Leyli Jafari Taghvasani, Soran Marzang
In this paper we show that a finite nonabelian characteristically simple group G satisfying n = |π(G)|+2 if and only if G is isomorphic to A5, where n is the number of isomorphism classes of derived subgroups of G and π(G) is the set of prime divisors of the group G.
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For primes $p,e>2$ there are at least $p^{e-3}/e$ groups of order $p^{2e+2}$ that have equal multisets of isomorphism types of proper subgroups and proper quotient groups, isomorphic character tables, and power maps. This obstructs recent speculation concerning a path towards efficient isomorphism tests for general finite groups. These groups have a special purpose polylogarithmic-time isomorphism test.
A subgroup of a Kac-Moody group is called bounded if it is contained in the intersection of two finite type parabolic subgroups of opposite signs. In this paper, we study the isomorphisms between Kac-Moody groups over arbitrary fields of cardinality at least 4, which preserve the set of bounded subgroups. We show that such an isomorphism between two such Kac-Moody groups induces an isomorphism between the respective twin root data of these groups. As a consequence, we obtain the solution of the isomorphism problem for Kac-Moody groups over finite fields of cardinality at least 4.
In this paper we discuss the isomorphism types of parabolic subgroups in Kac-Moody groups. The results have applications in the study of topology of Kac-Moody groups and their classifying spaces.
In this paper we prove that there does not exist a subgroup $H$ of a finite group $G$ such that the number of isomorphism classes of normalized right transversals of $H$ in $G$ is four.
In this paper, we prove that there does not exist a subgroup H of a finite group G such that the number of isomorphism classes of right transversals of H in G is two.
On the Modular Isomorphism Problem for 2-generated groups with cyclic derived subgroup
本文研究具有循环换位子群的2生成 p-群的模同构问题，证明特定商群同构类型由模群代数确定，并对阶至多 p^{11} 的群给出肯定答案。
The concept of configuration was first introduced by Rosenblatt and Willis to give a condition for amenability of groups. We show that if $G_1$ and $G_2$ have the same configuration sets and $H_1$ is a normal subgroup of $G_1$ with abelian quotient, then there is a normal subgroup $H_2$ of $G_2$ such that $\frac{G_1}{H_1}\cong\frac{G_2}{H_2}.$ Also configuration of FC-groups and isomorphism is studied.
In this paper the number of isomorphism classes of the right transversals of the symmetric group $Σ_{n-1}$ in the symmetric group $Σ_n$ is calculated. This result together with the techniques employed along the paper may be used to go forward in the study and classification of the subgroups with a certain number of isomorphism classes of transversals in the group.
The Subgroup Isomorphism Problem for Integral Group Rings asks for which finite groups U it is true that if U is isomorphic to a subgroup of V(ZG), the group of normalized units of the integral group ring of the finite group G, it must be isomorphic to a subgroup of G. The smallest groups known not to satisfy this property are the counterexamples to the Isomorphism Problem constructed by M. Hertweck. However the only groups known to satisfy it are cyclic groups of prime power order and elementary-abelian p-groups of rank 2. We prove the Subgroup Isomorphism Problem for C_4 x C_2. Moreover we p…
We consider the isomorphism problem for almost split Kac--Moody groups, which have been constructed by Rémy via Galois descent from split Kac-Moody groups as defined by Tits. We show that under certain technical assumptions, any isomorphism between two such groups must preserve the canonical subgroup structure, i.e. the twin root datum associated to these groups, which generalizes results of Caprace in the split case. An important technical tool we use is the existence of maximal split subgroups inside almost split Kac-Moody groups, which generalizes the corresponding result of Borel-Tits for …
Polynomial-time isomorphism test for solvable groups with abelian Sylow subgroups
本文针对具有阿贝尔Sylow子群的群（A-群）提出多项式时间同构测试算法，推广了Babai和Qiao关于可解群的结果，并利用了其计算方法和A-群的结构性质。
Let V_* be the normalized unitary subgroup of the modular group algebra FG of a finite p-group G over a finite field F with the classical involution *. We investigate the isomorphism problem for the group V_*, that asks when the group V_* is determined by its group algebra FG. We confirm it for classes of finite abelian p-groups, 2-groups of maximal class and non-abelian 2-groups of order at most 16.
