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

Let G be a finite group. A group G is called a T group if its every subnormal subgroup is normal. A subgroup H of G is called Hall normally embedded in G if H is a Hall subgroup of \(H^G\), where \(H^G\) is the normal closure of H in G. Using the notion of Hall normally embedded subgroups, we characterize supersolvable groups and solvable T-group. First, we prove that if every cyclic subgroup of G of order prime or 4 is Hall 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 subgroups are products of cyclic Hall normally embedded subgroups of G. Final, we show that G is a solvable T-group if and only if every p-subgroup of G is Hall normally embedded in G, for all primes \(p\in \pi (G)\).