Prerequisites of π-closed-Sylow-tower Groups

Group G is called a π-closed-Sylow-tower group if there exists a normal Hall π-subgroup in G, which is a Sylow-tower group. Discussion is made on the properties of π-closed-Sylow-tower group as follows through the use of maximal subgroups and s-normal subgroups: Let G be a π-closed-Sylow-tower group, (1) If the index of every maximal subgroup containing Hall π'- subgroup is a prime, then G is π-supersolvable. (2) If every primary subgroup contained in π'-Hall subgroup is -normal in G, then G is solvable.