Random generation of subgroups of the modular group with a fixed isomorphism type
该研究提出了高效计数与均匀随机生成模群$\textsf{PSL}(2,\u2124)$中给定同构类型有限生成子群的方法，借助关联子群轮廓图的自然映射实现，该映射本身也具有独立研究价值。
We continue the classification of isomorphism classes of k-involutions of exceptional algebraic groups. In this paper we classify k-involutions for split groups of type F4 over certain fields, and their fixed point groups.The classification of k-involutions is equivalent to the classification of symmetric k-varieties.
Finite $p$-groups with a unique $\mathcal{A}_2$-subgroup are classified up to isomorphism. A problem proposed by Berkovich and Janko is solved.
Consider the mapping class group $\Mod_{g,p}$ of a surface $Σ_{g,p}$ of genus $g$ with $p$ punctures, and a finite collection $\{f_1,...,f_k\}$ of mapping classes, each of which is either a Dehn twist about a simple closed curve or a pseudo-Anosov homeomorphism supported on a connected subsurface. In this paper we prove that for all sufficiently large $N$, the mapping classes $\{f_1^N,...,f_k^N\}$ generate a right-angled Artin group. The right-angled Artin group which they generate can be determined from the combinatorial topology of the mapping classes themselves. When $\{f_1,...,f_k\}$ are a…
Let $p$ be a prime number, $G$ be a finite $p$-group and $K$ be a field of characteristic $p$. The Modular Isomorphism Problem (MIP) asks whether the group algebra $KG$ determines the group $G$. Dealing with MIP, we investigated a question whether the nilpotency class of a finite $p$-group is determined by its modular group algebra over the field of $p$ elements. We give a positive answer to this question provided one of the following conditions holds: (i) $\exp G=p$; (ii) $\cl(G)=2$; (iii) $G'$ is cyclic; (iv) $G$ is a group of maximal class and contains an abelian subgroup of index $p$.
We prove that an isomorphism between saturated fusion systems over the same finite p-group is detected on the elementary abelian subgroups of the hyperfocal subgroup if p is odd, and on the abelian subgroups of the hyperfocal subgroup of exponent at most 4 if p = 2. For odd p, this has implications for mod p group cohomology.
Let $G$ be a finite group and $H$ a subgroup of $G$. Each left transversal (with identity) of $H$ in $G$ has a left loop (left quasigroup with identity) structure induced by the binary operation of $G$. We say two left transversals are isomorphic if they are isomorphic with respect to the induced left loop structures. In this paper, we develop a method to calculate the number of isomorphism classes of transversals of $H$ in $G$. Also with the help of this we calculate the number of non-isomorphic left loops of a given order.
Let G be an 1-connected simple Lie group, and let x\inG be a group element. We determine the isomorphism type of the centralizer C_{x} in term of a minimal geodesic joinning the group unit e\inG to x. This result is applied to classify the isomorphism types of maximal subgroups of maximal rank of G, and the isomorphism types of parabolic subgroups of G.
In this paper we describe a family of isomorphism invariants of a finitely generated Coxeter group W. Each of these invariants is the isomorphism type of a quotient group W/N of W by a characteristic subgroup N. The virtue of these invariants is that W/N is also a Coxeter group. For some of these invariants, the isomorphism problem of W/N is solved and so we obtain isomorphism invariants that can be effectively used to distinguish isomorphism types of finitely generated Coxeter groups.
In this paper we introduce and study the poset of equivalence classes of subgroups of a finite group $G$, induced by the isomorphism relation. This contains the well-known lattice of solitary subgroups of $G$. We prove that in several particular cases it determines the structure of $G$.
