Every finite group is represented by a finite incidence geometry

Antonio Díaz Ramos; Rémi Molinier; Antonio Viruel

Paper

Every finite group is represented by a finite incidence geometry

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

, where

, as pairs of correlation--automorphism groups of suitable incidence geometries. Specifically, we prove that for every such pair

, there exists a finite incidence geometry

satisfying that the pair

of correlation--automorphism groups of

is isomorphic to

. Our construction proceeds in two main steps: first, we realize

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

for all