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

## TL;DR

This paper establishes a criterion based on the sum of element orders to determine when a finite group is nilpotent, providing a specific inequality and characterizing the equality case.

## Contribution

It introduces a new criterion for nilpotency in finite groups using the sum of element orders and characterizes the equality case explicitly.

## Key findings

- If rac{13}{21}rac{13}{21} ext{ of the sum of element orders exceeds a threshold, the group is nilpotent.
- Equality holds iff the group is isomorphic to } S_3 imes C_m ext{ with specific conditions on } m.
- The result offers new insights into the structure of finite groups via element order sums.

## 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 $|G|=n$ 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.