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

Marius Tărnăuceanu

paper

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

arXiv:1903.09744

Abstract

Denote the sum of element orders in a finite group $G$ by $ψ (G)$ and let $C_{n}$ denote the cyclic group of order $n$ . In this paper, we prove that if $∣ G ∣ = n$ and $ψ (G) > \frac{13}{21} ψ (C_{n})$ , then $G$ is nilpotent. Moreover, we have $ψ (G) = \frac{13}{21} ψ (C_{n})$ if and only if $n = 6 m$ with $(6, m) = 1$ and $G ≅ S_{3} \times C_{m}$ . Two interesting consequences of this result are also presented.