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Detecting structural properties of finite groups by the sum of element orders. (English) Zbl 1483.20048

Summary: “In this paper, we introduce a new function related to the sum of element orders of finite groups. It is used to give some criteria for a finite group to be cyclic, abelian, nilpotent, supersolvable and solvable, respectively.”
Reviewer’s remarks: To be more explicit, the formation \(\psi(G)= \sum_{x\in G}|x|\) (where \(|x|\) stands for the order of the element \(x\) of the finite group \(G\)) has been the subjects of recent investigations in several papers; see the References but also the paper by M.-S. Lazorec [Bull. Malays. Math. Sci. Soc. (2) 44, No. 2, 941–951 (2021; Zbl 1479.20018)]. In the paper under review, the author introduces, for reasons given in the beginning of the paper, the function \(\psi''(G)= \psi(G)/|G|^2\), from which he is able to show his main Theorem 1.1, namely
(a)
If \(\psi''(G)>\alpha= \psi''(C_2\times C_2)\), then \(G\) is cyclic;
(b)
if \(\psi''(G)>\beta=\psi''(Q_8)\), then \(G\) is abelian;
(c)
if \(\psi''(G)>\gamma=\psi''(S_3)\), then \(G\) is nilpotent;
(d)
if \(\psi''(G)>\delta= \psi''(A_4)\), then \(G\) is supersolvable;
(e)
if \(\psi''(G)>\varepsilon= \psi''(A_5)\), then \(G\) is solvable.
(Of course, \(C_2\times C_2\) stands for the Klein-four group, \(Q_8\) for the quaternion group of order 8, \(S_3\) for the symmetric group on three symbols, \(A_4\) for the alternating group on four symbols and \(A_5\) for the alternating group on five symbols.) The ordered set of values \(\{\alpha,\beta,\gamma, \delta,\varepsilon\}\) is equal to the corresponding ordering values \[ \Biggl\{\frac{7}{16}, \frac{27}{64}, \frac{13}{36}, \frac{31}{144}, \frac{211}{3600}\Biggr\}. \] The author proves also, that the converses of the implications in (a), (b), (c), (d) and (e) are not true.
In order to obtain the justification of Theorem 1.1, several lemmas and numerical properties had to be shown, and that is, what the author did too.

MSC:

20D60 Arithmetic and combinatorial problems involving abstract finite groups

Citations:

Zbl 1479.20018
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References:

[1] Amiri, H.; Jafarian Amiri, S. M.; Isaacs, I. M., Sums of element orders in finite groups, Communications in Algebra, 37, 2978-2980 (2009) · Zbl 1183.20022 · doi:10.1080/00927870802502530
[2] Amiri, H.; Jafarian Amiri, S. M., Sums of element orders on finite groups of the same order, Journal of Algebra and its Applications, 10, 187-190 (2011) · Zbl 1217.20015 · doi:10.1142/S0219498811004057
[3] Jafarian Amiri, S. M., Second maximum sum of element orders on finite nilpotent groups, Communications in Algebra, 41, 2055-2059 (2013) · Zbl 1273.20021 · doi:10.1080/00927872.2011.653070
[4] Jafarian Amiri, S. M.; Amiri, M., Second maximum sum of element orders on finite groups, Journal of Pure and Applied Algebra, 218, 531-539 (2014) · Zbl 1283.20018 · doi:10.1016/j.jpaa.2013.07.003
[5] Baniasad Asad, M.; Khosravi, B., A criterion for solvability of a finite group by the sum of element orders, Journal of Algebra, 516, 115-124 (2018) · Zbl 1446.20037 · doi:10.1016/j.jalgebra.2018.09.009
[6] Herzog, M.; Longobardi, P.; Maj, M., An exact upper bound for sums of element orders in non-cyclic finite groups, Journal of Pure and Applied Algebra, 222, 1628-1642 (2018) · Zbl 1486.20030 · doi:10.1016/j.jpaa.2017.07.015
[7] Herzog, M.; Longobardi, P.; Maj, M., Two new criteria for solvability of finite groups in finite groups, Journal of Algebra, 511, 215-226 (2018) · Zbl 1436.20034 · doi:10.1016/j.jalgebra.2018.06.015
[8] Herzog, M.; Longobardi, P.; Maj, M., Sums of element orders in groups of order 2m with m odd, Communications in Algebra, 47, 2035-2048 (2019) · Zbl 1476.20024 · doi:10.1080/00927872.2018.1527924
[9] M. Herzog, P. Longobardi and M. Maj, The second maximal groups with respect to the sum of element orders, https://arxiv.org/abs/1901.09662. · Zbl 1476.20024
[10] Isaacs, I. M., Finite Group Theory (2008), Providence, RI: American Mathematical Society, Providence, RI · Zbl 1169.20001
[11] Scott, W. R., Group Theory (1964), Englewood Cliffs, NJ: Prentice-Hall, Englewood Cliffs, NJ · Zbl 0126.04504
[12] Shen, R.; Chen, G.; Wu, C., On groups with the second largest value of the sum of element orders, Communications in Algebra, 43, 2618-2631 (2015) · Zbl 1317.20027 · doi:10.1080/00927872.2014.900686
[13] Suzuki, M., Group Theory, Vols. I, II (1982), Berlin: Springer, Berlin
[14] M. Tărnăuceanu A criterion for nilpotency of a finite group by the sum of element orders, https://arxiv.org/abs/1903.09744. · Zbl 1449.20017
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