具有唯一极大抛物的极小融合系统

一般线性群的抛物子群中的共轭类

Conjugacy classes in parabolic subgroups of general linear groups

本文证明了有限一般线性群的极大抛物子群中的幂幺共轭类数与较小维数的各种抛物子群中的幂幺共轭类数之间的关系式。我们推广了这个公式，并推导出一些推论，特别是，我们表示的数量共轭类的unitriangular矩阵在一个有限域的数量幂单共轭类的极大抛物子群在同一领域。我们证明了如何计算小维数的抛物子群中幂幺共轭类的个数。

Fusion categories containing a fusion subcategory with maximal rank

In this paper, we study fusion categories which contain a proper fusion subcategory with maximal rank. They can be viewed as generalizations of near-group fusion categories. We first prove that they admit spherical structure. We then classify those which are non-degenerate or symmetric. Finally, we classify such fusion categories of rank 4.

扩展模群中的 Neumann 性质与模群的极大非抛物子群

Neumann property in the extended modular group and maximal nonparabolic subgroups of the modular group

本文引入扩展模群及其 Neumann 子群概念，并通过取其中正行列式元素给出模群的极大非抛物非 Neumann 子群的新构造。

Saturated fusion systems on $p$-groups of maximal class

For a prime number $p$, a finite $p$-group of order $p^n$ has maximal class if it has nilpotency class $n-1$. Here we examine saturated fusion systems on maximal class $p$-groups and, in particular, we describe all the reduFor a prime number $p$, a finite $p$-group of order $p^n$ has maximal class if and only if it has nilpotency class $n-1$. Here we examine saturated fusion systems $\mathcal F$ on maximal class $p$-groups $S$ of order at least $p^4$. The Alperin-Goldschmidt Theorem for saturated fusion systems yields that $\mathcal F$ is entirely determined by the $\mathcal F$-automorphisms o…

任意秩Coxeter群的局部抛物子群

Locally parabolic subgroups in Coxeter groups of arbitrary ranks

尽管在有限秩的考克斯特群中抛物闭包的概念很重要，但在一般情况下，抛物闭包不能保证作为一个抛物子群存在。本文首先给出一个具体的例子，说明即使是秩为可数的不可约反射子群的抛物闭包也不一定作为抛物子群存在。然后通过引入“局部抛物子群”的概念，将抛物子群作为一种特殊情况，提出了“局部抛物闭包”的广义概念，并证明了局部抛物闭包总是作为局部抛物子群存在.它是抛物闭包的一个子群，并给出另一个例子说明它的包含在一般情况下是严格的。我们的结果表明，局部抛物型闭包比抛物型闭包具有更自然的性质，并提供更多的信息。我们还给出了Coxeter群中极大局部有限局部抛物子群的一个结果，推广了极大有限抛物子群的一个类似的已知事实.

Weakly maximal subgroups in regular branch groups

Let $G$ be a finitely generated regular branch group acting by automorphisms on a regular rooted tree $T$. It is well-known that stabilizers of infinite rays in $T$ (aka parabolic subgroups) are weakly maximal subgroups in $G$, that is, maximal among subgroups of infinite index. We show that, given a finite subgroup $Q\leq G$, $G$ possesses uncountably many automorphism equivalence classes of weakly maximal subgroups containing $Q$. In particular, for Grigorchuk-Gupta-Sidki type groups this implies that they have uncountably many automorphism equivalence classes of weakly maximal subgroups tha…

Rank 2 Amalgams and Fusion Systems

We classify fusion systems $\mathcal{F}$ in which $O_p(\mathcal{F})=\{1\}$, and there are two $\mathrm{Aut}_{\mathcal{F}}(S)$-invariant essential subgroups whose normalizer systems generate $\mathcal{F}$. We employ the amalgam method and, as a bonus, obtain $p$-local characterizations of certain rank $2$ group amalgams whose parabolic subgroups involve strongly $p$-embedded subgroups.

从 Monster 群的四个极大子群进行的共轭类融合

Conjugacy class fusion from four maximal subgroups of the Monster

本文确定了 Monster 群的四个极大子群到 Monster 群的共轭类融合，以证明将这些数据添加到计算代数系统 GAP 的 Character Table Library 中是合理的，并通过 Python 包 mmgroup 的可重现计算支持了相关证明。

Direct embeddings of relatively hyperbolic groups with optimal $\ell^p$ compression exponent

We prove that for all $p>1$, every relatively hyperbolic group has $\ell^p$ compression exponent equal to the minimum of the exponents of its maximal parabolic subgroups. Moreover, this embedding can be explicitly described in terms of given embeddings of the maximal parabolic subgroups.

自动结构、有理增长与几何有限双曲群

Automatic structures, rational growth and geometrically finite hyperbolic groups

我们证明了几何有限双曲群$G$上的同步自动结构的等价类的集合$SA(G)$在所有极大抛物子群$P$上的集合$SA(P)$的乘积中是稠密的. $G$上的双自动结构的等价类的集合$BSA(G)$同构于$G$的尖点（极大抛物子群的共轭类）上的集合$BSA(P)$的乘积.每个最大抛物线$P$是一个虚拟阿贝尔群，因此$SA(P)$和$BSA(P)$是由M.Shapiro和W.Neumann在"等价自动结构及其边界“中计算的。J.的Alg。Comp.2（1992）我们证明了任何几何有限双曲群都有一个生成集，对于该生成集，$G$的测地线的全语言是正则的。此外，$G$关于该生成集的增长函数是有理的。我们还确定这样一个组的自动结构是相当于测地线的。不是所有的都是，尽管所有的双自动结构都是。

典型群中抛物子群的幂幺根的多项式性质

Polynomial properties of unipotent radicals of parabolic subgroups in classical groups

本文研究辛群、正交群和酉群中极大标准抛物子群的幂幺根，给出其次数为$q^e$的不可约特征标数量的显式公式。

证明一类群上融合系统的猜想

Proving a conjecture for fusion systems on a class of groups

本文证明了奇异与块奇异融合系统重合的猜想对于极大幂零类例外 p-群（p ≥ 5）成立，推广了作者先前结果并结合 Grazian-Parker 结论，同时验证了小素数情形。

Parabolic subgroups and Automorphism groups of Schubert varieties

Let $G$ be a simple algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers, $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $w$ be an element of the Weyl group $W$ and $X(w)$ be the Schubert variety in $G/B$ corresponding to $w$. In this article we show that given any parabolic subgroup $P$ of $G$ containing $B$ properly, there is an element $w\in W$ such that $P$ is the connected component, containing the identity element of the group of all algebraic automorphisms of $X(w).$