Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The second maximal groups with respect to the sum of element orders   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
$\frac {((q^2-1)q+1)(q+1)}{q^5+1}=\frac 7{11}$.

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The categories of this paper are: nucl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: cs
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The categories of this paper are: hep-th
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The categories of this paper are: gr-qc
4
The categories of this paper are: cond-mat
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The categories of this paper are: hep-lat
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The categories of this paper are: cmp-lg
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The categories of this paper are: eess
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The categories of this paper are: stat
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-ph
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The categories of this paper are: econ
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The categories of this paper are: ao-sci
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The categories of this paper are: nlin
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The categories of this paper are: funct-an