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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2509.21781 |
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| _version_ | 1866912607761334272 |
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| author | Zhan, Xiaoqin |
| author_facet | Zhan, Xiaoqin |
| contents | Let $\mathcal{D} = (\mathcal{P}, \mathcal{B})$ be a $2$-$(v, k, λ)$ design, and let $G$ be a half-flag-transitive automorphism group of ${\cal D}$. In this article, we first establish three sufficient conditions for $G$ to be point-primitive: (i) $λ\geq (r, 2λ)^2$, (ii) $r > 4λ(k-2)$, (iii) $(v-1,2k-2)\le2$. Next, we prove that for $λ\geq (r, 2λ)^2$, the group $G$ is either of affine type, almost simple type, or product type. Finally, we analyze the case where $G$ is of almost simple type and prove that if the socle of $G$ is a sporadic simple group then $G \cong \text{HS}$ and $\cal D$ is either the unique $2$-$(176, 128, 15240)$ design or the unique $2$-$(176, 160, 19080)$ design. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_21781 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On primitivity and reduction for half-flag-transitive block designs Zhan, Xiaoqin Group Theory 05B25, 20B25 Let $\mathcal{D} = (\mathcal{P}, \mathcal{B})$ be a $2$-$(v, k, λ)$ design, and let $G$ be a half-flag-transitive automorphism group of ${\cal D}$. In this article, we first establish three sufficient conditions for $G$ to be point-primitive: (i) $λ\geq (r, 2λ)^2$, (ii) $r > 4λ(k-2)$, (iii) $(v-1,2k-2)\le2$. Next, we prove that for $λ\geq (r, 2λ)^2$, the group $G$ is either of affine type, almost simple type, or product type. Finally, we analyze the case where $G$ is of almost simple type and prove that if the socle of $G$ is a sporadic simple group then $G \cong \text{HS}$ and $\cal D$ is either the unique $2$-$(176, 128, 15240)$ design or the unique $2$-$(176, 160, 19080)$ design. |
| title | On primitivity and reduction for half-flag-transitive block designs |
| topic | Group Theory 05B25, 20B25 |
| url | https://arxiv.org/abs/2509.21781 |