University of Isfahan International Journal of Group Theory 2251-7650 14 3 2024 01 01 A characterization of $A_5$ by its average order 117 123 28078 10.22108/ijgt.2023.138637.1866 EN Marius Tarnauceanu Faculty of Mathematics, "Al.I. Cuza" University of Iasi Journal Article 2023 08 06 Let $o(G)$ be the average order of a finite group $G$. M. Herzog, P. Longobardi and M. Maj [M. Herzog, P. Longobardi and M. Maj, Another criterion for solvability of finite groups, J. Algebra, 597 (2022) 1-23.] showed that if $G$ is non-solvable and $o(G)=o(A_5)$, then $G\cong A_5$. In this note, we prove that the equality $o(G)=o(A_5)$ does not hold for any finite solvable group $G$. Consequently, up to isomorphism, $A_5$ is determined by its average order.

https://ijgt.ui.ac.ir/article_28078_39bbf4fdefa1697ab68304edc8acd500.pdf