Upper bounds for the product of element orders of finite groups

TL;DR

This paper investigates upper bounds for the product of element orders in finite groups, relating these bounds to the group's order and its smallest prime divisor, especially within certain non-cyclic group classes.

Contribution

It introduces new upper bounds for the product of element orders in finite groups based on group order and prime divisors, focusing on non-cyclic groups.

Findings

Derived bounds depend on group order and least prime divisor.

Results apply to specific classes of non-cyclic groups.

Provides theoretical limits for element order products.

Abstract

Let $G$ be a finite group of order $n$, and denote by $\rho(G)$ the product of element orders of $G$. The aim of this work is to provide some upper bounds for $\rho(G)$ depending only on $n$ and on its least prime divisor, when $G$ belongs to some classes of non-cyclic groups.