On the automorphism group of a possible symmetric $(81,16,3)$ design
A. Abdollahi, H. R. Maimani, R. Torabi
本文研究了一个可能的对称$(81,16,3)$设计的自同构群。
In this paper we study the automorphism group of a possible symmetric $(81,16,3)$ design.
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Sporadic simple groups as flag-transitive automorphism groups of symmetric designs
本文分类了以散在单群为基座且旗传递的对称设计，证明此类设计恰有七个，其中一个点非本原、其余点本原。
On symmetric 2-designs of prime order with almost simple flag-transitive automorphism groups
本文对允许旗传递几乎单自同构群的素数阶对称2-设计进行了完全分类，基于先前点本原性结果，给出了此类设计的完整刻画。
An automorphism of a graph is called quasi-semiregular if it fixes a unique vertex of the graph and its remaining cycles have the same length. This kind of symmetry of graphs was first investigated by Kutnar, Malnič, Martínez and Marušič in 2013, as a generalization of the well-known semiregular automorphism of a graph. Symmetric graphs of valency three or four, admitting a quasi-semiregular automorphism, have been classified in recent two papers. Let $p\geq 5$ be a prime and $Γ$ a connected symmetric graph of valency $p$ admitting a quasi-semiregular automorphism. In this paper, we first prov…
Automorphism groups of cyclic codes
本文研究了二元循环码的自同构群。特别是，我们提供了明确的构造代码的自同构群可以描述为（a）直积的两个对称群或（b）迭代圈积的几个对称群。有趣的是，我们考虑的一些代码也出现在规则格图和置换解码的上下文中。
Affine groups as flag-transitive and point-primitive automorphism groups of symmetric designs
本文研究了容许旗传递且点本原仿射自同构群的对称设计，证明了当λ为素数时自同构群的结构，并给出了此类对称设计的完整分类。
The symmetric $2$-$(v,k,λ)$ designs, with $k>λ\left(λ-3 \right)/2$, admitting a flag-transitive, point-imprimitive automorphism group are completely classified: they are the known $2$-designs with parameters $(16,6,2),(45,12,3),(15,8,4)$ or $(96,20,4)$.
It was proved in [Y.-Q. Feng, C. H. Li and J.-X. Zhou, Symmetric cubic graphs with solvable automorphism groups, {\em European J. Combin.} {\bf 45} (2015), 1-11] that a cubic symmetric graph with a solvable automorphism group is either a Cayley graph or a $2$-regular graph of type $2^2$, that is, a graph with no automorphism of order $2$ interchanging two adjacent vertices. In this paper an infinite family of non-Cayley cubic $2$-regular graphs of type $2^2$ with a solvable automorphism group is constructed. The smallest graph in this family has order 6174.
In this note we report the classification of all symmetric 2-(36,15,6) designs that admit an automorphism of order 2 and their incidence matrices generate an extremal ternary self-dual code. It is shown that up to isomorphism, there exists only one such design, having a full automorphism group of order 24, and the ternary code spanned by its incidence matrix is equivalent to the Pless symmetry code.
On Automorphism Group of a Family of Symmetric Graphs
本文研究Fu-Tao Hu等人2010年引入的顶点传递图族$H(n,k)$的自同构群与测地线传递性，并解答了原论文中遗留的未解问题。
When dealing with symmetry properties of mathematical objects, one of the fundamental questions is to determine their full automorphism group. In this paper this question is considered in the context of even/odd permutations dichotomy. More precisely: when is it that existence of automorphisms acting as even permutations on the vertex set of a graph, called {\em even automorphisms}, forces existence of automorphisms that act as odd permutations, called {\em odd automorphisms}. As a first step towards resolving the above question, a complete information on existence of odd automorphisms in cubi…
The full automorphism groups of the five symmetric $(15,8,4)$-designs
本文利用点线几何方法，确定了五个对称$(15,8,4)$-设计的全自同构群，并描述了它们在点集与区组集上的作用。
The structure of automorphism groups of semigroup inflations
证明了一个半群的自同构群是它的真子半群的一个膨胀，它分解为两个群的半直积，其中一个群是全对称群的直和。
Cubic symmetric graphs having an abelian automorphism group with two orbits
围长为6的有限连通三次对称图已被K. Kutnar和D. Marušiši，特别是，这些图中的每一个都有一个阿贝尔自同构群，在顶点集上有两个轨道。本文确定了所有具有后一性质的三次对称图。特别地，除图K_4，K_{3,3}，Q_3，GP（5，2），GP（10，2），F40和GP（24，5）外，所得到的图都是围长为6的。
Symmetric 2-(35,17,8) designs with an automorphism of order 2
本文分类所有容许阶2自同构的对称2-(35,17,8)设计，证明其互不同构者共11,642,495个，并统计相关3-(36,18,8)设计共1,015,225个。
The dual stable Grothendieck polynomials $g_λ$ and their sums $\sum_{μ\subsetλ} g_μ$ (which represent $K$-homology classes of boundary ideal sheaves and structure sheaves of Schubert varieties in the Grassmannians) have the same product structure constants. In this paper we first explain that the ring automorphism $g_λ\mapsto\sum_{μ\subsetλ} g_μ$ on the ring of symmetric functions is described as the operator $F^\perp$, the adjoint of the multiplication $(F\cdot)$, by a "group-like" element $F=\sum_{i} h_i$ where $h_i$ is the complete symmetric function. Next we give a generalization: starting…
In this paper we construct two new symmetric designs with parameters 2-(176,50,14) as designs invariant under certain subgroups of the full automorphism group of the Higman design. One is self-dual and has the full automorphism group of size 11520 and other is not self-dual and has the full automorphism group of size 2520.
