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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$H \trianglelefteq G$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo>⊴</mo> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation>, 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> satisfying that the pair <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(\operatorname {Aut}(\Gamma ), \operatorname {Aut}_I(\Gamma ))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mo>Aut</mo> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <msub> <mo>Aut</mo> <mi>I</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of correlation–automorphism groups of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> is isomorphic to (G, H). Our construction proceeds in two main steps: first, we realize (G, H) as the correlation and automorphism groups of an incidence system; then, we refine this system into a genuine incidence geometry preserving the same pair of automorphisms groups. We also provide explicit examples, including a family of geometries realizing <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(S_n, A_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mi>n</mi> </msub> <mo>,</mo> <msub> <mi>A</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$n \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.

Every finite group is represented by a finite incidence geometry

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2026年7月4日

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Antonio Díaz Ramos,
Rémi Molinier,
Antonio Viruel

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 $H \trianglelefteq G$ $H ⊴ G$ , 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 $ $Γ$ satisfying that the pair $(\operatorname {Aut}(\Gamma ), \operatorname {Aut}_I(\Gamma ))$ $(Aut (Γ), {Aut}_{I} (Γ))$ of correlation–automorphism groups of $\Gamma $ $Γ$ 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 $(S_n, A_n)$ $(S_{n}, A_{n})$ for all $n \ge 2$ $n \geq 2$ .