A Characterization of Group Through Isomorphism Classes of Transversals
本文证明：若群 G 中有限指数子群 H 的右横截线同构类数为 5，则 H 在 G 中的指数为 6，且 G 在 H 右陪集上的置换表示同构于四元交错群。
Let $S_\infty$ denote the topological group of permutations of the natural numbers. We study the complexity of the isomorphism relation on classes of closed subgroups $S_\infty$ in the setting of Borel reducibility between equivalence relations on Polish spaces. Given a closed subgroup $G$ of $S_\infty$, the coarse group $\mathcal M(G)$ is the structure with domain the cosets of open subgroups of $G$, and a ternary relation $AB \sqsubseteq C$. If $G$ has only countably many open subgroups, then $\mathcal M(G)$ is a countable structure. Coarse groups form our main tool in studying such closed s…
For any prime $p$ and $S$ a $p$-group isomorphic to a Sylow $p$-subgroup of $\mathrm{G}_2(p^n)$ or $\mathrm{PSU}_4(p^n)$ with $n\in\mathbb{N}$, we determine all saturated fusion systems supported on $S$ up to isomorphism.
We consider random subgroups of Thompson's group $F$ with respect to two natural stratifications of the set of all $k$ generator subgroups. We find that the isomorphism classes of subgroups which occur with positive density are not the same for the two stratifications. We give the first known examples of {\em persistent} subgroups, whose isomorphism classes occur with positive density within the set of $k$-generator subgroups, for all sufficiently large $k$. Additionally, Thompson's group provides the first example of a group without a generic isomorphism class of subgroup. Elements of $F$ are…
This paper describes how to use subgroups to parameterize unipotent classes in the classical algebraic group in characteristic 2. These results can be viewed as an extension of the Bala-Carter Theorem, and give a convenient way to compare unipotent classes in a group $G$ with unipotent classes of a subgroup $X$ where $G$ is exceptional and $X$ is a Levi subgroup of classical type.
We prove a formula connecting the number of unipotent conjugacy classes in a maximal parabolic subgroup of a finite general linear group with the numbers of unipotent conjugacy classes in various parabolic subgroups in smaller dimensions. We generalise this formula and deduce a number of corollaries; in particular, we express the number of conjugacy classes of unitriangular matrices over a finite field in terms of the numbers of unipotent conjugacy classes in maximal parabolic subgroups over the same field. We show how the numbers of unipotent conjugacy classes in parabolic subgroups of small …
The N-prime graph and the Subgroup Isomorphism Problem
本文引入群的N-素图（经典Gruenberg-Kegel图的细化），证明有限可解群与其整群环增广1单位群N-素图相同，将问题约化到几乎单群并验证特定几乎单群情形，还为可解群子群同构问题提供结果。
Tournaments are graphs obtained by assigning a direction for every edge in an undirected complete graph. We give a formula for the number of isomorphism classes of vertex-transitive tournaments with prime order. For that, we introduce Cayley tournaments, which are special Cayley digraphs, and show that there is a one to one identification between the isomorphism classes of vertex-transitive tournaments with prime order and the isomorphism classes of Cayley tournaments with prime order.
We construct a hyperbolic group with a finitely presented subgroup, which has infinitely many conjugacy classes of finite-order elements. We also use a version of Morse theory with high dimensional horizontal cells and use handle cancellation arguments to produce other examples of subgroups of CAT(0) groups with infinitely many conjugacy classes of finite-order elements.
In this note, we determine the finite groups whose poset of conjugacy classes of subgroups has breaking points. This leads to a new characterization of the generalized quaternion $2$-groups. A generalization of this property is also studied.
Finding an Isomorphism between the Riordan Group and a Subgroup of the Double Riordan Group
本文研究了 Riordan 群与 Double Riordan 群子群之间的同构关系，通过构造性方法回答了 Davenport 等人提出的开放性问题，证实了该同构的存在性。
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.
Landau's Theorem on conjugacy classes for normal subgroups
本文推广Landau定理，证明对任意正整数n和s，具有指数为n且含s个非中心G-共轭类的正规子群的有限群在同构意义下仅有有限多个，并给出阶的上界及GAP分类结果。
In this paper we consider the {\em conjugacy stability} property of subgroups and provide effective procedures to solve the problem in several classes of groups. In particular, we start with free groups, that is, we give an effective procedure to find out if a finitely generated subgroup of a free group is conjugacy stable. Then we further generalize this result to quasi-convex subgroups of torsion-free hyperbolic groups and finitely generated subgroups of limit groups.
