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

Document Type : Research Paper

Authors

School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing, P. R. China

Abstract

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 \(U_4(2)\).

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References

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International Journal of Group Theory
Volume 15, Issue 4 - Serial Number 4
December 2026
Pages 207-214

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History

  • Receive Date: 16 April 2025
  • Revise Date: 28 December 2025
  • Accept Date: 02 January 2026
  • Published Online: 27 January 2026

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