Finite Groups All of Whose Subgroups are $$\mathbb {P}$$-Subnormal or $${{\,\textrm{TI}\,}}$$-Subgroups

Abstract

AbstractLet $$\mathbb {P}$$ P be the set of all prime numbers. A subgroup H of a finite group G is said to be $$\mathbb {P}$$ P -subnormal in G if there exists a chain of subgroups $$\begin{aligned} H = H_0 \subseteq H_1 \subseteq \cdots \subseteq H_{n-1} \subseteq H_n = G \end{aligned}$$ H = H 0 ⊆ H 1 ⊆ ⋯ ⊆ H n - 1 ⊆ H n = G such that either $$H_{i-1}$$ H i - 1 is normal in $$H_i$$ H i or $$|H_i{:}\, H_{i-1}|$$ | H i : H i - 1 | is a prime number for every $$i = 1, 2, \ldots , n$$ i = 1 , 2 , … , n . A subgroup H of G is called a $${{\,\textrm{TI}\,}}$$ TI -subgroup if every pair of distinct conjugates of H has trivial intersection. The aim of this paper is to give a complete description of all finite groups in which every non-$$\mathbb {P}$$ P -subnormal subgroup is a $${{\,\textrm{TI}\,}}$$ TI -subgroup.