素数 $p\geq 7$ 时 $p^6$ 阶群的表示

Permutation-based presentations for Brin's higher-dimensional Thompson groups $nV$

The higher-dimensional Thompson groups $nV$, for $n \geq 2$, were introduced by Brin in 2005. We provide new presentations for each of these infinite simple groups. The first is an infinite presentation, analogous to the Coxeter presentation for the finite symmetric group, with generating set equal to the set of transpositions in $nV$ and reflecting the self-similar structure of $n$-dimensional Cantor space. We then exploit this infinite presentation to produce further finite presentations that are considerably smaller than those previously known.

导子群为素数阶的p-群的阿贝尔化

Abelianization of p-groups with derived subgroup of prime order

在本文中，我们到达的结构$G^{ab}(= G/G')$从介绍$G$。我们还证明了阿贝尔化的弗拉蒂尼子群是$G^p$或$G^p$与导出子群$G'$的商.

满足双素假设且合成因子同构于$q\geq 7$的${\rm PSL}_2(q)$的群

Groups satisfying the two-prime hypothesis with a composition factor isomorphic to ${\rm PSL}_2(q)$ for $q\geq 7$

设$G$是一个有限群，记${\rm cd}(G)$为$G$的复不可约特征标的度集。如果对于任何不同的次数$a, b \in {\rm cd}(G)$，最大公约数${\rm gcd}(a, b)$的素数（不一定不同）的总数至多为$2$，则称群$G$满足{ \it双素数假设}。本文证明了具有与$q \geq 7$同构的${\rm PSL}_2 (q)$的合成因子的不可解群的不可约特征标次数的一个上界.

有限单群的表示：一种定量方法

Presentations of finite simple groups: a quantitative approach

每个非交换的有限单群秩$n$在一个域的大小$q$，与可能的例外的Ree组$^2G_2(3^{2e+1})$，有一个表示与有限数量的发电机和关系和总长度$O(\log n +\log q)$。作为推论，我们推出了Holt的一个猜想：存在一个常数$C$，使得对每个有限单群$G$，每个素数$p$和每个不可约$F_p [G]$ -模$M$，有$\dim H^2(G,M)\leq C\dim M$.

Short presentations of finite simple groups

Guralnick, Kantor, Kassabov and Lubotzky (J. Eur. Math. Soc. 13.2, 2011, 391-458) [GKKL] give 3-generator 7-relator presentations of $A_n$ and $S_n$ with bit-length $O(\log n)$ for $n\geq5$. This is the best possible bit-length, since $Ω(\log n)$ bits are required to specify the integer $n$ in the input. However, the generators do not satisfy the relations. This paper considers the relevant arguments given in [GKKL], identifies where the errors occur, and shows how they can be fixed in order to recover this result. The presentations are available in Magma.

群作用、变形、多群扩张和群表示

Group actions, deformations, polygroup extensions, and group presentations

推广经典扩张理论，我们解决了一个Schreier型扩张问题的多群群。因此，我们得到了一个方法来计算一个组的介绍，从它的行动在一个集合。该方法的有用性是通过推导出明确的介绍为集团$GL_2$在赋值环和超过价值的领域，集团$SL_3$在任意领域，以及为五个马蒂厄集团。此外，我们勾勒出一些方面的一个新的变形技术的群体，他们的行动，并介绍，并将其应用到计算介绍的急剧$3$ -传递Zassenhaus群$M(q^2)$（在符号的Huppert和Blackburn）的任何奇素数权力$q$。这种计算用于演示如何适当的变形的群体和他们的行动相互作用，从而提高，演示方法。

