$${\mathbb {P}}$$-subnormal subgroups and the structure of finite groups

Abstract

Let G be a finite group. A subgroup H of G is called a $${\mathbb {P}}$$ -subnormal subgroup whenever $$H=G$$ or there exists a chain of subgroups $$H=H_0\le H_1\le \cdots \le H_t=G$$ such that $$|H_i:H_{i-1}|$$ is a prime for every $$i\in \{1, 2, \ldots t\}$$ . In the present paper, we study finite groups in which some 2-maximal subgroups or cyclic subgroups are $${\mathbb {P}}$$ -subnormal subgroups. Several conditions for G to possess an ordered Sylow tower of supersolvable type are given.