Inequalities detecting structural properties of a finite group

TL;DR

This paper establishes inequalities involving element orders in finite groups to detect properties like cyclicity and nilpotency, revealing that cyclic groups minimize cyclic subgroups and maximize the product of element orders.

Contribution

It introduces new inequalities that characterize cyclicity and nilpotency in finite groups based on element order distributions.

Findings

Cyclic groups minimize the number of cyclic subgroups among groups of the same order.

Cyclic groups maximize the product of element orders among groups of the same order.

The paper provides criteria to detect structural properties using inequalities involving element orders.

Abstract

We prove several results detecting ciclicity or nilpotency of a finite group $G$ in terms of inequalities involving the orders of the elements of $G$ and the orders of the elements of the cyclic group of order $|G|$. We prove that, among the groups of the same order, the number of cyclic subgroups is minimal for the cyclic group and the product of the orders of the elements is maximal for the cyclic group.