Kirkman triple systems (KTSs) are among the most popular combinatorial designs and their existence has been settled a long time ago. Yet, in comparison with Steiner triple systems, little is known about their automorphism groups. In particular, there is no known congruence class representing the orders of a KTS with a number of automorphisms at least close to the number of points. We fill this gap by proving that whenever $v \equiv 39$ (mod 72), or $v \equiv 4^e48 + 3$ (mod $4^e96$) and $e \geq 0$, there exists a KTS on $v$ points having at least $v-3$ automorphisms. This is only one of the co…
On symmetric 2-(70,24,8) designs with an automorphism of order 6
该研究对满足特定不动点条件的6阶循环自同构群作用下的对称2-(70,24,8)设计进行完全分类，得到同构意义下3718个此类设计，显著扩充了该类设计的已知数量。
Symmetric 2$-$(36,15,6) designs with an automorphism of order two
本文完成了所有容许二阶自同构的对称 2-(36,15,6) 设计的分类，共发现 1,547,701 个非同构设计，并计算了其关联矩阵张成的三元线性码，发现了具有未知重量分布的近极值自对偶码。
The exceptional symmetry
本文给出了对称群只具有一个例外对称的初等证明。我指的是对称群$S_n$的外自同构群是平凡的，除非$n=6$和$S_6$的外自同构群有唯一的非平凡元素。
Let $S_n$ denote the symmetric group of degree $n$ with $n\geq 3$. Set $S=\{c_n=(1\ 2\ldots \ n),c_n^{-1},(1\ 2)\}$. Let $Γ_n=\mathrm{Cay}(S_n,S)$ be the Cayley graph on $S_n$ with respect to $S$. In this paper, we show that $Γ_n$ ($n\geq 13$) is a normal Cayley graph, and that the full automorphism group of $Γ_n$ is equal to $\mathrm{Aut}(Γ_n)=R(S_n)\rtimes \langle\mathrm{Inn}(ϕ)\rangle\cong S_n\rtimes \mathbb{Z}_2$, where $R(S_n)$ is the right regular representation of $S_n$, $ϕ=(1\ 2)(3\ n)(4\ n-1)(5\ n-2)\cdots$ $(\in S_n)$, and $\mathrm{Inn}(ϕ)$ is the inner isomorphism of $S_n$ induced b…
Automorphisms of the cycle prefix digraph
循环前缀有向图是并行结构中对称互连网络的有效模型。人们已经发现，循环前缀网络具有许多吸引人的通信特性。本文确定了圈前缀有向图的自同构群。证明了圈前缀有向图的自同构群与其基础字母表上的对称群同构。我们的方法可以应用于其他类的图表建立在字母表，包括超立方体，考茨图，和de Bruijn图。
Maximally symmetric stable curves II
我们找到了一个尖锐的界限的阶的自同构群的稳定曲线属$g$与$3g-3$节点，和一个尖锐的界限的阶的自同构群的这种曲线与所有光滑组件。结合我们的文章math.CO/0608645的结果，我们发现：从图的理论上讲，具有给定顶点数和最多自同构的三次图是简单的，但从代数几何上讲，具有最多自同构的$3g-3$节点的稳定曲线是非简单的对偶图。
Quantum automorphism groups of vertex-transitive graphs of order $\leq 11$
本文研究了顶点数小于小于11的点传递图的量子自同构群。除了一个可能的例外，这些可以通过使用各种乘积运算从循环群${\mathbb Z}_n$、对称群$S_n$和量子对称群$\mathcal Q_n$获得。例外的情况是彼得森图，我们提出了一些问题。
Constructions of graphs with any possible two-fold automorphism and automorphism groups
本文构造连通非二部顶点决定图，使其标准二重覆盖的自同构群可实现为任意抽象群与Z₂的任意半直积，并证明非对称图的二重覆盖自同构群为奇数阶阿贝尔群，同时给出具有任意数量和类型同构二重覆盖的图。
The $S_3$-symmetric $q$-Onsager algebra and its Lusztig automorphisms
本文研究 S_3-对称 q-Onsager 代数，为六个标准生成元构造 Lusztig 自同构，刻画其相互关系及其对有限维不可约模的作用，并给出维数 5 的实例。
We give a new construction of the outer automorphism of the symmetric group on six points. Our construction features a complex Hadamard matrix of order six containing third roots of unity and the algebra of split quaternions over the real numbers.
