局部挠超群、亚幂零群与$T$-可解群的素图
Prime Graphs of Locally Torsion Ultragroups, Metanilpotent Groups, and $T$-Solvable Groups
- 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总结：
本文利用模型论工具，继续分类$T$-可解局部挠群和局部挠亚幂零群的素图，并分类具有特定一阶性质的超群的素图，同时证明实现给定素图的群类数量与类型的结果。
AI中文摘要：
给定一个群$G$，$G$的素图，记为$\Gamma(G)$，是一个图，其顶点集为所有素数$p$的集合，使得$G$中存在阶为$p$的元素，并且当且仅当存在阶为$pq$的元素时，素数$p$和$q$之间有一条边。设$T$为非阿贝尔有限单群。一个$T$-可解群$G$是一个群，使得$G$存在一个有限次正规列，其中每个因子要么是阿贝尔群，要么同构于$T$。在本文中，我们继续对$T$-可解局部挠群的素图进行分类，并利用模型论的工具对局部挠亚幂零群的素图进行分类。我们还对具有某些一阶性质的超群的素图进行分类，并证明了关于能够将给定图实现为素图的某些群类的数量和类型的结果。
英文摘要：
Given a group $G$, the prime graph of $G$, denoted $Γ(G)$, is the graph with vertex set the set of primes $p$ such that there exists an element of order $p$ in $G$, and such that there is an edge between primes $p$ and $q$ if and only if there exists an element of order $pq$ in $G$. Let $T$ be nonabelian finite simple group. A $T$-solvable group $G$ is a group such that there exists a finite subnormal series of $G$ such that each factor is either abelian or isomorphic to $T$. In this paper, we continue the classification of the prime graphs of $T$-solvable locally torsion groups and classify prime graphs of locally torsion metanilpotent groups using tools from model theory. We also classify the prime graphs of ultragroups with certain first-order properties and prove results on the number and type of certain classes of groups that can realize a given graph as a prime graph.
