Abstract
Let G be a finite group and \(\psi (G)=\sum _{g\in {G}}{o(g)}\). There are some results about the relation between \(\psi (G)\) and the structure of G. For instance, it is proved that if G is a group of order n and \(\psi (G)>\dfrac{211}{1617}\psi (C_n)\), then G is solvable. Herzog et al. in (J Algebra 511:215–226, 2018) put forward the following conjecture:
Conjecture. If G is a non-solvable group of order n, then
with equality if and only if \(G \cong A_5\). In particular, this inequality holds for all non-Abelian simple groups. In this paper, we prove a modified version of Herzog’s Conjecture.
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Bahri, A., Khosravi, B. & Akhlaghi, Z. A result on the sum of element orders of a finite group. Arch. Math. 114, 3–12 (2020). https://doi.org/10.1007/s00013-019-01385-8
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DOI: https://doi.org/10.1007/s00013-019-01385-8