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

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 $\frac {((q^2-1)q+1)(q+1)}{q^5+1}=\frac 7{11}$.