Cluster automorphisms have been shown to have links to the mapping class groups of surfaces, maximal green sequences and to exchange graph automorphisms for skew-symmetric cluster algebras. In this paper we aim to generalise these results to the skew-symmetrizable case by introducing a marking on the exchange graph. Many skew-symmetrizable matrices unfold to skew-symmetric matrices and we consider how cluster automorphisms behave under this unfolding with applications to coverings of orbifolds by surfaces.
Nilpotent dessins: Decomposition theorem and classification of the abelian dessins
映射是可定向闭曲面的2-胞腔剖分。一个图是一个顶点着色固定的二分图。一个dessin是正则的，如果它的一组颜色和方向保持自同构的作用传递的边缘，和一个正则dessin是对称的，如果它允许一个额外的外部对称转置顶点颜色。研究了具有幂零自同构群正则设计。我们证明了每一个这样的Dessin是一个平行积的正则Dessins的自同构群的Sylow子群。对具有交换自同构群的正则和对称系统进行了分类和列举。
A regular cover of a connected graph is called {\em cyclic} or {\em dihedral} if its transformation group is cyclic or dihedral respectively, and {\em arc-transitive} (or {\em symmetric}) if the fibre-preserving automorphism subgroup acts arc-transitively on the regular cover. In this paper, we give a classification of arc-transitive cyclic and dihedral covers of a connected pentavalent symmetric graph of order twice a prime. All those covers are explicitly constructed as Cayley graphs on some groups, and their full automorphism groups are determined.
Asymptotic Automorphism Groups of Circulant Graphs and Digraphs
我们证明了几乎所有的循环图都有尽可能小的自同构群。对于不具有尽可能小的自同构群的循环图，我们给出了一些整数族，使得几乎所有阶在这些整数族中的循环图都不是正规的.第二作者证明了几乎所有自同构群不是尽可能小的Cayley（di）图都是正规的，这些结果为这一猜想提供了反例。然后，它表明，有一个大家庭的整数，几乎所有的循环有向图，其顺序在于在这个家庭，并没有自同构群尽可能小，是正常的。此外，我们还探讨了非正规循环（di）图的自同构群的渐近行为，并表明没有得到一般性的结论。
Automorphism groups of rational circulant graphs through the use of Schur rings
研究了具有有理谱的循环群上的Cayley图（简称有理循环图）的自同构群.借助于Schur环的技巧，证明了该问题等价于考虑循环群的正交群块结构的自同构群.利用这种观察，所需的群体表示的广义圈积的对称群。
Complex Representations of Groups and Involutions of its Automorphisms
本文建立了不可约特征标次数和与有限群自同构相关的扭曲对合数之间的关系，并开发了内自同构和对称群情形下的算法框架，进而给出了识别群具有复（非实）不可约表示的条件并探讨了结构推论。
These lecture notes are on automorphism groups of Cayley graphs and their applications to optimal fault-tolerance of some interconnection networks. We first give an introduction to automorphisms of graphs and an introduction to Cayley graphs. We then discuss automorphism groups of Cayley graphs. We prove that the vertex-connectivity of edge-transitive graphs is maximum possible. We investigate the automorphism group and vertex-connectivity of some families of Cayley graphs that have been considered for interconnection networks; we focus on the hypercubes, folded hypercubes, Cayley graphs gener…
A connected symmetric graph of prime valency is {\em basic} if its automorphism group contains no nontrivial normal subgroup having more than two orbits. Let $p$ be a prime and $n$ a positive integer. In this paper, we investigate properties of connected pentavalent symmetric graphs of order $2p^n$, and it is shown that a connected pentavalent symmetric graph of order $2p^n$ is basic if and only if it is either a graph of order $6$, $16$, $250$, or a graph of three infinite families of Cayley graphs on generalized dihedral groups -- one family has order $2p$ with $p=5$ or $5 \mid (p-1)$, one f…
Full automorphism groups of association schemes based on attenuated spaces
衰减空间中给定维数的子空间集合具有对称结合格式的结构，它是Grassmann格式和双线性型格式的推广. In [K. Wang，J. Guo，F.李，基于衰减空间的关联方案，欧洲J. Combin。31（2010）297- 305]，计算其相交数。本文确定了它的全自同构群。
Automorphism groups of graphs of bounded Hadwiger number
本文研究有界哈德维格数有限图的自同构群结构，通过分析边传递图证明非阿贝尔合成因子阶有界，并表明这类自同构群可由阿贝尔群、对称群和有界阶群反复扩张得到。
