散发单群的内平凡模及其覆盖

零散群及其扩张的强 Gelfand 对

Strong Gelfand pairs of the sporadic groups and their extensions

本文确定了零散群、其自同构群与覆盖群的强 Gelfand 对，并找出了广义 Mathieu 群、Tits 群及 Leech Lattice 自同构群的（强）Gelfand 对。

Lyons群的上同调与交错群的双覆盖

The cohomology of the Lyons group and double covers of alternating groups

计算了散在单群Ly的模2上同调以及交错群A_8和A_10的双覆盖的模2上同调.

Regular orbits of sporadic simple groups

Given a finite group $G$ and a faithful irreducible $FG$-module $V$ where $F$ has prime order, does $G$ have a regular orbit on $V$? This problem is equivalent to determining which primitive permutation groups of affine type have a base of size 2. Let $G$ be a covering group of an almost simple group whose socle $T$ is sporadic, and let $V$ be a faithful irreducible $FG$-module where $F$ has prime order dividing $|G|$. We classify the pairs $(G,V)$ for which $G$ has no regular orbit on $V$, and determine the minimal base size of $G$ in its action on $V$. To obtain this classification, for each…

Normal $2$-coverings of the finite simple groups and their generalizations

Given a finite group $G$, we say that $G$ has weak normal covering number $γ_w(G)$ if $γ_w(G)$ is the smallest integer with $G$ admitting proper subgroups $H_1,\ldots,H_{γ_w(G)}$ such that each element of $G$ has a conjugate in $H_i$, for some $i\in \{1,\ldots,γ_w(G)\}$, via an element in the automorphism group of $G$. We prove that the weak normal covering number of every non-abelian simple group is at least $2$ and we classify the non-abelian simple groups attaining $2$. As an application, we classify the non-abelian simple groups having normal covering number $2$. We also show that the weak…

关于零星群的归纳Giannelli-McKay条件

On the inductive Giannelli-McKay condition for sporadic groups

本文提出计算框架，验证零星群及其泛覆盖上Giannelli-McKay细化猜想，通过严格条件（等变性、导子保持、中心特征相容）及理论约简，利用Clifford理论和特征表构造不增加特征度的双射。

A Class of Efficient Presentations of Finite Simple Groups

We exhibit a new presentation of the (equilateral) Von Dyck groups $D(2,3,n), \ n\ge 3$, in terms of two generators of order $n$ satisfying three relations, one of which is Artin's braid relation. By dropping the relation which fixes the order of the generators we obtain the universal covering groups of the corresponding Von Dyck groups. In the cases $n=3,\, 4,\,5$, these are respectively the double covers of the finite rotational tetrahedral, octahedral and icosahedral groups. When $n\ge 6$ we obtain infinite covers of the corresponding infinite Von Dyck groups. The interesting cases arise fo…

The Covering Numbers of the McLaughlin Group and some Primitive Groups of Low Degree

A \emph{finite cover} of a group $G$ is a finite collection $\mathcal{C}$ of proper subgroups of $G$ with the property that $\bigcup \mathcal{C} = G$. A finite group admits a finite cover if and only if it is noncyclic. More generally, it is known that a group admits a finite cover if and only if it has a finite, noncyclic homomorphic image. If $\mathcal{C}$ is a finite cover of a group $G$, and no cover of $G$ with fewer subgroups exists, then $\mathcal{C}$ is said to be a \emph{minimal cover} of $G$, and the cardinality of $\mathcal{C}$ is called the \emph{covering number} of $G$, denoted by…

CAT(0)立方复形接触图的自同构

Automorphisms of contact graphs of ${\rm CAT(0)}$ cube complexes

研究CAT(0)立方复形的自同构群与Hagen接触图的自同构群的等价性，揭示接触图与Kim-Koberda扩展图的差异。

每对非共轭极大子群之积构成的有限群是可解群

Finite groups that are the product of every pair of non-conjugate maximal subgroups are soluble

该研究证明每对非共轭极大子群之积构成的有限群必可解，否定解决了Kourovka笔记的问题10.34，通过可除性准则及GAP验证消除一般情形的基S^k障碍。

双Hurwitz Beauville群

Doubly Hurwitz Beauville groups

如果$\mathcal S$是Beauville曲面$({\mathcal C}_1\times{\mathcal C}_2)/G$，那么Hurwitz界意味着$|G|\le 1764\,χ({\mathcal S})$，当且仅当Beauville群$G$在两条曲线${\mathcal C}_i$上充当Hurwitz群。等价地，$G$有两个生成类型为$(2,3,7)$的三元组，使得一个三元组中没有生成元与另一个三元组中的生成元的幂共轭。我们证明了，如果$n$足够大，交替群$A_n$，它们的双覆盖$2.A_n$和特殊线性群$SL_n(q)$都满足这个性质，但没有零星单群或单群$L_n(q)$（$n\le 7$），${}^2G_2(3^e)$，${}^2F_4(2^e)$，${}^2F_4(2)'$，$G_2(q)$或${}^3D_4(q)$的小李秩。