Every finite group is represented by a finite incidence geometry
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914202378043392 |
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| author | Ramos, Antonio Díaz Molinier, Rémi Viruel, Antonio |
| author_facet | Ramos, Antonio Díaz Molinier, Rémi Viruel, Antonio |
| contents | 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 $n \ge 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02410 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Every finite group is represented by a finite incidence geometry Ramos, Antonio Díaz Molinier, Rémi Viruel, Antonio Group Theory Combinatorics 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 $n \ge 2$. |
| title | Every finite group is represented by a finite incidence geometry |
| topic | Group Theory Combinatorics |
| url | https://arxiv.org/abs/2511.02410 |