The influence of the prime omega function on the product of element orders in finite groups

TL;DR

This paper explores how the prime omega function applied to the product of element orders in finite groups behaves similarly to derivatives, establishing rules and inequalities that relate group structure to this function.

Contribution

It introduces the prime omega function of the product of element orders in finite groups and establishes calculus-like rules and bounds for this function.

Findings

Product rule for coprime direct products

Quotient rule involving central Sylow p-subgroups

Inequality comparing cyclic and non-cyclic groups of same order

Abstract

Let $G$ be a finite group and define $\rho(G) = \prod_{x \in G} o(x)$, where $o(x)$ denotes the order of the element $x \in G$. Let $\Omega$ be the prime omega function giving the number of (not necessarily distinct) prime factors of a natural number. In this paper, we consider the function $\Omega_{\rho}(G):= \Omega(\rho(G))$. We show that, under certain conditions, this function exhibits behavior analogous to the derivative in calculus. We establish the following results: \textbf{(Product rule)} If $A$ and $B$ are finite groups, where $\operatorname{gcd}(|A|,|B|)=1$, then $\Omega_{\rho}(A\times B) = \Omega_{\rho}(A) \cdot |B|+\Omega_{\rho}(B) \cdot |A|$. \\ \textbf{(Quotient rule)} If $P$ is a central cyclic normal Sylow $p$-subgroup of a finite group $G$, then $ \Omega_{\rho}(\dfrac{G}{P}) = \dfrac{\Omega_{\rho}(G)\cdot|P|-\Omega_{\rho}(P)\cdot |G|}{{|P|}^2}.$ \\ Moreover, we show that if $C$ is a cyclic group and $G$ is a non-cyclic group of the same order, then $\Omega_{\rho}(G) \leq \Omega_{\rho}(C)$. Finally, we show that if $G$ is a group of order $|L_2(p)|$, then $\Omega_{\rho}(G) \geqslant \Omega_{\rho}(L_2(p))$, where $p \in \{5, 11, 13\} $.