Every finite group is represented by a finite incidence geometry

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 $n \ge 2$.

💡 Research Summary

The paper establishes that every pair consisting of a finite group G and a normal subgroup H can be realized as the correlation‑automorphism group and the automorphism group of a finite incidence geometry. The authors build on the classical result that any finite group can appear as the full automorphism group of a suitably coloured graph, and they extend this to incidence geometries, which are richer combinatorial structures consisting of elements of various types (points, lines, planes, etc.) together with an incidence relation.

Two main constructions are presented. In the first stage (Theorem 3.1) the authors start from the Cayley directed graph of G, where each directed edge from vertex P_i to P_j is coloured by the group element g_k = g_j g_i⁻¹. They replace each directed edge by a small undirected gadget consisting of vertices S_{i,j,0}, S_{i,j,k+2}, T_{i,j} and appropriate edges, thereby obtaining an undirected simple graph G in which every vertex has degree at least 2 and every clique has size at most 2. Next they colour the vertices with a type function t:V(G)→I, where I = {0,1,…,