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Glasnik matematički, Vol. 58 No. 2, 2023.

Original scientific paper

https://doi.org/10.3336/gm.58.2.10

On groups with average element orders equal to the average element order of the alternating group of degree \(5\)

Marcel Herzog ; School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv, Tel-Aviv, Israel
Patrizia Longobardi ; Dipartimento di Matematica, Università di Salerno, via Giovanni Paolo II, 132, 84084 Fisciano (Salerno), Italy
Mercede Maj ; Dipartimento di Matematica, Università di Salerno, via Giovanni Paolo II, 132, 84084 Fisciano (Salerno), Italy

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Abstract

Let \(G\) be a finite group. Denote by \(\psi(G)\) the sum
\(\psi(G)=\sum_{x\in G}|x|,\) where \(|x|\) denotes the order of the element \(x\), and
by \(o(G)\) the average element orders, i.e. the quotient \(o(G)=\frac{\psi(G)}{|G|}.\)
We prove that \(o(G) = o(A_5)\) if and only if \(G \simeq A_5\), where \(A_5\) is the alternating group of degree \(5\).

Keywords

Group element orders, alternating group

Hrčak ID:

312017

URI

https://hrcak.srce.hr/312017

Publication date:

22.12.2024.

Visits: 441 *

Publication date: 27 December 2023

Volume: Vol 58

Issue: Svezak 2

Pages: 307-315

DOI: 10.3336/gm.58.2.10

Article Information (continued)

Categories:

Subject: Izvorni znanstveni članak

Keywords:

Keyword: Group element orders, alternating group

On groups with average element orders equal to the average element order of the alternating group of degree \(5\)

Marcel Herzog[1]

Email: herzogm@tauex.tau.ac.il

Patrizia Longobardi[2]

Email: plongobardi@unisa.it

Mercede Maj[3]

Email: mmaj@unisa.it

 

[1] School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv, Tel-Aviv, Israel

[2] Dipartimento di Matematica, Università di Salerno, via Giovanni Paolo II, 132, 84084 Fisciano (Salerno), Italy

[3] Dipartimento di Matematica, Università di Salerno, via Giovanni Paolo II, 132, 84084 Fisciano (Salerno), Italy

Abstract

Let \(G\) be a finite group. Denote by \(\psi(G)\) the sum \(\psi(G)=\sum_{x\in G}|x|,\) where \(|x|\) denotes the order of the element \(x\), and by \(o(G)\) the average element orders, i.e. the quotient \(o(G)=\frac{\psi(G)}{|G|}.\) We prove that \(o(G) = o(A_5)\) if and only if \(G \simeq A_5\), where \(A_5\) is the alternating group of degree \(5\).

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