The Number of Isomorphism Classes of Beauville Surfaces with Beauville $p$-Group
本文研究了Beauville曲面的同构类数目，针对非阿贝尔混合循环p-群或nilpotency类为2的p-群扩展了先前关于阿贝尔p-群的结果。
We investigate the relationship between various isomorphism invariants for finite groups. Specifically, we use the Weisfeiler-Leman dimension (WL) to characterize, compare and quantify the effectiveness and complexity of invariants for group isomorphism. It turns out that a surprising number of invariants and characteristic subgroups that are classic to group theory can be detected and identified by a low dimensional Weisfeiler-Leman algorithm. These include the center, the inner automorphism group, the commutator subgroup and the derived series, the abelian radical, the solvable radical, the …
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…
P. Hall constructed a universal countable locally finite group U, determined up to isomorphism by two properties: every finite group C is a subgroup of U, and every embedding of C into U is conjugate in U. Every countable locally finite group is a subgroup of U. We prove that U is a subgroup of the abstract commensurator of a finite-rank nonabelian free group.
We construct a CAT(0) group containing a finitely presented subgroup with infinitely many conjugacy classes of finite-order elements. Unlike previous examples (which were based on right-angled Artin groups) our ambient CAT(0) group does not contain any rank 3 free abelian subgroups. We also construct examples of groups of type F_n inside mapping class groups, Aut(F), and Out(F) which have infinitely many conjugacy classes of finite-order elements.
We show that locally solvable subgroups of PLo(I) are countable. Then for each countable ordered set, we construct a locally solvable subgroup of Thompson's Group F. We develop machinery for understanding embeddings from solvable subgroups into solvable subgroups. Finally, we apply this machinery to show the ordered sets used in our construction are invariant under isomorphisms between the groups constructed. Therefore, we effectively distinguish the groups and provide uncountably many non-isomorphic locally solvable, hence elementary amenable, subgroups of Thompson's Group F.
In this paper we classify the isomorphism classes of four dimensional nilpotent associative algebras over a field F, studying regular subgroups of the affine group AGL_4(F). In particular we provide explicit representatives for such classes when F is a finite field, the real field R or an algebraically closed field.
We show that in the category of groups, every singly-generated class which is closed under isomorphisms, direct limits and extensions is also singly-generated under isomorphisms and direct limits, and in particular is co-reflective. We also establish several new relations between singly-generated closed classes.
In this paper, we set $η(G)$ to be the number of conjugacy classes of maximal cyclic subgroups of $G$. We prove that if $G$ is a $p$-group of order $p^n$ and nilpotence class $l$, then $η(G)$ is bounded below by a linear function in $n/l$.
We consider the question: When do two finite abelian groups have isomorphic lattices of characteristic subgroups? An explicit description of the characteristic subgroups of such groups enables us to give a complete answer to this question in the case where at least one of the groups has odd order. An "exceptional" isomorphism, which occurs between the lattice of characteristic subgroups of $Z_p\times Z_{p^2}\times Z_{p^4}$ and $Z_{p^2} \times Z_{p^5}$, for any prime $p$, is noteworthy.
In this paper we find the number of conjugate $π$-Hall subgroups in all finite almost simple groups. We also complete the classification of $π$-Hall subgroups in finite simple groups and correct some mistakes from our previous paper.
Given a finite group $ G $, we study certain regular subgroups of the group of permutations of $ G $, which occur in the classification theories of two types of algebraic objects: skew left braces with multiplicative group isomorphic to $ G $ and Hopf-Galois structures admitted by a Galois extension of fields with Galois group isomorphic to $ G $. We study the questions of when two such subgroups yield isomorphic skew left braces or Hopf-Galois structures involving isomorphic Hopf algebras. In particular, we show that in some cases the isomorphism class of the Hopf algebra giving a Hopf-Galois…
For $S$ a Sylow $p$-subgroup of the group $\mathrm{G}_2(p)$ for $p$ odd, up to isomorphism of fusion systems, we determine all saturated fusion systems $\mathcal{F}$ on $S$ with $O_p(\mathcal{F})=1$. For $p \ne 7$, all such fusion systems are realized by finite groups whereas for $p=7$ there are $29$ saturated fusion systems of which $27$ are exotic.
