A nilpotency criterion for finite groups

TL;DR

This paper presents a new criterion for determining the nilpotency of finite groups by examining the existence of elements of specific orders within each section of the group.

Contribution

It introduces a novel nilpotency criterion based on element orders in all sections of finite groups, advancing understanding of group structure.

Findings

Provides a new criterion for nilpotency based on element orders.

Applicable to all finite groups, offering a practical test.

Enhances theoretical understanding of group structure.

Abstract

Let $G$ be a finite group. In this short note, we give a criterion of nilpotency of $G$ based on the existence of elements of certain order in each section of $G$.