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

By Morteza Baniasad Azad, B. Khosravi. Journal of Algebra, 2018

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Let $G$ be a finite group and $\psi(G) = \sum_{g \in G} o(g)$, where $o(g)$ denotes the order of $g \in G$. In [M. Herzog, et. al., Two new criteria for solvability of finite groups, J. Algebra, 2018], the authors put forward the following conjecture: \textbf{Conjecture.} \textit{If $G$ is a group of order $n$ and $\psi(G)>211\psi(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.

Subfields: Discrete Mathematics and Combinatorics, Organic Chemistry, Polymers and Plastics

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