The infinite cyclohedron and its automorphism group
多面体是一个著名的无限族的有限维多面体，可以构造从中心对称三角剖分的偶数边多边形。本文引入了一个无限维的类比，证明了我们构造的对称群是由单群T给出的Thompson无限群的2次中心扩张与2阶循环群的半直积.这些结果的灵感来自于一个类似的最近分析的第一作者的自同构群的无限维associahedron。
Let $\mathrm{G}$ be a subgroup of the symmetric group $\mathfrak S(U)$ of all permutations of a countable set $U$. Let $\overline{\mathrm{G}}$ be the topological closure of $\mathrm{G}$ in the function topology on $U^U$. We initiate the study of the poset $\overline{\mathrm{G}}[U]:=\{f[U]\mid f\in \overline{\mathrm{G}}\}$ of images of the functions in $\overline{\mathrm{G}}$, being ordered under inclusion. This set $\overline{\mathrm{G}}[U]$ of subsets of the set $U$ will be called the \emph{poset of copies for} the group $\mathrm{G}$. A denomination being justified by the fact that for every …
A graph is said to be symmetric if its automorphism group is transitive on its arcs. Guo et al. (Electronic J. Combin. 18, \#P233, 2011) and Pan et al. (Electronic J. Combin. 20, \#P36, 2013) determined all pentavalent symmetric graphs of order $4pq$. In this paper, we shall generalize this result by determining all connected pentavalent symmetric graphs of order four times an odd square-free integer. It is shown in this paper that, for each of such graphs $\itΓ$, either the full automorphism group ${\sf Aut}\itΓ$ is isomorphic to ${\sf PSL}(2,p)$, ${\sf PGL}(2,p)$, ${\sf PSL}(2,p){\times}\mat…
Full automorphism groups of association schemes based on isotropic subspaces
有限经典极空间中给定维数的所有子空间的集合具有对称结合概型的结构。如果维数为零，这是空间的共线图的方案;如果维数为最大，这是对偶极方案。本文确定了该方案的全自同构群。
Unitary graphs and classification of a family of symmetric graphs with complete quotients
一个有限图$Γ$称为$G$ -对称的，如果$G$是$Γ$的一个自同构群，且该自同构群在$Γ$的邻顶点的序偶的集合上是传递的.本文研究一类对称图，称为酉图，其顶点是厄米酉标的标志，其邻接关系由基础有限域的某些元素决定。这样的图允许酉群作为自同构群，并且它们在具有完全关联的对称图族的分类中起着重要的作用，使得相关的关联结构是一个双重点传递的线性空间。本文给出了酉图的分类，并研究了酉图的组合性质。
On Automorphism Groups of Networks
我们认为各种经验的“现实世界”的网络的自同构群的大小和结构，并发现，在经典的随机图模型相比，许多现实世界的网络是丰富的对称。我们将自同构群结构与网络拓扑结构联系起来，并讨论了对称性的一般形式及其在现实网络中的起源。
Let $d, n \in \mathbb{Z}^+$ such that $1\leq d \leq n$. A $d$-code $\mathcal{C} \subset \mathbb{F}_q^{n \times n}$ is a subset of order $n$ square matrices with the property that for all pairs of distinct elements in $\mathcal{C}$, the rank of their difference is greater than or equal to $d$. A $d$-code with as many as possible elements is called a maximum $d$-code. The integer $d$ is also called the minimum distance of the code. When $d<n$, a classical example of such an object is the so-called generalized Gabidulin code. There exist several classes of maximum $d$-codes made up respectively o…
Let $Γ$ be a graph and let $G$ be a group of automorphisms of $Γ$. The graph $Γ$ is called $G$-normal if $G$ is normal in the automorphism group of $Γ$. Let $T$ be a finite non-abelian simple group and let $G = T^l$ with $l\geq 1$. In this paper we prove that if every connected pentavalent symmetric $T$-vertex-transitive graph is $T$-normal, then every connected pentavalent symmetric $G$-vertex-transitive graph is $G$-normal. This result, among others, implies that every connected pentavalent symmetric $G$-vertex-transitive graph is $G$-normal except $T$ is one of $57$ simple groups. Furthermo…
We construct Hopf algebras whose elements are representations of combinatorial automorphism groups, by generalising a theorem of Zelevinsky on Hopf algebras of representations of wreath products. As an application we attach symmetric functions to representations of graph automorphism groups, generalising and refining Stanley's chromatic symmetric function.