素数 $p\geq 5$ 时 $p^6$ 阶例外群

Exceptional groups of order $p^6$ for primes $p\geq 5$

研究素数 $p\geq 5$ 时 $p^6$ 阶群中例外群的比例，证明其渐近为0，并识别出 $(11p+107)/2$ 个此类群。

由完美差集编码的三角表示

Triangle Presentations Encoded by Perfect Difference Sets

本文揭示了Singer完美差集与Cartwright等人三角表示的新内在联系，提升了三角表示生成算法的效率，并通过三角表示术语统一了两类正则群的理解。

相对酉Steinberg群的表示

A presentation of relative unitary Steinberg groups

本文通过奇形式环构造了相对奇酉Steinberg群和双倍棱Steinberg群，揭示了B_ℓ、C_ℓ、F_4型Chevalley群方案的关联。

群表示上Aut（F_n）作用的传递性和（不）顺从性

On transitivity and (non)amenability of Aut(F_n) actions on group presentations

对于一个自由生成群$G$，Nielsen图$N_n(G)$，$n\geq \operatorname{rank}(G)$描述了自由群$F_n$的自同构群$\operatorname{Aut}F_n$通过初等Nielsen移动生成G的$n$ -元组的作用. Nielsen图的（非）顺从性问题与关于$\operatorname{Aut}F_n$的性质$(T)$，$n\geq 4$的开放问题有关，特别令人感兴趣。当$G$可指示时，证明了Nielsen图$N_n(G)$对所有$n\ge \max\{2,\operatorname{rank}(G)\}$的不可顺应性;当$G$初等可顺应时，证明了Nielsen图$N_n(G)$对所有足够大的$n$的不可顺应性.我们给出了秩为$d$的相对自由群（在某些簇中）的$N_d(G)$的一个明确的描述，并讨论了它们的连通性和不可变性.考虑的例子包括自由多幂零群和自由伯恩赛德群。

Full rank presentations and nilpotent groups: structure, Diophantine problem, and genericity

We study finitely generated nilpotent groups $G$ given by full rank finite presentations $\langle A \mid R\rangle$ in the variety $\mathcal{N}_c$ of nilpotent groups of class at most $c$, where $c \geq 2$. We prove that if the deficiency $|A| - |R| $ is at least $2$ then the group $G$ is virtually free nilpotent, it is quasi finitely axiomatizable (in particular, first-order rigid), and it is almost (up to finite factors) directly indecomposable. One of the main results of the paper is that the Diophantine problem in nilpotent groups given by full rank finite presentations $\langle A \mid R\ra…

Metabelian groups: full-rank presentations, randomness and Diophantine problems

We study metabelian groups $G$ given by full rank finite presentations $\langle A \mid R \rangle_{\mathcal{M}}$ in the variety $\mathcal{M}$ of metabelian groups. We prove that $G$ is a product of a free metabelian subgroup of rank $\max\{0, |A|-|R|\}$ and a virtually abelian normal subgroup, and that if $|R| \leq |A|-2$ then the Diophantine problem of $G$ is undecidable, while it is decidable if $|R|\geq |A|$. We further prove that if $|R| \leq |A|-1$ then in any direct decomposition of $G$ all, but one, factors are virtually abelian. Since finite presentations have full rank asymptotically a…

来自一个128阶群的卡尔森关联素数深度猜想的精确反例

An Exact Counterexample to Carlson's Associated-Prime Depth Conjecture from a Group of Order 128

研究卡尔森关于有限群上同调环深度与关联素数维数关系的猜想，通过列举特定128阶群的二阶初等阿贝尔子群及相关计算，给出该猜想的否定答案，证明该群上同调环无二维关联素数。

单关联子相对表示的相对双曲性

The relative hyperbolicity of one-relator relative presentations

证明了：若$G$是自由扭群，且$w(t)$是字母表$G \sqcup \{t^{\pm 1}\}$中指数和为1的字，则群$<G,t|(w(t))^k = 1>$，其中$k \geq 2$，相对于$G$是双曲的.

多圈群的q-张量平方的多圈表示

A Polycyclic Presentation for the q-Tensor Square of a Polycyclic Group

设$G$为群，$q$为非负整数.我们用$ν^q(G)$表示由$G \times G$对$q$ -张量平方$G \otimes^q G$的某种扩张.在本文中，我们得到了一个多环表示的$G \otimes^q G$，当$G$是多环的，通过其嵌入到$ν^q(G)$。此外，我们还建立了一个计算多环群$G, $的$q$ -外方$q$ MR和第二同调群$H_2(G, \mathbb{Z}_q).$的外中心$Z_q^\wedge (G)$的准则，这有助于判定$G$是否能模$q$。这些结果扩展到所有$q \geq 0$现有的方法，由于Eick和镍的情况下$q = 0$。

中心扩张映射类群的有限表示

Finite presentations of centrally extended mapping class groups

我们描述了$g \geq 3$的$ \mathcal{T}_{g,r} $的有限表示。%或$(g,r)=(2,0)$。这里$\mathcal{T}_{g,r} $是亏格为$g$的曲面的映射类群在$r$ -边界上的泛中心扩张。我们还调查了$g=2$的情况，

亏格为p+1的Riemann surfaces的自同构群，其中p是素数

Automorphism groups of Riemann surfaces of genus p+1, where p is prime

证明了如果$\cal S$是亏格为$g = p+1$的紧致黎曼曲面，其中$p$是素数，有一组自同构$G$使得对于某个实数$λ>6$有$|G|\geqλ(g-1)$，那么对于所有足够大的$p$（依赖于$λ$），$\cal S$和$G$位于六个无穷序列的例子之一中.特别是，如果$λ=8$，那么这对所有的$p\geq 17$成立，我们得到了最大的群的黎曼曲面的自同构genenera $g=p+1$。

The automorphism group of certain 3-generator $p$-groups

A presentation as well as a structural description of the automorphism group of a family of 3-generator finite $p$-groups is given, $p$ being an odd prime.

