射影线性群PGL(2, p)的一种新刻画

A New Characterization of the Projective Linear Group PGL(2, p)

摘要: 不依赖有限单群分类定理，证明了中心平凡的有限不可解群$G$ 若满足$\left|G\right|=p\left(p^2-1\right)$ 且$\left|G\left(p\right)\right|=p^2-1$ ，则$G\cong PGL\left(2,p\right)$ ，其中$p>3$ 为素数，$G\left(p\right)$ 为$G$ 中所有$p$ 阶元的集合。这一结论在Moretó猜想的框架下，通过引入中心平凡和不可解条件，成功将其推广并验证几乎单群$PGL\left(2,p\right)$ 。

Abstract: Without relying on the classification of finite simple groups, we prove that if $G$ is a finite insoluble group with a trivial center satisfying $\left|G\right|=p\left(p^2-1\right)$ and $\left|G\left(p\right)\right|=p^2-1$ , then $G\cong PGL\left(2,p\right)$ , where $p>3$ is a prime and $G\left(p\right)$ is the set of elements of order $p$ in $G$ . This conclusion, within the framework of Moretó's conjecture and through the introduction of central triviality and non-solvability conditions, successfully generalizes the conjecture and verifies it for the almost simple group $PGL\left(2,p\right)$ .