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PROPERTIES OF FINITE GROUPS DETERMINED BY THE PRODUCT OF THEIR ELEMENT ORDERS

Part of: Representation theory of groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  01 June 2020

MORTEZA BANIASAD AZAD Open the ORCID record for MORTEZA BANIASAD AZAD [Opens in a new window]  and
BEHROOZ KHOSRAVI Open the ORCID record for BEHROOZ KHOSRAVI [Opens in a new window]
MORTEZA BANIASAD AZAD*
Affiliation:
Department of Pure Mathematics, Faculty of Mathematics and Computer Science,Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Ave., Tehran 15914, Iran email baniasad84@gmail.com
BEHROOZ KHOSRAVI
Affiliation:
Department of Pure Mathematics, Faculty of Mathematics and Computer Science,Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Ave., Tehran 15914, Iran email khosravibbb@yahoo.com
*
✉baniasad84@gmail.com
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Abstract

For a finite group $G$, define $l(G)=(\prod _{g\in G}o(g))^{1/|G|}/|G|$, where $o(g)$ denotes the order of $g\in G$. We prove that if $l(G)>l(A_{5}),l(G)>l(A_{4}),l(G)>l(S_{3}),l(G)>l(Q_{8})$ or $l(G)>l(C_{2}\times C_{2})$, then $G$ is solvable, supersolvable, nilpotent, abelian or cyclic, respectively.

Keywords

product of element orderssolvable groupssupersolvable groupsnilpotent groupsabelian groupscyclic groups

MSC classification

Primary: 20D60: Arithmetic and combinatorial problems
Secondary: 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks 20D15: Nilpotent groups, $p$-groups 20F16: Solvable groups, supersolvable groups 20F18: Nilpotent groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 1 , February 2021 , pp. 88 - 95
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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References

Amiri, H., Jafarian Amiri, S. M. and Isaacs, I. M., ‘Sums of element orders in finite groups’, Comm. Algebra 37(9) (2009), 2978–2980.CrossRefGoogle Scholar
Bahri, A., Khosravi, B. and Akhlaghi, Z., ‘A result on the sum of element orders of a finite group’, Arch. Math. 114(1) (2020), 3–12.CrossRefGoogle Scholar
Baniasad Azad, M. and Khosravi, B., ‘A criterion for solvability of a finite group by the sum of element orders’, J. Algebra 516 (2018), 115–124.CrossRefGoogle Scholar
Baniasad Azad, M. and Khosravi, B., ‘On the sum of element orders of PSL(2, p) for some p’, Ital. J. Pure Appl. Math. 2 (2019), 18–24.Google Scholar
Baniasad Azad, M. and Khosravi, B., ‘On two conjectures about the sum of element orders’, Preprint, 2019, arXiv:1905.00815.Google Scholar
Garonzi, M. and Patassini, M., ‘Inequalities detecting structural properties of a finite group’, Comm. Algebra 45(2) (2017), 677–687.CrossRefGoogle Scholar
Herzog, M., Longobardi, P. and Maj, M., ‘An exact upper bound for sums of element orders in non-cyclic finite groups’, J. Pure Appl. Algebra 222(7) (2018), 1628–1642.CrossRefGoogle Scholar
Herzog, M., Longobardi, P. and Maj, M., ‘Two new criteria for solvability of finite groups’, J. Algebra 511 (2018), 215–226.CrossRefGoogle Scholar
Herzog, M., Longobardi, P. and Maj, M., ‘Properties of finite and periodic groups determined by their element orders (A survey)’, in: Group Theory and Computation (eds. Sastry, N. and Yadav, M.) (Springer, Berlin, 2018), 59–90.CrossRefGoogle Scholar
Herzog, M., Longobardi, P. and Maj, M., ‘Sums of element orders in groups of order 2m with m odd’, Comm. Algebra 47(5) (2019), 2035–2048.CrossRefGoogle Scholar
Herzog, M., Longobardi, P. and Maj, M., ‘The second maximal groups with respect to the sum of element orders’, Preprint, 2019, arXiv:1901.09662.Google Scholar
Isaacs, I. M., Finite Group Theory (American Mathematical Society, Providence, RI, 2008).Google Scholar
Jafarian Amiri, S. M. and Amiri, M., ‘Second maximum sum of element orders on finite groups’, J. Pure Appl. Algebra 218(3) (2014), 531–539.CrossRefGoogle Scholar
Khosravi, B. and Baniasad Azad, M., ‘Recognition by the product element orders’, Bull. Malays. Math. Sci. Soc. 43 (2020), 1183–1193.CrossRefGoogle Scholar
Tărnăuceanu, M., ‘A note on the product of element orders of finite Abelian groups’, Bull. Malays. Math. Sci. Soc. 36(4) (2013), 1123–1126.Google Scholar
Tărnăuceanu, M., ‘A criterion for nilpotency of a finite group by the sum of element orders’, Preprint, 2019, arXiv:1903.09744.CrossRefGoogle Scholar
Tărnăuceanu, M., ‘Detecting structural properties of finite groups by the sum of element orders’, Preprint, 2019, arXiv:1904.03340.CrossRefGoogle Scholar
Xu, H., Chen, G. and Yan, Y., ‘A new characterization of simple K 3 -groups by their orders and large degrees of their irreducible characters’, Comm. Algebra 42(12) (2014), 5374–5380.CrossRefGoogle Scholar