单群PSL(2,p)的数量刻画<sup>*</sup>

单群PSL(2,p)的数量刻画^*

摘要: 证明了以下２个结果.设 p\geq 5 为素数，若单群 G 的阶为 \dfrac(p+1)p(p-1)2 ，则 G\cong PSL(2,p) .若群 G 的阶为 \dfrac(p+1)p(p-1)2 ，且群中元素最高阶数为 p ，则 G\cong PSL(2,p) 或 p=7 以及 G\cong \mathbfZ_2^3\colon (\mathbfZ_7\colon \mathbfZ_3) .后一个结论是Bull. Korean Math. Soc., 2015, 52(6): 2 025-2 034中的一个主要成果，但本文提供了一种完全不同的证明.本文的证明仅采用初等群论和置换群论的基础知识，不依赖于有限单群分类定理.

Abstract: This paper establishes the following two results. Let p\geq 5 be a prime number, if the order of a simple group G is \dfrac(p+1)p(p-1)2 , then G\cong PSL(2,p) . If the order of the group G is \dfrac(p+1)p(p-1)2 , and the maximal order of elements in G is p , then either G\cong PSL(2,p) or p=7 and G\cong \mathrmZ_2^3\colon (\mathrmZ_7\colon \mathrmZ_3) . The latter result is a main finding of Bull. Korean Math. Soc., 2015, 52(6): 2025-2034 and we present a completely different method for its proof. The proofs in this paper utilize elementary group theory and basic knowledge of permutation groups, without relying on the classification theorem of finite simple groups.