Every finite group is represented by a finite incidence geometry

TL;DR

This paper proves that every finite group can be represented as the automorphism group of a finite incidence geometry, extending classical realizability results to a broader geometric context.

Contribution

It introduces a general framework to realize pairs of finite groups as automorphism groups of finite incidence geometries, including explicit constructions for symmetric and alternating groups.

Findings

Every finite group can be realized as an automorphism group of a finite incidence geometry.

Constructs explicit examples for symmetric and alternating groups.

Provides a method to refine incidence systems into genuine geometries.

Abstract

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 $\Gamma$ satisfying that the pair $(\operatorname{Aut}(\Gamma), \operatorname{Aut}_I(\Gamma))$ of correlation--automorphism groups of $\Gamma$ 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 $n \ge 2$.