Affine groups as flag-transitive and point-primitive automorphism groups of symmetric designs
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913544028553216 |
|---|---|
| author | Alavi, Seyed Hassan Bayat, Mohsen Daneshkhah, Ashraf Montinaro, Alessandro |
| author_facet | Alavi, Seyed Hassan Bayat, Mohsen Daneshkhah, Ashraf Montinaro, Alessandro |
| contents | In this article, we investigate symmetric designs admitting a flag-transitive and point-primitive affine automorphism group. We prove that if an automorphism group $G$ of a symmetric $(v,k,λ)$ design with $λ$ prime is point-primitive of affine type, then $G=2^{6}{:}\mathrm{S}_{6}$ and $(v,k,λ)=(16,6,2)$, or $G$ is a subgroup of $\mathrm{AΓL}_{1}(q)$ for some odd prime power $q$. In conclusion, we present a classification of flag-transitive and point-primitive symmetric designs with $λ$ prime, which says that such an incidence structure is a projective space $\mathrm{PG}(n,q)$, it has parameter set $(15,7,3)$, $(7, 4, 2)$, $(11, 5, 2)$, $(11, 6, 2)$, $(16,6,2)$ or $(45, 12, 3)$, or $v=p^d$ where $p$ is an odd prime and the automorphism group is a subgroup of $\mathrm{AΓL}_{1}(q)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_04790 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Affine groups as flag-transitive and point-primitive automorphism groups of symmetric designs Alavi, Seyed Hassan Bayat, Mohsen Daneshkhah, Ashraf Montinaro, Alessandro Group Theory Combinatorics 20B25, 05B05, 05B25 In this article, we investigate symmetric designs admitting a flag-transitive and point-primitive affine automorphism group. We prove that if an automorphism group $G$ of a symmetric $(v,k,λ)$ design with $λ$ prime is point-primitive of affine type, then $G=2^{6}{:}\mathrm{S}_{6}$ and $(v,k,λ)=(16,6,2)$, or $G$ is a subgroup of $\mathrm{AΓL}_{1}(q)$ for some odd prime power $q$. In conclusion, we present a classification of flag-transitive and point-primitive symmetric designs with $λ$ prime, which says that such an incidence structure is a projective space $\mathrm{PG}(n,q)$, it has parameter set $(15,7,3)$, $(7, 4, 2)$, $(11, 5, 2)$, $(11, 6, 2)$, $(16,6,2)$ or $(45, 12, 3)$, or $v=p^d$ where $p$ is an odd prime and the automorphism group is a subgroup of $\mathrm{AΓL}_{1}(q)$. |
| title | Affine groups as flag-transitive and point-primitive automorphism groups of symmetric designs |
| topic | Group Theory Combinatorics 20B25, 05B05, 05B25 |
| url | https://arxiv.org/abs/2409.04790 |