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

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 $\mathrm{\Omega }$ 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 ${\mathrm{\Omega }}_{\rho }(G):=\mathrm{\Omega }(\rho (G))$. We show that, under certain conditions, this function exhibits behavior analogous to the derivative in calculus. We establish the following results: $\mathbf{\text{(Product}}\mathbf{\text{rule)}}$ If $A$ and $B$ are finite groups, where $\gcd (\mathrm{\mid }A\mathrm{\mid },\mathrm{\mid }B\mathrm{\mid })=1$, then ${\mathrm{\Omega }}_{\rho }(A\times B)={\mathrm{\Omega }}_{\rho }(A)\cdot \mathrm{\mid }B\mathrm{\mid }+{\mathrm{\Omega }}_{\rho }(B)\cdot \mathrm{\mid }A\mathrm{\mid }$. \ $\mathbf{\text{(Quotient}}\mathbf{\text{rule)}}$ If $P$ is a central cyclic normal Sylow $p$-subgroup of a finite group $G$, then ${\mathrm{\Omega }}_{\rho }({\displaystyle \frac{G}{P}})={\displaystyle \frac{{\mathrm{\Omega }}_{\rho }(G)\cdot \mathrm{\mid }P\mathrm{\mid }-{\mathrm{\Omega }}_{\rho }(P)\cdot \mathrm{\mid }G\mathrm{\mid }}{{\mathrm{\mid }P\mathrm{\mid }}^2}}\mathrm{.}$ \ Moreover, we show that if $C$ is a cyclic group and $G$ is a non-cyclic group of the same order, then ${\mathrm{\Omega }}_{\rho }(G)\le {\mathrm{\Omega }}_{\rho }(C)$. Finally, we show that if $G$ is a group of order $\mathrm{\mid }{L}_2(p)\mathrm{\mid }$, then ${\mathrm{\Omega }}_{\rho }(G)⩾{\mathrm{\Omega }}_{\rho }(L_2(p))$, where $p\in \{5,11,13\}$.