Finite factorised groups with partially solvable $$\mathbb{P}$$ -subnormal subgroups

Abstract

A subgroup A of a group G is called \(\mathbb{P}\)-subnormal in G whenever either A = G or there exists a chain of subgroups A = A0 ⊂ A1 ⊂ ... ⊂ An = G such that |Ai : Ai−1| is a prime number for all i. We study a finite group G = AB on assuming that A and B are \(\mathbb{P}\)-subnormal subgroups with fixed properties. In particular, we show that G is r-solvable if A and B are r-solvable.