Finite groups in which normality or quasinormality is transitive

A subgroup H of a finite group G is called pronormal in G if and only if, for all x in G, H and H-x are conjugate in (H, H-x), the subgroup generated by H and H-x. We say that a finite group G satisfies the property Y-p, where p is a prime, if and only if each p-subgroup of G is pronormal in G. We say, following Beidleman et al., a finite group G, satisfies X-p, where p is a prime, if and only if each subgroup of a Sylow p-subgroup P of G is quasinormal in N-G (P) A finite T-group is a group G whose subnormal subgroups are all normal in G. A finite PT-group is a group G whose subnormal subgroups are all quasinormal in G. It is shown that (i) a finite group G is a solvable T-group if and only if it satisfies Y-p for all primes p dividing the order of the generalized Fitting subgroup F* (G) of G; (ii) a finite group G is a solvable PT-group if and only if it satisfies X-p for all primes p dividing the order of the generalized Fitting subgroup F* (G) of G.