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

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\} $.