Abstract
Let G be a finite group and is non-cyclic} . In this paper, we show that some arithmetical conditions of influence the structure of G. Firstly, we prove that if , then G is solvable. Secondly, we determine the structure of finite groups with . Moreover, we prove that if , then G is supersolvable, and we also determine the structure of finite groups G with . Finally, we show that does not imply the supersolvability of G for any constant .
Acknowledgments
The authors are grateful to the referee, who provided her/his valuable suggestions and detailed reports.
Disclosure statement
The authors declare that they have non conflict of interest.