A characterization of $A_5$ by its average order

Document Type : Research Paper

Author

Faculty of Mathematics, "Al.I. Cuza" University of Iasi

Abstract

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.

Keywords

Main Subjects


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Volume 14, Issue 3 - Serial Number 3
September 2025
Pages 117-123
  • Receive Date: 06 August 2023
  • Revise Date: 27 December 2023
  • Accept Date: 30 December 2023
  • Published Online: 01 January 2024