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On two conjectures about the sum of element orders

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

Published online by Cambridge University Press:  26 January 2021

Morteza Baniasad Azad  and
Behrooz Khosravi
Morteza Baniasad Azad
Affiliation:
Department of Pure Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), 424, Hafez Avenue, Tehran15914, Iran e-mail: baniasad84@gmail.com
Behrooz Khosravi*
Affiliation:
Department of Pure Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), 424, Hafez Avenue, Tehran15914, Iran e-mail: baniasad84@gmail.com
*
e-mail: khosravibbb@yahoo.com
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Abstract

Let G be a finite group and $\psi (G) = \sum _{g \in G} o(g)$ , where $o(g)$ denotes the order of $g \in G$ . There are many results on the influence of this function on the structure of a finite group G.

In this paper, as the main result, we answer a conjecture of Tărnăuceanu. In fact, we prove that if G is a group of order n and $\psi (G)>31\psi (C_n)/77$ , where $C_n$ is the cyclic group of order n, then G is supersolvable. Also, we prove that if G is not a supersolvable group of order n and $\psi (G) = 31\psi (C_n)/77$ , then $G\cong A_4 \times C_m$ , where $(m, 6)=1$ .

Finally, Herzog et al. in (2018, J. Algebra, 511, 215–226) posed the following conjecture: If $H\leq G$ , then $\psi (G) \unicode[stix]{x02A7D} \psi (H) |G:H|^2$ . By an example, we show that this conjecture is not satisfied in general.

Keywords

Sum of element orderssupersolvable groupelement orders

MSC classification

Primary: 20D60: Arithmetic and combinatorial problems
Secondary: 20F16: Solvable groups, supersolvable groups
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 1 , March 2022 , pp. 30 - 38
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Copyright
© Canadian Mathematical Society 2021

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