Characterization of‎ ‎$A_5$ and $PSL(2,7)$ by sum of element orders

Document Type: Research Paper

Author

Department of Mathematics, Faculty of Sciences, University of Zanjan

Abstract

Let $G$ be a finite group‎. ‎We denote by $\psi(G)$ the integer $\sum_{g\in G}o(g)$‎, ‎where $o(g)$ denotes the order of $g \in G$‎. ‎Here we show that‎ ‎$\psi(A_5)< \psi(G)$ for every non-simple group $G$ of order $60$‎, ‎where $A_5$ is the alternating group of degree $5$‎. ‎Also we prove that $\psi(PSL(2,7))<\psi(G)$ for all non-simple‎ ‎groups $G$ of order $168$‎. ‎These two results confirm the conjecture‎ ‎posed in [J‎. ‎Algebra Appl.‎, ‎{\bf 10} No‎. ‎2 (2011) 187-190] for simple groups $A_5$ and $PSL(2,7)$‎.

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H. Amiri, S. M. Jafarian Amiri and I. M. Isaacs (2009). Sums of element orders in finite groups. Comm. Algebra. 37 (9), 2978-2980
H. Amiri and S. M. Jafarian Amiri (2011). Sum of element orers on finite groups of the same order. J. Algebra Appl.. 10 (2), 187-190