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 $ρ(G) = \prod_{x \in G} o(x)$, where $o(x)$ denotes the order of the element $x \in G$. Let $Ω$ 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 $Ω_ρ(G):= Ω(ρ(G))$. 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 $Ω_ρ(A\times B) = Ω_ρ(A) \cdot |B|+Ω_ρ(B) \cdot |A|$. \\ \textbf{(Quotient rule)} If $P$ is a central cyclic normal Sylow $p$-subgroup of a finite group $G$, then $ Ω_ρ(\dfrac{G}{P}) = \dfrac{Ω_ρ(G)\cdot|P|-Ω_ρ(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 $Ω_ρ(G) \leq Ω_ρ(C)$. Finally, we show that if $G$ is a group of order $|L_2(p)|$, then $Ω_ρ(G) \geqslant Ω_ρ(L_2(p))$, where $p \in \{5, 11, 13\} $.