On flag-transitive automorphism groups of $2$-designs with $λ$ prime
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| Format: | Preprint |
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2025
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| _version_ | 1866915277789200384 |
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| author | Alavi, Seyed Hassan Daneshkhah, Ashraf Montinaro, Alessandro |
| author_facet | Alavi, Seyed Hassan Daneshkhah, Ashraf Montinaro, Alessandro |
| contents | In this article, we study $2$-$(v,k,λ)$ designs $\mathcal{D}$ with $λ$ prime admitting flag-transitive and point-primitive almost simple automorphism groups $G$ with socle $T$ a finite exceptional simple group or a sporadic simple groups. If the socle of $G$ is a finite exceptional simple group, then we prove that $\mathcal{D}$ is isomorphic to one of two infinite families of $2$-designs with point-primitive automorphism groups, one is the Suzuki-Tits ovoid design with parameter set $(v,b,r,k,λ)=(q^{2}+1,q^{2}(q^{2}+1)/(q-1),q^{2},q,q-1)$ design, where $q-1$ is a Mersenne prime, and the other is newly constructed in this paper and has parameter set $(v,b,r,k,λ)=(q^{3}(q^{3}-1)/2,(q+1)(q^{6}-1),(q+1)(q^{3}+1),q^{3}/2,q+1)$, where $q+1$ a Fermat prime. If $T$ is a sporadic simple group, then we show that $\mathcal{D}$ is isomorphic to a unique design admitting a point-primitive automorphism group with parameter set $(v,b,r,k,λ)=(176,1100,50,2)$, $(12,22,11,6,5)$ or $(22,77,21,6,5)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_04985 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On flag-transitive automorphism groups of $2$-designs with $λ$ prime Alavi, Seyed Hassan Daneshkhah, Ashraf Montinaro, Alessandro Group Theory Combinatorics 05B05, 05B25, 20B25, 20D08 In this article, we study $2$-$(v,k,λ)$ designs $\mathcal{D}$ with $λ$ prime admitting flag-transitive and point-primitive almost simple automorphism groups $G$ with socle $T$ a finite exceptional simple group or a sporadic simple groups. If the socle of $G$ is a finite exceptional simple group, then we prove that $\mathcal{D}$ is isomorphic to one of two infinite families of $2$-designs with point-primitive automorphism groups, one is the Suzuki-Tits ovoid design with parameter set $(v,b,r,k,λ)=(q^{2}+1,q^{2}(q^{2}+1)/(q-1),q^{2},q,q-1)$ design, where $q-1$ is a Mersenne prime, and the other is newly constructed in this paper and has parameter set $(v,b,r,k,λ)=(q^{3}(q^{3}-1)/2,(q+1)(q^{6}-1),(q+1)(q^{3}+1),q^{3}/2,q+1)$, where $q+1$ a Fermat prime. If $T$ is a sporadic simple group, then we show that $\mathcal{D}$ is isomorphic to a unique design admitting a point-primitive automorphism group with parameter set $(v,b,r,k,λ)=(176,1100,50,2)$, $(12,22,11,6,5)$ or $(22,77,21,6,5)$. |
| title | On flag-transitive automorphism groups of $2$-designs with $λ$ prime |
| topic | Group Theory Combinatorics 05B05, 05B25, 20B25, 20D08 |
| url | https://arxiv.org/abs/2505.04985 |