Addendum to "A generalization of a result on the sum of element orders of a finite group" [arXiv:2001.07275]

TL;DR

This paper investigates the sum of element orders relative to a subgroup in finite groups, showing that a known inequality for nilpotent groups does not extend to infinitely many solvable groups.

Contribution

It demonstrates that the inequality relating element order sums in nilpotent groups fails in infinitely many solvable groups, extending the understanding of subgroup element order sums.

Findings

The inequality holds for nilpotent groups.

Counterexamples exist in infinitely many solvable groups.

The inequality does not generalize to all solvable groups.

Abstract

Let $G$ be a group of order $n$ and $H$ be a subgroup of order $m$ of $G$. Denote by $\psi_H(G)$ the sum of element orders relative to $H$ of $G$. It is known that if $G$ is nilpotent, then $\psi_H(G)\leq\psi_{H_m}(G)$, where $H_m$ is the unique subgroup of order $m$ of $C_n$. In this note, we show that this inequality does not hold for infinitely many finite solvable groups.