Anton A. Klyachko
An adjoint Chevalley group of rank at least 2 over a rational algebra (or a similar ring), its elementary subgroup, and the corresponding Lie ring have the same automorphism group. These automorphisms are explicitly described.
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In the paper we prove that every automorphism of any adjoint Chevalley group of types B_2 or G_2 is standard, i.e., it is a composition of the ``inner'' automorphism, ring automorphism and central automorphism.
In the paper we prove that every automorphism of a Chevalley group of type $F_4$ over a commutative local ring with~1/2 is standard, i. e., it is a composition of ring and inner automorphisms.
We prove that every isomorphism of Chevalley groups of type $G_2$ over commutative local rings with 1/2 and 1/3 is standard, i. e., it is a composition of a ring isomorphism and an inner automorphism.
In the given paper we prove that every automorphism of a Chevalley group of type $B_l$, $l\geqslant 2$, over a commutative local ring with 1/2 is standard, i. e., it is a composition of ring, inner and central automorphisms.
In the given paper we prove that every automorphism of a Chevalley group of type $A_l$, $D_l$, or $E_l$, $l\geqslant 3$, over a commutative local ring without 1/2 is standard, i. e., it is a composition of ring, inner, central and graph automorphisms.
In the paper we prove that every automorphism of a (elementary) Chevalley group of type $A_l, D_l$, or $E_l$, $l\geqslant 2$, over a commutative local ring with 1/2 is standard, i. e., is the composition of inner, ring, graph and central automorphisms.
The irreducible Brauer characters of the automorphism group of the Chevalley group F_4(2)
本研究利用计算方法推导谢瓦莱群F_4(2)自同构群的不可约Brauer特征标，结果仍存在一个参数和一处一致性问题待完善。
In this paper we prove that every automorphism of (elementary) adjoint Chevalley group with root system of rank $>1$ over a commutative ring (with 1/2 for the systems $A_2$, $F_4$, $B_l$, $C_l$; with 1/2 and 1/3 for the system $G_2$) is standard, i.\,e., it is a composition of ring, inner, central and graph automorphisms.
Automorphisms of a Chevalley group of type G_2 over a commutative ring R with 1/3 generated by the all invertible elements and 2R
该研究刻画了一类含 1/3 交换环上 G_2 型 Chevalley 群的自同构结构，证明其均可分解为环自同构与内自同构的复合。
Automorphisms of Chevalley groups over commutative rings
本文证明交换环上秩大于1的 Chevalley 群及其初等子群的所有自同构均为环、内、中心与图自同构的标准复合，并给出模型论应用。
In the paper we prove that every automorphism of and elementary adjoint Chevalley group of types $A_l, D_l$, or $E_l$, over local commutative ring with 1/2 is a composition of a ring automorphism and conjugation by some matrix from the normalizer of the Chevalley group in $\GL(V)$ ($V$ is the adjoint representation space).
Automorphisms of Twisted Chevalley Groups of type ${}^2 D_\ell \ (\ell \geq 4)$ over Local Rings
本文作为交换环上扭Chevalley群同构与自同构分类系列研究的一部分，证明了含1/2的局部环上${}^2 D_\ell$（$\ell \geq 4$）型扭Chevalley群的所有自同构均为标准自同构。
Automorphisms of Twisted Chevalley Groups of type ${}^2 A_\ell \ (\ell \geq 5)$ over Local Rings
研究局部环上${}^2 A_\ell \ (\ell \geq 5)$型扭曲谢瓦莱群自同构，证明在2可逆局部环上其自同构是标准的，为后续相关群的自同构和同构分类研究奠定基础。
We provide an explicit construction for a class of commutative, non-associative algebras for each of the simple Chevalley groups of simply laced type. Moreover, we equip these algebras with an associating bilinear form, which turns them into Frobenius algebras. This class includes a 3876-dimensional algebra on which the Chevalley group of type E8 acts by automorphisms. We also prove that these algebras admit the structure of (axial) decomposition algebras.
Let $M^*(q)$ be the unique nonassociative finite simple Moufang loop constructed over $GF(q)$. We prove that $Aut(M^*(2))$ is the Chevalley group $G_2(2)$, by extending multiplicative automorphism of $M^*(2)$ into linear automorphisms of the unique split octonion algebra over GF(2). Many of our auxiliary results apply in the general case. In the course of the proof we show that every element of a split octonion algebra can be written as a sum of two elements of norm one.
