An exact upper bound for sums of element orders in non-cyclic finite groups

TL;DR

This paper establishes sharp upper bounds for the sum of element orders in non-cyclic finite groups, improving previous inequalities and providing new criteria related to group solvability.

Contribution

It introduces the best possible upper bounds for (G) in non-cyclic groups and derives new solvability conditions based on these bounds.

Findings

(G)  rac{7}{11}(C_n) for certain groups

(G) < rac{1}{q-1}(C_n) for groups of odd order

Improved inequalities over previous results from 2009, 2014, and 2015

Abstract

Denote the sum of element orders in a finite group $G$ by $\psi(G)$ and let $C_n$ denote the cyclic group of order $n$. Suppose that $G$ is a non-cyclic finite group of order $n$ and $q$ is the least prime divisor of $n$. We proved that $\psi(G)\leq\frac 7{11}\psi(C_n)$ and $\psi(G)<\frac 1{q-1}\psi(C_n)$. The first result is best possible, since for each $n=4k$, $k$ odd, there exists a group $G$ of order $n$ satisfying $\psi(G)=\frac 7{11}\psi(C_n)$ and the second result implies that if $G$ is of odd order, then $\psi(G)<\frac 12\psi(C_n)$. Our results improve the inequality $\psi(G)<\psi(C_n)$ obtained by H. Amiri, S.M. Jafarian Amiri and I.M. Isaacs in 2009, as well as other results obtained by S.M. Jafarian Amiri and M. Amiri in 2014 and by R. Shen, G. Chen and C. Wu in 2015. Furthermore, we obtained some $\psi(G)$-based sufficient conditions for the solvability of $G$.