Finite supersolvable groups and Hall normally embedded subgroups of prime power order

Let G be a finite group. A groupGis called a T group if its every subnormal subgroup is normal. A subgroup H of G is called Hall normally embedded in G if His a Hallsub group of H-G, where H-G is the normal closure of H in G. Using the notion of Hallnormally embedded subgroups, we characterize supersolvable groups and solvableT-group. First, we prove that if every cyclic subgroup ofGof order prime or 4 isHall normally embedded in G, then G is supersolvable with a well defined structure.Second, we prove that an A-group G is supersolvable if and only if its Sylow subgroupsare products of cyclic Hall normally embedded subgroups ofG. Final, we show that G is a solvable T-group if and only if everyp-subgroup of Gis Hall normally embeddedinG, for all primes p is an element of pi(G)