A density result on the sum of element orders of a finite group

TL;DR

This paper proves that the normalized sum of element orders in finite groups densely covers the interval [0,1], and explores the properties of the associated function.

Contribution

It establishes the density of the function's image in [0,1] and analyzes its injectivity and surjectivity.

Findings

The image of the normalized sum of element orders is dense in [0,1].

The paper characterizes when the function is injective or surjective.

Provides new insights into the distribution of element order sums in finite groups.

Abstract

Let $\mathcal{G}$ be the class of all finite groups and consider the function $\psi'':\mathcal{G}\longrightarrow(0,1]$, given by $\psi''(G)=\frac{\psi(G)}{|G|^2}$, where $\psi(G)$ is the sum of element orders of a finite group $G$. In this paper, we show that the image of $\psi''$ is a dense set in $[0, 1]$. Also, we study the injectivity and the surjectivity of $\psi''$.