Automorphisms of twisted Chevalley groups of type ${}^2E_6$ over local rings
本文分类了带对合且2可逆的交换局部环上，两种等义类型的${}^2E_6$型扭Chevalley群的抽象自同构，证明其均为标准型，明确了伴随与单连通情形下自同构的结构，通过基变换替换等步骤完成分类。
Automorphisms of Twisted Chevalley Groups of type ${}^2 A_3$ and ${}^2 A_4$ over Local Rings
本文对含1/2的局部环上${}^2A_3$、${}^2A_4$型扭Chevalley群的自同构完成完整分类，是交换环上扭Chevalley群自同构与同构分类系列研究的一部分，填补了低秩情形的空白。
Let $p$ be an odd prime. We construct a $p$-group $P$ of nilpotency class two, rank seven and exponent $p$, such that $\mathrm{Aut}(P)$ induces $N_{\mathrm{GL}(7,p)}(G_2(p)) = Z(\mathrm{GL}(7,p)) G_2(p)$ on the Frattini quotient $P/Φ(P)$. The constructed group $P$ is the smallest $p$-group with these properties, having order $p^{14}$, and when $p = 3$, our construction gives two nonisomorphic $p$-groups. To show that $P$ satisfies the specified properties, we study the action of $G_2(q)$ on the octonion algebra over $\mathbb{F}_q$, for each power $q$ of $p$, and explore the reducibility of the…
We prove that Chevalley group over the field $F$ of zero characteristic possess $R_{\infty}$ property, if $F$ has torsion group of automorphisms or $F$ is an algebraically closed field which has finite transcendence degree over $\mathbb{Q}$. As a consequence we obtain that the twisted conjugacy class $[e]_φ$ of unit element is a subgroup of Chevalley group if and only if $φ$ is central automorphism.
Let $\hat G$ be the finite simply connected version of an exceptional Chevalley group, and let $V$ be a nontrivial irreducible module, of minimal dimension, for $\hat G$ over its field of definition. We explore the overgroup structure of $\hat G$ in $\mathrm{GL}(V)$, and the submodule structure of the exterior square (and sometimes the third Lie power) of $V$. When $\hat G$ is defined over a field of odd prime order $p$, this allows us to construct the smallest (with respect to certain properties) $p$-groups $P$ such that the group induced by $\mathrm{Aut}(P)$ on $P/Φ(P)$ is either $\hat G$ or…
We prove that Chevalley groups of the classical series $B_l, C_l, D_l$ over an integral domain of zero characteristic, which has torsion automorphism group, possess the $R_{\infty}$-property.
Twisted conjugacy classes in twisted Chevalley groups
该研究聚焦扭共轭类与R_{∞}-性质相关问题，证明了满足特定域条件的特征零域上的扭Chevalley群同时具备R_{∞}-性质和S_{∞}-性质。
In this article, we show that if G is a simply connected Chevalley group of either classical type of rank bigger than 1 or type E6, and q > 9 is a power of a prime number p > 5, then G = G(F_q((1/t))), up to an automorphism, has a unique lattice of minimum covolume, which is G(F_q[t]).
To each automorphism of a spherical building there is naturally associated an "opposition diagram", which encodes the types of the simplices of the building that are mapped onto opposite simplices. If no chamber (that is, no maximal simplex) of the building is mapped onto an opposite chamber then the automorphism is called "domestic". In this paper we give the complete classification of domestic automorphisms of split spherical buildings of types $\mathsf{E}_6$, $\mathsf{F}_4$, and $\mathsf{G}_2$. Moreover, for all split spherical buildings of exceptional type we classify (i) the domestic homo…
Reduction of the group isomorphism problem to the group automorphism problem
本研究类比图同构与图自同构问题的多项式时间等价关系，证明群同构问题可多项式时间归约为群自同构问题，并给出向自同构计数等相关问题的归约方法。
This survey on the automorphism groups of finite p-groups focuses on three major topics: explicit computations for familiar finite p-groups, such as the extraspecial p-groups and Sylow p-subgroups of Chevalley groups; constructing p-groups with specified automorphism groups; and the discovery of finite p-groups whose automorphism groups are or are not p-groups themselves. The material is presented with varying levels of detail, with some of the examples given in complete detail.
