A criterion for solvability of a finite group by the sum of element orders

DOI10.1016/J.JALGEBRA.2018.09.009zbMATH Open1446.20037arXiv1808.00253OpenAlexW2962964556WikidataQ129251093 ScholiaQ129251093MaRDI QIDQ1799013

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Publication date: 18 October 2018

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Abstract: Let $G$ be a finite group and $psi(G)=su{m}_{ginG}o(g)$, where $o(g)$ denotes the order of $ginG$. In [M. Herzog, et. al., Two new criteria for solvability of finite groups, J. Algebra, 2018], the authors put forward the following conjecture: extbf{Conjecture.} extit{If $G$ is a group of order $n$ and $psi(G)>211psi(C_n)/1617$, where $C_n$ is the cyclic group of order $n$, then $G$ is solvable.} In this paper we prove the validity of this conjecture.

Full work available at URL: https://arxiv.org/abs/1808.00253

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