A criterion for p-nilpotency and p-closedness by the sum of element orders

Abstract

Let G be a finite group, $\psi (G)=\sum _{g\in G}o(g)$ and ${\psi }^{″}(G)=\psi (G)/|G{|}^2,$ where o(g) denotes the order of g. In this paper, we prove that if p is a prime number and $\psi ″(G)>(p^2+p+1)/(4p^2),$ then $G=O_p(G)\times O_{p^{\prime }}(G),$ where $O_p(G)$ is cyclic. In addition, G is supersolvable and $G^{″}\le Z(G).$

Keywords:

• Sum of element orders
• p-closedness
• p-nilpotency
• supersolvable

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20D15
• 20D10

Acknowledgement

The authors appreciate the referee’s valuable and profound comments.