Each quiver appearing in a seed of a cluster algebra determines a corresponding group, which we call a cluster group, which is defined via a presentation. Grant and Marsh showed that, for quivers appearing in seeds of cluster algebras of finite type, the associated cluster groups are isomorphic to finite reflection groups. As for finite Coxeter groups, we can consider parabolic subgroups of cluster groups. We prove that, in the type $A_{n}$ case, there exists an isomorphism between the lattice of subsets of the defining generators of the cluster group and the lattice of its parabolic subgroups…
In this note, we prove that if G is a countable group that contains a nonabelian free subgroup then every pair of nontrivial Bernoulli shifts over G are weakly isomorphic.
We extend the notion of induced conjugacy classes in reductive groups, introduced by Lusztig and Spaltenstein for unipotent classes, to arbitrary classes. We study properties of equivariant fibrations of prehomogeneous affine spaces, especially the existence of relative invariants. We also detect prehomogeneous affine spaces as subquotients of canonical parabolic subgroups attached to elements of reductive groups in the sense of Jacobson-Morozov. These results are prerequisites for making the geometric expansion of the Arthur-Selberg trace formula more explicit.
Let $G$ be a finite group and $M(G)$ be the subgroup of $G$ generated by all non-central elements of $G$ that lie in the conjugacy classes of the smallest size. Recently several results have been proved regarding the nilpotency class of $M(G)$ and $F(M(G))$, where $F(M(G))$ denotes the Fitting subgroup of $M(G)$. We prove some conditional results regarding the nilpotency class of $M(G)$.
Let ${\frak F}$ be a class of group and $G$ a finite group. Then a set $Σ$ of subgroups of $G$ is called a \emph{$G$-covering subgroup system} for the class ${\frak F}$ if $G\in {\frak F}$ whenever $Σ\subseteq {\frak F}$. We prove that: {\sl If a set of subgroups $Σ$ of $G$ contains at least one supplement to each maximal subgroup of every Sylow subgroup of $G$, then $Σ$ is a $G$-covering subgroup system for the classes of all $σ$-soluble and all $σ$-nilpotent groups, and for the class of all $σ$-soluble $PσT$-groups.} This result gives positive answers to questions 19.87 and 19.88 from the Ko…
In this paper, we set $η(G)$ to be the number of conjugacy classes of maximal cyclic subgroups of a finite group $G$. We compute $η(G)$ for all metacyclic $p$-groups. We show that if $G$ is a metacyclic $p$-group of order $p^n$ that is not dihedral, generalized quaternion, or semi-dihedral, then $η(G) \ge n-2$, and we determine when equality holds.
The McCool group has families of subgroups such as the ordinary pure braid group, the upper triangular McCool group and the partial inner automorphism group. The generalized Andreadakis conjecture holds for the ordinary pure braid group and the upper triangular McCool group. In this paper, we prove a similar result for the partial inner automorphism group and also establish an isomorphism between this group and the inner automorphism group of the upper triangular McCool group.
Triangles in the graph of conjugacy classes of normal subgroups
研究有限群正规子群共轭类图无三角形或恰有一个三角形时的结构，给出了相应的刻画。
In the 1940's Graham Higman initiated the study of finite subgroups of the unit group of an integral group ring. Since then many fascinating aspects of this structure have been discovered. Major questions such as the Isomorphism Problem and the Zassenhaus Conjectures have been settled, leading to many new challenging problems. In this survey we review classical and recent results, sketch methods and list questions relevant for the state of the art.
In this paper, we classify conjugacy classes of centralizers of irreducible subgroups in $PSL(n,\mathbb{C})$ using alternate modules a.k.a. finite abelian groups with an alternate bilinear form. When $n$ is squarefree, we prove that these conjugacy classes are classified by their isomorphism classes. More generally, we define a finite graph related to this classification whose combinatorial properties are expected to help us describe the stratification of the singular (orbifold) locus in some character varieties.
Isomorphisms and commensurability of surface Houghton groups
本文对曲面Houghton群及其纯子群在同构、可公度性和拟等距意义下进行了完整分类。
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