Symmetric and Kähler--Einstein Fano polygons
本文研究奇异对称与 Kähler--Einstein Fano 多胞体，证明所有对称 Fano 多胞体均为 Kähler--Einstein，并详细刻画了对称与 Kähler--Einstein Fano 多边形的自同构群。
Hadamard Structures with Associated Automorphisms
本文给出了新的Hadamard矩阵及其相关的组合结构。特别是，它构造了5202个36阶的不等价Hadamard矩阵以及180538个Hadamard对称设计，除了那些允许3阶自同构的结构之外，还有35个点。因此，至少有272116个Hadamard 3-设计，36个点和70条线。我们发现这里构造的所有Hadamard矩阵都等价于正则Hadamard矩阵。这一事实有助于推测，阿达玛矩阵的秩序36，并可能秩序$4m^2$，是正规的。
Automorphism group of the complete transposition graph
完全转置图是指其顶点是对称群$S_n$的元素的图，且该图的两个顶点$α$和$β$相邻当且仅当存在某个转置$(i,j)$使得$α=(i,j) β$ .因此，完全转置图是由所有转置的集合$S$生成的对称群的凯莱图$\Cay(S_n,S)$。文献中的一个公开问题是确定哪些Cayley图是正规的。最近证明了由4个循环相邻转置生成的Cayley图是不正规的。本文证明了对所有$n \ge 3$，完全换位图不是正规Cayley图.此外，证明了完全转置图的自同构群等于\[ \Aut(\Cay(S_n,S)) = (R(S_n) \rtimes \Inn(S_n)) \rtimes \mathbb{Z}_2, \]，其中$R(S_n)$是$S_n$的右正则表示，$\Inn(S_n)$是$S_n$的内自同构群，而$\mathbb{Z}_2 = \langle h \rangle$，其中$h$是映射$α\mapsto α^{-1}$ .
Let $X$ be a connected, locally finite graph with symmetric growth. We prove that there is a vertex coloring $ϕ\colon X\to\{0,1\}$ and some $R\in\mathbb{N}$ such that every automorphism $f$ preserving $ϕ$ is $R$-close to the identity map; this can be seen as a coarse geometric version of symmetry breaking. We also prove that the infinite motion conjecture is true for graphs where at least one vertex stabilizer $S_x$ satisfies the following condition: for every non-identity automorphism $f\in S_x$, there is a sequence $x_n$ such that $\lim d(x_n,f(x_n))=\infty$.
Brill-Noether Theory of Maximally Symmetric Graphs
我们分析了三价图和在固定亏格中具有最大可能自同构群的重图的Brill-Noether理论。对于亏格至少为3的三价重图，我们证明了存在一个具有极大自同构群的图是Brill-Noether特殊的。对于亏格至少为6的无圈多重图和亏格至少为7的单图，我们证明了类似的结果。这种分析在任何足够大的亏格中产生了卡波拉索猜想的反例。
The automorphism group of a self-dual [72,36,16] code does not contain S_3, A_4, or D_8
用Magma进行的计算机计算表明，不存在长度为72的极值自对偶二元码C，其自同构群包含3次对称群、4次交错群或8次二面体群。结合文献中已知的结果，得到Aut（C）的阶数至多为5或同构于阶数为8的初等阿贝尔群。
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