On the product of element orders of some finite groups

Abstract

In a finite group G, $\rho (G)$ denotes the product of element orders of G. Let $p,q,r,s$ and t be prime numbers. Recently, it was proved that $C_{p^3}$, C_pq, $C_{p^{2}q}$ and C_pqrs are characterizable by the product of element orders, and if $\rho (G)=\rho (C_{\mathit{pqr}})$, then $G\cong A_5$ or $C_{\mathit{pqr}}$. In this paper, we continue this work and we show that $C_p\times A_5$ (for p > 5) and C_pqrst are characterizable by the product of element orders. In the rest of the paper, we show that if $\rho (G)=\rho {(C_{\mathit{pqr}})}^s\rho {(C_s)}^{p^{2}q},$ then $(p,q,r,s)=(2,3,5,11)$ and $G\cong L_2(11)$. This shows that the structure of $\rho (G)$ uniquely determined $L_2(11)$ and so $L_2(11)$ is the only group with this form of $\rho (G)$.

KEYWORDS:

• Characterization
• cyclic group
• element order
• product of element orders

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20D60
• 20D05

Acknowledgments

The authors would like to thank the referee for valuable comments and suggestions which improved the manuscript seriously.