用共轭类长相同的元素个数刻画PSL (2, 5)

A Characterization of PSL (2, 5) by Number of Elements with the Same Conjugacy Class Size

摘要: 设$G$ 是一个有限群，$U\left(G\right)$ 表示群$G$ 的共轭类长相同的元素个数的集合。证明了若$U\left(G\right)=\left\{1,pq,r^{2}q,r^{3}p\right\}$ ($p,q,r$ 为素数)，则$p,q,r$ 是互不相同的素数。进一步，若$G$ 是有限单群，则$G\cong PSL\left(2,5\right)$ 。

Abstract: Let $G$ be a finite group and $U\left(G\right)$ be the set of numbers of elements with the same conjugacy class size of $G$ . In this paper, we proved that if $U\left(G\right)=\left\{1,pq,r^{2}q,r^{3}p\right\}$ ($p,q,r$ are prime numbers), then $p,q,r$ are distinct prime numbers. Moreover, if $G$ is a finite simple group, then $G\cong PSL\left(2,5\right)$ .