# The second maximal groups with respect to the sum of element orders

## TL;DR

This paper characterizes the groups that maximize the sum of element orders relative to cyclic groups, identifying the second maximal groups and generalizing the bounds based on the smallest prime divisor.

## Contribution

It proves the exact conditions for when non-cyclic groups reach the second maximum of the sum of element orders and extends the result to a broader class of groups.

## Key findings

- Equality holds for specific non-cyclic groups as characterized.
- Established bounds for the sum of element orders based on prime divisors.
- Generalized previous results to wider group classes.

## Abstract

Denote by $G$ a finite group and let $\psi(G)$ denote the sum of element orders in $G$. In 2009, H.Amiri, S.M.Jafarian Amiri and I.M.Isaacs proved that if $|G|=n$ and $G$ is non-cyclic, then $\psi(G)<\psi(C_n)$, where $C_n$ denotes the cyclic group of order $n$. In 2018 we proved that if $G$ is non-cyclic group of order $n$, then $\psi(G)\leq \frac 7{11}\psi(C_n)$ and equality holds if $n=4k$ with $(k,2)=1$ and $G=(C_2\times C_2)\times C_k$. In this paper we proved that equality holds if and only if $n$ and $G$ are as indicated above. Moreover we proved the following generalization of this result: Theorem 4. Let $q$ be a prime and let $G$ be a non-cyclic group of order $n$, with $q$ being the least prime divisor of $n$. Then $\psi(G)\leq \frac {((q^2-1)q+1)(q+1)}{q^5+1}\psi(C_n)$, with equality if and only if $n=q^2k$ with $(k,q)=1$ and $G=(C_q\times C_q)\times C_k$. Notice that if $q=2$, then $\frac {((q^2-1)q+1)(q+1)}{q^5+1}=\frac 7{11}$.