained in the class of all $$\mathrm {w}$$ -supersoluble groups.