Abstract
Let \({\mathscr {G}}\) be the class of all finite groups and consider the function \(\psi '':{\mathscr {G}}\longrightarrow (0,1]\), given by \(\psi ''(G)=\frac{\psi (G)}{|G|^2}\), where \(\psi (G)\) is the sum of element orders of a finite group G. In this paper, we show that the image of \(\psi ''\) is a dense set in [0, 1]. Also, we study the injectivity and the surjectivity of \(\psi ''\).
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References
Amiri, H., Jafarian Amiri, S.M., Isaacs, I.M.: Sums of element orders in finite groups. Comm. Algebra 37, 2978–2980 (2009)
Amiri, H., Jafarian Amiri, S.M.: Sum of element orders on finite groups of the same order. J. Algebra Appl. 10(2), 187–190 (2011)
Apostol, T.M.: Calculus. Vol. I: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd edn. Wiley, New York (1967)
Baniasad Azad, M., Khosravi, B.: A criterion for solvability of a finite group by the sum of element orders. J. Algebra 516, 115–124 (2018)
Azad, M. B., Khosravi, B.: On two conjectures about the sum of element orders. arXiv:1905.00815
Herzog, M., Longobardi, P., Maj, M.: An exact upper bound for sums of element orders in non-cyclic finite groups. J. Pure Appl. Algebra 222(7), 1628–1642 (2018)
Herzog, M., Longobardi, P., Maj, M.: Two new criteria for solvability of finite groups. J. Algebra 511, 215–226 (2018)
Herzog, M., Longobardi, P., Maj, M.: The second maximal groups with respect to the sum of element orders. arXiv:1901.09662
Lazorec, M.S.: A connection between the number of subgroups and the order of a finite group. arXiv:1901.06425
Microsoft Visual Studio 2019. https://visualstudio.microsoft.com/vs/
Nitecki, Z.: Cantorvals and subsum sets of null sequences. Am. Math. Mon. 122, 862–870 (2015)
Tărnăuceanu, M.: Detecting structural properties of finite groups by the sum of element orders, accepted for publication in Israel J. Math
Tărnăuceanu, M.: A criterion for nilpotency of a finite group by the sum of element orders. arXiv:1903.09744
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Lazorec, MS., Tărnăuceanu, M. A density result on the sum of element orders of a finite group. Arch. Math. 114, 601–607 (2020). https://doi.org/10.1007/s00013-020-01437-4
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DOI: https://doi.org/10.1007/s00013-020-01437-4