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

Let P P be the set of all prime numbers. A subgroup H of a finite group G is said to be P P - subnormal in G if there exists a chain of subgroups aligned H = H₀ ⊆ H₁ ⊆ ⋯ ⊆ Hₙ₋₁ ⊆ Hₙ = G aligned H = H 0 ⊆ H 1 ⊆ ⋯ ⊆ H n - 1 ⊆ H n = G such that either Hᵢ₋₁ H i - 1 is normal in Hᵢ H i or |Hᵢ:\, Hᵢ₋₁| | H i : H i - 1 | is a prime number for every i = 1, 2, … , n i = 1 , 2 , … , n . A subgroup H of G is called a \,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- P P -subnormal subgroup is a \,TI\, TI -subgroup.