arXiv subjects

Xiaoqin Zhan

Publications and source records attributed to Xiaoqin Zhan.

4 recordsLinked to original sources

arXiv2025

On primitivity and reduction for half-flag-transitive block designs

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.

math.GR↗

arXiv2025

Block-Transitive Automorphism Groups of $2$-$(v,5,λ)$ Designs

This paper investigates $2$-$(v,5,λ)$ designs $\mathcal{D}$ admitting a block-transitive automorphism group $G$. We first prove that if $G$ is point-imprimitive, then $v$ must be one of 16, 21, or 81. We further provide a complete classification of all such designs for $v=16$ and $v=21$. Secondly, we demonstrate that if $G$ is point-primitive, then it must be of affine type, almost simple type, or product type. Additionally, we present a classification of pairs $(\mathcal{D},G)$ where $G$ is of product type.

math.GR↗

arXiv2022

Block designs with $\gcd(r,λ)=1$ admitting flag-transitive automorphism groups

In this paper, we present a classification of $2$-designs with $\gcd(r,λ)=1$ admitting flag-transitive automorphism groups. If $G$ is a flag-transitive automorphism group of a non-trivial $2$-design $\mathcal{D}$ with $\gcd(r,λ)=1$, then either $(\mathcal{D},G)$ is one of the known examples described in this paper, or $\mathcal{D}$ has $q = p^{d}$ points with $p$ prime and $G$ is a subgroup of $AΓL_{1}(q)$.

math.GR↗

arXiv2020

Block-transitive automorphism groups on 3-designs with small block size

The paper is an investigation of the structure of block-transitive automorphism groups of a 3-design with small block size. Let $G$ be a block-transitive automorphism group of a nontrivial $3$-$(v,k,λ)$ design $\mathcal{D}$ with $k\le 6$. We prove that if $G$ is point-primitive then $G$ is of affine or almost simple type. If $G$ is point-imprimitive then $\mathcal{D}$ is a $3$-$(16,6,λ)$ design with $λ\in\{4, 12, 16, 24, 28, 48, 56, 64, 84, 96, 112, 140\}$, and $rank(G)=3$.

math.GR↗