# Detecting structural properties of finite groups by the sum of element orders

## TL;DR

This paper introduces a new function based on the sum of element orders in finite groups, providing criteria to identify various structural properties such as cyclic, abelian, nilpotent, supersolvable, and solvable groups.

## Contribution

The paper proposes a novel function related to element orders that helps determine key structural properties of finite groups, advancing group classification methods.

## Key findings

- Provides criteria for cyclic groups
- Offers conditions for abelian groups
- Characterizes nilpotent, supersolvable, and solvable groups

## Abstract

In this paper, we introduce a new function related to the sum of element orders of finite groups. It is used to give some criteria for a finite group to be cyclic, abelian, nilpotent, supersolvable and solvable, respectively.