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

Let G be a finite group and define ρ(G) = ∏x ∈ G o(x), where $o(x)$ denotes the order of the element x ∈ 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: (Product rule) If A and B are finite groups, where gcd(|A|,|B|)=1, then Ω_ρ(A× B) = Ω_ρ(A) · |B|+Ω_ρ(B) · |A|. \\ (Quotient rule) If P is a central cyclic normal Sylow p-subgroup of a finite group G, then Ω_ρ(G/P) = Ω_ρ(G)·|P|-Ω_ρ(P)· |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) ≤ Ω_ρ(C). Finally, we show that if G is a group of order |L₂(p)|, then Ω_ρ(G) Ω_ρ(L₂(p)), where p ∈ \5, 11, 13\.