Finite Groups All of Whose Subgroups are P-Subnormal or TI-Subgroups

Let P be the set of all prime numbers. A subgroup H of a finite group G is said to be P - subnormal in G if there exists a chain of subgroups H = H 0 ⊆ H 1 ⊆ ⋯ ⊆ H n - 1 ⊆ H n = G such that either H i - 1 is normal in H i or | H i : H i - 1 | is a prime number for every i = 1 , 2 , … , n . A subgroup H of G is called a 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- P -subnormal subgroup is a TI -subgroup.