Isomorphisms and automorphisms of graph products of groups
研究群的图积的同构问题，通过给出其自同构群的生成集来解决，此生成集推广了直角阿廷群的相关成果。
Sha-rigidity of adjoint Chevalley groups of types $A_1$, $A_2$, $B_2$, $G_2$ over commutative rings
本文证明了在特定环上，A1、A2、B2和G2类型的伴随Chevalley群及其初等子群的局部内自同构是内自同构，从而这些群具有Sha-刚性。
Let $G$ be a finitely generated group and let $C^*$ denote the group of all central automorphisms of $G$ fixing the center of $G$ elementwise. Azhdari and Malayeri [J. Algebra Appl., {\bf 6}(2011), 1283-1290] gave necessary and sufficient conditions on $G$ such that $C^* \simeq \mathrm{Inn}(G)$. We prove a technical lemma and, as a consequence, obtain a short and easy proof of this result of Azhdari and Malayeri. Subsequently, we also obtain short proofs of some other existing and some new related results.
We prove that the isomorphism problem for finitely generated fully residually free groups is decidable. We also show that each finitely generated fully residually free group G has a decomposition that is invariant under automorphisms of G, and obtain a structure theorem for the group of outer automorphisms Out(G).
Let $(G,*)$ and $(G',\cdot)$ be groupoids. A bijection $f: G \rightarrow G'$ is called a half-isomorphism if $f(x*y)\in\{f(x)\cdot f(y),f(y)\cdot f(x)\}$, for any $ x, y \in G$. A half-isomorphism of a groupoid onto itself is a half-automorphism. A half-isomorphism $f$ is called special if $f^{-1}$ is also a half-isomorphism. In this paper, necessary and sufficient conditions for the existence of special half-isomorphisms on groupoids and quasigroups are obtained. Furthermore, some examples of non-special half-automorphisms for loops of infinite order are provided.
We show that any isomorphism between mapping class groups of orientable infinite-type surfaces is induced by a homeomorphism between the surfaces. Our argument additionally applies to automorphisms between finite-index subgroups of these `big' mapping class groups and shows that each finite-index subgroup has finite outer automorphism group. As a key ingredient, we prove that all simplicial automorphisms between curve complexes of infinite-type orientable surfaces are induced by homeomorphisms.
We describe the structure of the algebraic group of automorphisms of all simple finite dimensional Lie superalgebras. Using this and étale cohomology considerations, we list all different isomorphism classes of the corresponding twisted loop superalgebras.
Automorphism groups of axial algebras
本文针对轴向代数中因维数重复出现而可能存在的意外同构问题，提出计算完全自同构群的方法，并在最高151维的若干例子上进行应用验证。
For every odd prime $p$ and every integer $n\geq 12$ there is a Heisenberg group of order $p^{5n/4+O(1)}$ that has $p^{n^2/24+O(n)}$ pairwise nonisomorphic quotients of order $p^{n}$. Yet, these quotients are virtually indistinguishable. They have isomorphic character tables, every conjugacy class of a non-central element has the same size, and every element has order at most $p$. They are also directly and centrally indecomposable and of the same indecomposability type. The recognized portions of their automorphism groups are isomorphic, represented isomorphically on their abelianizations, an…
We construct a nonuniform lattice and an infinite family of uniform lattices in the automorphism group of a hyperbolic building with all links a fixed finite building of rank 2 associated to a Chevalley group. We use complexes of groups and basic facts about spherical buildings.
