A criterion for nilpotency of a finite group by the sum of element orders

Abstract

Denote the sum of element orders in a finite group G by $\psi (G)$ and let C_n denote the cyclic group of order n. In this paper, we prove that if $\left|G\right|=n$ and $\psi (G)>\frac{13}{21}\psi (C_n)$, then G is nilpotent. Moreover, we have $\psi (G)=\frac{13}{21}\psi (C_n)$ if and only if $n=6m$ with $(6,m)=1$ and $G\cong S_3\times C_m$. Two interesting consequences of this result are also presented.

Keywords:

• Group element orders
• nilpotent groups

MSC2000:

• Primary 20D60
• Secondary 20D15
• 20F18

Notes

1 See Theorem 6 of [4] for an alternative argument.

2 Note that we have equality if and only if $n_1=1.$