Glasnik Matematicki, Vol. 58, No. 2 (2023), 307-315. \( \)

ON GROUPS WITH AVERAGE ELEMENT ORDERS EQUAL TO THE AVERAGE ELEMENT ORDER OF THE ALTERNATING GROUP OF DEGREE \(5\)

Marcel Herzog, Patrizia Longobardi and Mercede Maj

School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv, Tel-Aviv, Israel
e-mail:herzogm@tauex.tau.ac.il

Dipartimento di Matematica, Università di Salerno, via Giovanni Paolo II, 132, 84084 Fisciano (Salerno), Italy
e-mail:plongobardi@unisa.it

Dipartimento di Matematica, Università di Salerno, via Giovanni Paolo II, 132, 84084 Fisciano (Salerno), Italy
e-mail:mmaj@unisa.it

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 \(5\).

2020 Mathematics Subject Classification.   20D60, 20F16, 20E34

Key words and phrases.   Group element orders, alternating group

Full text (PDF) (access from subscribing institutions only)

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

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