We prove that the automorphism group of the dihedral quandle with n elements is isomorphic to the affine group of the integers mod n, and also obtain the inner automorphism group of this quandle. In [9], automorphism groups of quandles (up to isomorphisms) of order less than or equal to 5 were given. With the help of the software Maple, we compute the inner and automorphism groups of all seventy three quandles of order six listed in the appendix of [4]. Since computations of automorphisms of quandles relates to the problem of classification of quandles, we also describe an algorithm implemente…
This paper studies the rigidity properties of the abstract commensurator of the outer automorphism group of a universal Coxeter group of rank $n$, which is the free product $W_n$ of $n$ copies of $\mathbb{Z}/2\mathbb{Z}$. We prove that for $n\geq 5$ the natural map $\mathrm{Out}(W_n) \to \mathrm{Comm}(\mathrm{Out}(W_n))$ is an isomorphism and that every isomorphism between finite index subgroups of $\mathrm{Out}(W_n)$ is given by a conjugation by an element of $\mathrm{Out}(W_n)$.
It is well known that the automorphism group of a regular dessin is a two-generator finite group, and the isomorphism classes of regular dessins with automorphism groups isomorphic to a given finite group $G$ are in one-to-one correspondence with the orbits of the action of $\Aut(G)$ on the ordered generating pairs of $G$. If there is only one orbit, then up to isomorphism the regular dessin is uniquely determined by the group $G$ and it is called uniquely regular. In the paper we investigate the classification of uniquely regular dessins with a nilpotent automorphism group. The problem is red…
In this article, we study isomorphisms between simple current extensions of a simple VOA. For example, we classify the isomorphism classes of simple current extensions of some VOAs. Moreover, we consider the same simple current extension and describe the normalizer of the abelian automorphism group associated with this extension. In particular, we regard the moonshine module as simple current extensions of five subVOAs V_L^+ for 2-elementary totally even lattices L, and describe corresponding five normalizers of elementary abelian 2-group in the automorphism group of the moonshine module in te…
In this paper we look at the automorphisms of the multiplicative group of finite nearfields. We find partial results for the actual automorphism groups. We find counting techniques for the size of all finite nearfields. We then show that these results can be used in order to count the number of near vector spaces of a given dimension over a given nearfield, up to isomorphism.
We show that each half-automorphism of a finite automorphic Moufang loop is trivial. In general this is not true for finite left automorphic Moufang loops and for finite automorphic loops.
Automorphisms of order $2$ are studied in order to understand generalized symmetric spaces. The groups of type $E_6$ we consider here can be realized as both the group of linear maps that leave a certain determinant invariant, and also as the identity component of the automorphism group of a class of structurable algebras known as Brown algebras. We will classify the $k$-involutions of these groups of type $E_6$ using aspects of both descriptions, and give detailed descriptions of representatives over certain fields including algebraically closed fields, $\mathbb{R}$, $\mathbb{F}_p$, and $\mat…
Let S be a finite generating set of a torsion-free, nilpotent group G. We show that every automorphism of the Cayley graph Cay(G;S) is affine. (That is, every automorphism of the graph is obtained by composing a group automorphism with multiplication by an element of the group.) More generally, we show that if Cay(G;S) and Cay(G';S') are connected Cayley graphs of finite valency on two nilpotent groups G and G', then every isomorphism from Cay(G;S) to Cay(G';S') factors through to a well-defined affine map from G/N to G'/N', where N and N' are the torsion subgroups of G and G', respectively. F…
In this paper we consider the class of 2-dimensional Artin groups with connected, large type, triangle-free defining graphs (type CLTTF). We classify these groups up to isomorphism, and describe a generating set for the automorphism group of each such Artin group. In the case where the defining graph has no separating edge or vertex we show that the Artin group is not abstractly commensurable to any other CLTTF Artin group. If, moreover, the defining graph satisfies a further `vertex rigidity' condition, then the abstract commensurator group of the Artin group is isomorphic to its automorphism…
Families preserving isomorphisms via techniques in anabelian geometry: with an application to a generalized Neukirch-Uchida theorem
本文从算术代数几何视角推广保持自同构的族的概念，证明一大类profinit群的相关同构族源自内自同构，并将其应用得到ℓ-拟数域的广义Neukirch-Uchida定理。
Rigidity and automorphisms of groups constructed using Jones' technology
本文推广 Brothier 的结果，给出 Jones 技术产生的 Thompson 群 V 分裂扩张同构的结构定理，并据此分类无限制扭曲置换式圈积、分解其自同构群，揭示协变与反变函子构造之间的对偶性。
On automorphism groups of metric reduced products of symmetric groups
在OCA和马丁公理下，研究对称群度量约化积的同构，刻画其自同构群。
In this paper, we show the existence of a polynomial time graph isomorphism algorithm for all graphs excluding graphs that are locally trianglefree. This particular class of graphs allows to divide the graph into neighbourhood sub-graph where each of induced sub-graph (neighbourhood) has at least 2 vertices. We construct all possible permutations for each induced sub-graph using a search tree. We construct automorphisms of subgraphs based on these permutations. Finally, we decide isomorphism through automorphisms . The author expects that the solution, present in this paper, may lead to a fast…
More Automorphism Groups of Quandles
该研究证明了二面体广群位移群的同构结构与含平凡列广群的位移群和内自同构群的等价性，验证并计算了10阶及以内所有广群的自同构群、内自同构群与位移群。
We prove that there are groups in the constructible universe whose automorphism towers are highly malleable by forcing. This is a consequence of the fact that, under a suitable diamond hypothesis, there are sufficiently many highly rigid non-isomorphic Souslin trees whose isomorphism relation can be precisely controlled by forcing.
