发表机构
Rose-Hulman Institute of Technology（罗斯-赫尔曼理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文定义无限群的素图，并将有限可解及T-可解群的素图结果推广至相应无限群类别。
AI 中文摘要
有限群 $G$ 的素图是图 $\Gamma(G)$，其顶点集为 $|G|$ 的素数因子集 $\pi(G)$，且当且仅当存在元素 $g\in G$ 满足阶 $o(g) = pq$ 时，顶点 $p, q\in\pi(G)$ 之间有一条边。给定有限非阿贝尔单群 $T$，若群 $G$ 存在一个合成列，使得每个合成因子要么是阿贝尔群，要么同构于 $T$，则称 $G$ 是 $T$-可解群。在本文中，我们引入了无限群的素图，即图 $\Gamma(G)$，其顶点集为 $\pi(G) = \{o(g):g\in G\text{ 且 }o(g)\text{ 为素数}\}$，且当且仅当存在元素 $g\in G$ 具有阶 $pq$ 时，$p,q\in\pi(G)$ 之间有一条边，并将有限可解群和 $T$-可解群的素图的若干结果推广到某些类别的无限可解群和 $T$-可解群的素图上。
英文摘要
The prime graph of a finite group $G$ is the graph $Γ(G)$ with vertex set the set of prime divisors $π(G)$ of $|G|$ and an edge between vertices $p, q\inπ(G)$ if and only if there exists an element $g\in G$ with order $o(g) = pq$. Given a finite nonabelian simple group $T$, a group $G$ is $T$-solvable if there exists a composition series of $G$ such that every composition factor is either abelian or isomorphic to $T$. In this paper, we introduce the prime graph of an infinite group, the graph $Γ(G)$ with vertex set $π(G) = \{o(g):g\in G\text{ and }o(g)\text{ is prime}\}$ and an edge between $p,q\inπ(G)$ if and only if there exists an element $g\in G$ with order $pq$, and generalize several results on the prime graphs of finite solvable and $T$-solvable groups to results on the prime graphs of members of certain classes of infinite solvable and $T$-solvable groups.
Comments41 pages
