On products of prime element orders in finite groups

Abstract

Let $G$ be a finite group. The functions $\psi(G)$ and $\psi_*(G)$ denote the sum of the element orders and the sum of the prime element orders of $G$, respectively. Significant results related to the study of these functions have been published recently. Further, the function $R(G)$ was introduced to denote the product of the element orders of $G$. We introduce $\nrs(G)$, which denotes the product of the prime element orders of a finite group $G$. We find a lower bound for $\nrs$ on the set of groups of the same order and deduce a result on nilpotent groups using $\nrs$.