每个有限群都可以用有限的 incidences 几何来表示
Every finite group is represented by a finite incidence geometry
AI总结：
本文提出了一种将有限群对 (G, H) 表示为有限 incidences 几何关联-自同构群对的通用框架，并给出了具体例子。
摘要：
我们通过其自同构结构研究有限群与 incidences 几何之间的关系。在经典结果的基础上，我们发展了一个通用框架，将有限群对 (G, H) 表示为适当 incidences 几何的关联-自同构群对。具体来说，我们证明对于每一个这样的群对 (G, H)，存在一个有限 incidences 几何 Γ，使得 Γ 的关联-自同构群对 (Aut(Γ), Aut_I(Γ)) 与 (G, H) 是同构的。我们的构造分为两个主要步骤：首先，我们将 (G, H) 作为 incidences 系统的关联和自同构群来实现；然后，我们将这个系统细化为一个真正的 incidences 几何，保持相同的自同构群对。我们还提供了具体的例子，包括一个实现 (S_n, A_n) 对于所有 n ≥ 2 的几何族。
英文摘要：
We investigate the relationship between finite groups and incidence geometries through their automorphism structures. Building upon classical results on the realizability of groups as automorphism groups of graphs, we develop a general framework to represent pairs of finite groups $(G, H)$, where $H \trianglelefteq G$, as pairs of correlation--automorphism groups of suitable incidence geometries. Specifically, we prove that for every such pair $(G, H)$, there exists a finite incidence geometry $Γ$ satisfying that the pair $(\operatorname{Aut}(Γ), \operatorname{Aut}_I(Γ))$ of correlation--automorphism groups of $Γ$ is isomorphic to $(G, H)$. Our construction proceeds in two main steps: first, we realize $(G, H)$ as the correlation and automorphism groups of an incidence system; then, we refine this system into a genuine incidence geometry preserving the same pair of automorphisms groups. We also provide explicit examples, including a family of geometries realizing $(S_n, A_n)$ for all $n \ge 2$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
