On characterization of simple $K_3$-groups by the number of elements of prime order

Liu, Lin; Chen, Yanheng; Jia, Songfang

doi:10.22108/ijgt.2026.144924.1955

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	On characterization of simple $K_3$-groups by the number of elements of prime order
International Journal of Group Theory
دوره 15، شماره 4، اسفند 2026، صفحه 207-214 اصل مقاله (400.57 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/ijgt.2026.144924.1955
نویسندگان
Lin Liu؛ Yanheng Chen^* ؛ Songfang Jia
School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing, P. R. China
چکیده
Since the classification theorem of finite simple groups was declared proven in the early 1980s, many group theorists have been attempting to delve deeper into the structure of simple groups from the perspective of group invariants, resulting in a series of research topics on the quantitative characterization of simple groups, such as spectral characterization, two-order characterization, and OD-characterization. In 2018, Moret'{o} proposed a new conjecture for the characterization of finite simple groups by the group order and the number of elements of the largest prime order. A specific group whose order is divisible by exactly three distinct prime numbers is called a simple $K_3$-group. These groups form a simple class of finite non-abelian simple groups. This paper establishes a characterization of $L_2(8)$ and $L_3(3)$ by combining the group order with the number of elements of the largest prime order, which shows that the conjecture holds for all simple $K_3$-groups except $L_2(7)$, $U_3(3)$ and $U_4(2)$. In addition, we also characterize $L_2(7)$, $U_3(3)$ and $U_4(2)$ under additional condition of non-solvability. Furthermore, we prove that a conjecture of Li and Shi holds for the alternating groups $A_8$, $A_{10}$, and $L_2(7)$. Thus the conjecture of Li and Shi is valid for sporadic simple groups, for alternating groups $A_n(n\geq5)$, and for all simple $K_3$-groups except $U_3(3)$ and $U_4(2)$.
کلیدواژه‌ها
Simple $K_3$-groups؛ Number of elements of prime order؛ Quantitative characterization

مراجع
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On characterization of simple $K_3$-groups by the number of elements of prime order