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 $\gcd (|A|,|B|)=1$, 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 ${\Omega }_{\rho }(\frac{G}{P})=\frac{{\Omega }_{\rho }(G)\cdot |P|-{\Omega }_{\rho }(P)\cdot |G|}{{|P|}^2}.$ \\ Moreover, we show that if $C$ is a cyclic group and $G$ is a non-cyclic group of the same order, then ${\Omega }_{\rho }(G)\le {\Omega }_{\rho }(C)$. Finally, we show that if $G$ is a group of order $|L_2(p)|$, then ${\Omega }_{\rho }(G)⩾{\Omega }_{\rho }(L_2(p))$, where $p\in \{5,11,13\}$.