Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Two Conjectures about the Sum of Element Orders   Let $G$ be a finite group and $\psi(G) = \sum_{g \in G} o(g)$, where $o(g)$
denotes the order of $g \in G$. First, we prove that if $G$ is a group of order
$n$ and $\psi(G) >31\psi(C_n)/77$, where $C_n$ is the cyclic group of order
$n$, then $G$ is supersolvable. This proves a conjecture of
M.~{T\u{a}rn\u{a}uceanu}. Moreover, M. Herzog, P. Longobardi and M. Maj put
forward the following conjecture: If $H\leq G$, then $\psi(G) \leqslant \psi(H)
|G:H|^2$. In the sequel, by an example we show that this conjecture is not
satisfied in general.

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The categories of this paper are: mtrl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: supr-con
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The categories of this paper are: ao-sci
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The categories of this paper are: solv-int
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The categories of this paper are: comp-gas
5
The categories of this paper are: gr-qc
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The categories of this paper are: hep-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-fin
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The categories of this paper are: econ
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The categories of this paper are: atom-ph
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The categories of this paper are: funct-an
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The categories of this paper are: alg-geom
0
The categories of this paper are: hep-ph
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The categories of this paper are: stat