A note on p-nilpotence and solvability of finite groups

In this note, we first give some examples to show that some hypotheses of some well-known results for a group G to be p-nilpotent, solvable and supersolvable are essential and cannot be removed. Second, we give some generalizations of two theorems in [A. Ballester-Bolinches, X. Guo, Some results on p-nilpotence and solubility of finite groups, J. Algebra 228 (2000) 491–496].