Sums of element orders in groups of odd order

Abstract

Denote by $G$ a finite group and by $\psi (G)$ the sum of element orders in $G$. If $t$ is a positive integer, denote by $C_t$ the cyclic group of order $t$ and write $\psi (t)=\psi (C_t)$. In this paper we proved the following Theorem A: Let $G$ be a non-cyclic group of odd order $n=qm$, where $q$ is the smallest prime divisor of $n$ and $(m,q)=1$. Then the following statements hold. (1) If $q=3$, then $\frac{\psi (G)}{\psi (|G|)}\le \frac{85}{301}$, and equality holds if and only if $n=3\cdot 7\cdot m_1$ with $(m_1,42)=1$ and $G=(C_7⋊C_3)\times C_{m_1}$, with $C_7⋊C_3$ non-abelian. (2) If $q>3$, then $\frac{\psi (G)}{\psi (|G|)}\le \frac{p^4+p^3-p^2+1}{p^5+1}$, where $p$ is the smallest prime bigger than $q$ and equality holds if and only if $n=q{p}^2{m}_1$ with $(m_1,p!)=1$ and $G=C_q\times C_p\times C_p\times C_{m_1}$.