辫群的同余子群

Congruence subgroups of braid groups

本文给出了辫群的素水平同余子群的生成元的刻划。给出了有限域上辛群的一个新的表示，并计算了辫群的素水平同余子群的对称群。最后，我们找到了3股辫群的3级同余子群的有限生成集。

Unit group of $\f_{p^k}\SL(3,2),p\geq 11$

We provide the structure of the unit group of $\f_{p^k}(\SL(3,2))$, where $p\geq 11$ is a prime and $\SL(3,2)$ denotes the $3\times 3$ invertible matrices over $\f_2$.

Group-theoretic remarks on Goldbach's conjecture

The famous strongly binary Goldbach's conjecture asserts that every even number $2n \geq 8$ can always be expressible as the sum of two distinct odd prime numbers. We use a new approach to dealing with this conjecture. Specifically, we apply the element order prime graphs of alternating groups of degrees $2n$ and $2n-1$ to characterize this conjecture, and present its six group-theoretic versions; and further prove that this conjecture is true for $p+1$ and $p-1$ whenever $p \geq 11$ is a prime number.

随机幂零群群、多环表示和丢番图问题

Random nilpotent groups, polycyclic presentations, and Diophantine problems

我们引入一个随机f.g.模型，无挠，$2$ -阶幂零群（简称$τ_2$ -群）。要做到这一点，我们表明，这些正是承认的形式$ \label{tau2pres_0}\langle A, C \mid [a_i, a_j]= \prod_t {\scriptstyle c_t^{\scriptscriptstyle λ_{t,i,j}}} \ (i< j), \ [A,C]=[C,C]=1\rangle,$，其中$A=\{a_1, \dots, a_n\}$，和$C=\{c_1, \dots, c_m\}$介绍的群体。因此，可以通过固定$A$和$C$，然后对于某个$\ell$随机选择指数$λ_{t,i,j}$和$|λ_{t,i,j}|\leq \ell$，来选择随机的$τ_2$ -组$G$.我们证明，如果$m\geq n-1\geq 1$，那么下面的成立渐近几乎肯定，作为$\ell\to \infty$：整数环$\mathbb{Z}$是e-可定义的$G$，系统的方程在$\mathbb{Z}$可约系统在$G$（因此，他们是不可判定的），最大环的标量的$G$是$\mathbb{Z}$，$G$是不可分解的直积的非阿贝尔因素，和$Z(G)=\langle C \rangle$。此外，如果$m \leq n(n-1)/2$，则$G$是规则的（即$Z(G)\leq {\it Is}(G')$）。如…

具有固定关系子长度的随机表示

On random presentations with fixed relator length

随机组的标准$(n, k, d)$模型是这样一种模型，其中从$n$元素生成集上的长度为$k$的循环缩减的字的集合中随机地选择关系子。Gromov的随机群密度模型考虑了$n$是固定的，而$k$趋于无穷大的情况。相反，我们固定$k$，并让$n$趋于无穷大。我们证明了，对于所有的$k \geq 2$密度$d > 1/2$的随机组在这个模型是平凡的或循环的顺序2，而对于$d < \frac{1}{2}$这样的随机组是无限的和双曲的。此外，我们表明，对于$d<\frac{1}{k}$这样一个随机组是免费的，这个阈值是尖锐的。这些结果推广了随机群的三角形（$k = 3$）和正方形（$k = 4)$）模型的已知结果.

四元数$S$ -单元群的表示

Presentations for quaternionic $S$-unit groups

本文的目的是给出有理域$\mathbb{Q}$上Hamilton四元数中Hurwitz阶射影$S$ -单位群的表示.据我们所知，这提供了一个$S$ -算术格在一个半单李群与$S$大的第一个明确的介绍。特别是，我们给介绍了集团的产品上的Bruhat-山雀树的作用不可约和cocompanies。我们还包括一些讨论和实验相关的同余子群问题，这是开放的，当$S$包含至少两个奇素数。在附录中，我们提供了代码，允许读者计算任意有限集$S$的表示。

Two-arc-transitive bicirculants

In this paper, we determine the class of finite 2-arc-transitive bicirculants. We show that a connected $2$-arc-transitive bicirculant is one of the following graphs: $C_{2n}$ where $n\geqslant 2$, $\K_{2n}$ where $n\geqslant 2$, $\K_{n,n}$ where $n\geqslant 3$, $ \K_{n,n}-n\K_2$ where $n\geqslant 4$, $B(\PG(d-1,q))$ and $B'(\PG(d-1,q))$ where $d\geq 3$ and $q$ is a prime power, $ X_1(4,q)$ where $q\equiv 3\pmod{4}$ is a prime power, $\K_{q+1}^{2d}$ where $q$ is an odd prime power and $d\geq 2$ dividing $q-1$, $ AT_Q(1+q,2d)$ where $d\mid q-1$ and $d\nmid \frac{1}{2}(q-1)$, $ AT_D(1+q,2d)$ whe…

