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

Let \(\mathbb {P}\) $\mathbb{P}$ be the set of all prime numbers. A subgroup H of a finite group G is said to be \(\mathbb {P}\) $\mathbb{P}$ -subnormal in G if there exists a chain of subgroups \(\begin{aligned} H = H_0 \subseteq H_1 \subseteq \cdots \subseteq H_{n-1} \subseteq H_n = G \end{aligned}\) $$\begin{array}{c}\hfill H=H_0\subseteq H_1\subseteq \cdots \subseteq H_{n-1}\subseteq H_n=G\end{array}$$ 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 \(i = 1, 2, \ldots , n\) $i=1,2,\dots ,n$ . A subgroup H of G is called a \({{\,\textrm{TI}\,}}\) $\text{TI}$ -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- \(\mathbb {P}\) $\mathbb{P}$ -subnormal subgroup is a \({{\,\textrm{TI}\,}}\) $\text{TI}$ -subgroup.