Abstract
Let G be a group. A subgroup H of G is called an -subgroup in G if
for all
Furthermore, a subgroup H of G is called a weakly
-subgroup in G if there exists a normal subgroup K of G such that G = HK and
is an
-subgroup in G. In this article, some new criteria for a group to be p-nilpotent and supersolvable are given.
Acknowledgements
The authors are indebted to the referees for valuable suggestions and very careful reading of the original manuscript.