Quotient graphs and amalgam presentations for unitary groups over cyclotomic rings

Suppose $4|n$, $n\geq 8$, $F=F_n=\mathbb{Q}(ζ_n+\barζ_n)$, and there is one prime $\mathfrak{p}=\mathfrak{p}_n$ above $2$ in $F_n$. We study amalgam presentations for $\operatorname{PU_{2}}(\mathbb{Z}[ζ_n, 1/2])$ and $\operatorname{PSU_{2}}(\mathbb{Z}[ζ_n, 1/2])$ with the Clifford-cyclotomic group in quantum computing as a subgroup. These amalgams arise from an action of these groups on the Bruhat-Tits tree $Δ=Δ_{\mathfrak{p}}$ for $\operatorname{SL_{2}}(F_\mathfrak{p})$ constructed via the Hamilton quaternions. We explicitly compute the finite quotient graphs and the resulting amalgams for $8…

映射类群的素数阶共轭类的标准辛表示

Canonical Symplectic Representations for Prime Order Conjugacy Classes of the Mapping-class Group

本文给出了映射类群中素阶元素共轭类的辛矩阵表示的唯一标准形。我们找到了一组生成元的基本群的表面与共形自同构的素数阶，反映了行动的自同构在一个最佳的方式。这被称为{ \sl adapted}同伦基，并且存在对应的{ \sl adapted presentation}。给出了一个素数阶辛矩阵是映射类群中素数阶元的象的充要条件。

Computing a spectral sequence of finite Heisenberg groups of prime power order

Let $p\geq 5$ be a prime number, let $n\geq 2$ be a natural number and let $\text{Heis}(p^n)$ denote the Heisenberg group modulo $p^n$. We study the Lyndon-Hochschild-Serre spectral sequence $E(\text{Heis}(p^n))$ associated to $\text{Heis}(p^n)$ considered as a split extension, and show that, $E(\text{Heis}(p^n))$ collapses in the third page. Moreover, for a fixed $p$, the spectral sequences $E(\text{Heis}(p^n))$ are isomorphic from the second page on.

关于具有复根的素次多项式的伽罗瓦群

On Galois Groups of Prime Degree Polynomials with Complex Roots

设$f$是$\QQ$上的一个素数阶不可约多项式，它具有精确的复根$k$对.利用Jens Höchsmann（1999）的结果，我们证明了如果$p\geq 4k+1$，则$\Gal(f/\QQ)$同构于$A_{p}$或$S_{p}$。这改进了A。作者声明：by T.如果这样的多项式$f$可由根式求解，则它的伽罗瓦群是p次的弗罗贝纽斯群。相反，任何p次的偶数阶弗罗贝纽斯群都可以实现为$\QQ$上的具有复根的不可约多项式$p$的伽罗瓦群。

正则图的分类：具有欧拉特征数$-p^4$的正则图分类

A classification of regular maps with Euler characteristic $-p^4$ for a prime $p\geq 5$

本文研究了欧拉特征数为$-p^4$的正则图，证明了其Sylow $p$-子群的阶数限制，并分类了所有此类正则图，表明当$p$不在{2,3,5,7,13}时，相关曲面不支持正则图。

映射类群的Zapriki稠密性与有限幂次

Zariski density and finite quotients of mapping class groups

我们的主要结果是：亏格$g\geq 3$闭可定向曲面的映射类群在素$p\geq 5$上的中心扩张的量子表示的象是定义在$\Q$上的某个高阶代数半单李群$\mathbb G_p$的Zeromki稠密离散子群.作为一个应用，我们发现，对于任何素数$p\geq 5$，亏格$g$映射类群的中心扩张都覆盖到有限群$\mathbb G_p(\Z/q\Z)$上，对于除$q$以外的所有素数.该方法给出了辛群域之外的一个给定映射类群的无穷多个有限直子。

Polyhedral products, and relations in the commutator subgroup of a right-angled Coxeter group

We give a criterion of the existence of a presentation with a single relation for the commutator subgroup $RC_{\mathscr{K}}'$ of a right-angled Coxeter group $RC_{\mathscr{K}}$. Namely, we prove that $RC_{\mathscr{K}}'$ is a one-relator group if and only if $\mathscr{K}$ is either a $p$-cycle with $p\geqslant4$ or has the form ($p$-cycle)$*Δ^q$ with $p\geqslant4$ and $g\geqslant0$, where $Δ^q$ is a $q$-simplex.