Part 1 : We remark that the conjugacy problem for pairs of hyperbolic au- tomorphisms of a finitely presented group (typically a free group) is decidable. The solution that we propose uses the isomorphism problem for the suspensions, and the study of their automorphism group. Part 2 : In a previous work (part 1), we remarked that the conjugacy problem for pairs of atoroidal automorphisms of a free group was solvable by mean of the isomorphism problem for hyperbolic groups and an orbit problem for the automorphism group of their suspensions (i.e. their semidirect product with Z for the relevant…
Oligomorphic groups, their automorphism groups, and the complexity of their isomorphism
本文研究Roelcke预紧的置换群子群，证明其内自同构群在自同构群中闭，且在外自同构群完全不连通局部紧；并给出方法证明两类寡态群的同构关系光滑，且其自同构群拓扑同构于寡态群，外自同构群为profinite。
The main result of this paper is a complete classification of the outer automorphism groups of two-generator, one-relator groups with torsion. To this classification we apply recent algorithmic results of Dahmani--Guirardel, which yields an algorithm to compute the isomorphism class of the outer automorphism group of a given two-generator, one-relator group with torsion.
We prove that if the group of fixed points of a generic automorphism of a simple group of finite Morley rank is pseudofinite, then this group is an extension of a (twisted) Chevalley group over a pseudofinite field. On the way to obtain this result, we classify non-abelian definably simple pseudofinite groups of finite centralizer dimension (using the ideas of John S. Wilson).
In this paper, we study the isomorphism problem for linear representations. A linear representation Tn*(K) of a point set K is a point-line geometry, embedded in a projective space PG(n+1,q), where K is contained in a hyperplane. We put constraints on K which ensure that every automorphism of Tn*(K) is induced by a collineation of the ambient projective space. This allows us to show that, under certain conditions, two linear representations Tn*(K) and Tn*(K') are isomorphic if and only if the point sets K and K' are PGammaL-equivalent. We also deal with the slightly more general problem of iso…
In this paper, given a split extension of an arbitrary Coxeter group by automorphisms of the Coxeter graph, we determine the involutions in that extension whose centralizer has finite index. Our result has applications to many problems such as the isomorphism problem of general Coxeter groups. In the argument, some properties of certain special elements and of the fixed-point subgroups by graph automorphisms in Coxeter groups, which are of independent interest, are also given.
The automorphism group of the vertex operator algebra $V_L^+$ is studied by using its action on isomorphism classes of irreducible $V_L^+$-modules. In particular, the shape of the automorphism group of $V_L^+$ is determined when $L$ is isomorphic to an even unimodular lattice without roots, $\sqrt2R$ for an irreducible root lattice $R$ of type $ADE$ and the Barnes-Wall lattice of rank 16.
Subperiodic groups and bounded automorphisms of periodic graphs
研究亚周期群在三维空间中的分类，通过开发易计算的不变量识别同构类，并探讨有限阶有界自同构在晶格图中的应用。
Classification of generalized Alexander quandles
本文利用基础群及其自同构，为广义Alexander quandle的同构类提供了新的刻画以推广先前结论，并计算了阶数不超过127的群及其群自同构导出的广义Alexander quandle数量。
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