Stefanos Aivazidis, Maria Loukaki, John Shareshian
We investigate number-theoretic properties of the collection of nilpotent injectors or nilpotent projectors containing certain subgroups of finite soluble (or ${mathcal N}$-constrained) groups.
{"title":"Counting in nilpotent injectors and Carter subgroups","authors":"Stefanos Aivazidis, Maria Loukaki, John Shareshian","doi":"arxiv-2408.15622","DOIUrl":"https://doi.org/arxiv-2408.15622","url":null,"abstract":"We investigate number-theoretic properties of the collection of nilpotent\u0000injectors or nilpotent projectors containing certain subgroups of finite\u0000soluble (or ${mathcal N}$-constrained) groups.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we study the dynamics of translations of an element of a semisimple Lie group $G$ acting on its maximal compact subgroup $K$. First, we extend to our context some classical results in the context of general flag manifolds, showing that when the element is hyperbolic its dynamics is gradient and its fixed points components are given by some suitable right cosets of the centralizer of the element in $K$. Second, we consider the dynamics of a general element and characterizes its recurrent set, its minimal Morse components and their stable and unstable manifolds in terms of the Jordan decomposition of the element, and we show that each minimal Morse component is normally hyperbolic.
{"title":"Dynamics of translations on maximal compact subgroups","authors":"Mauro Patrão, Ricardo Sandoval","doi":"arxiv-2408.16114","DOIUrl":"https://doi.org/arxiv-2408.16114","url":null,"abstract":"In this article, we study the dynamics of translations of an element of a\u0000semisimple Lie group $G$ acting on its maximal compact subgroup $K$. First, we\u0000extend to our context some classical results in the context of general flag\u0000manifolds, showing that when the element is hyperbolic its dynamics is gradient\u0000and its fixed points components are given by some suitable right cosets of the\u0000centralizer of the element in $K$. Second, we consider the dynamics of a\u0000general element and characterizes its recurrent set, its minimal Morse\u0000components and their stable and unstable manifolds in terms of the Jordan\u0000decomposition of the element, and we show that each minimal Morse component is\u0000normally hyperbolic.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206487","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A triangle presentation is a combinatorial datum that encodes the action of a group on a $2$-dimensional triangle complex with prescribed links, which is simply transitive on the vertices. We provide the first infinite family of triangle presentations that give rise to lattices in exotic buildings of type $widetilde{text{A}_2}$ of arbitrarily large order. Our method also gives rise to infinite families of triangle presentations for other link types, such as opposition complexes in Desarguesian projective planes.
{"title":"Infinite families of triangle presentations","authors":"Alex Loué","doi":"arxiv-2408.15763","DOIUrl":"https://doi.org/arxiv-2408.15763","url":null,"abstract":"A triangle presentation is a combinatorial datum that encodes the action of a\u0000group on a $2$-dimensional triangle complex with prescribed links, which is\u0000simply transitive on the vertices. We provide the first infinite family of\u0000triangle presentations that give rise to lattices in exotic buildings of type\u0000$widetilde{text{A}_2}$ of arbitrarily large order. Our method also gives rise\u0000to infinite families of triangle presentations for other link types, such as\u0000opposition complexes in Desarguesian projective planes.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206484","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ali Shehper, Anibal M. Medina-Mardones, Bartłomiej Lewandowski, Angus Gruen, Piotr Kucharski, Sergei Gukov
Using a long-standing conjecture from combinatorial group theory, we explore, from multiple angles, the challenges of finding rare instances carrying disproportionately high rewards. Based on lessons learned in the mathematical context defined by the Andrews-Curtis conjecture, we propose algorithmic improvements that can be relevant in other domains with ultra-sparse reward problems. Although our case study can be formulated as a game, its shortest winning sequences are potentially $10^6$ or $10^9$ times longer than those encountered in chess. In the process of our study, we demonstrate that one of the potential counterexamples due to Akbulut and Kirby, whose status escaped direct mathematical methods for 39 years, is stably AC-trivial.
{"title":"What makes math problems hard for reinforcement learning: a case study","authors":"Ali Shehper, Anibal M. Medina-Mardones, Bartłomiej Lewandowski, Angus Gruen, Piotr Kucharski, Sergei Gukov","doi":"arxiv-2408.15332","DOIUrl":"https://doi.org/arxiv-2408.15332","url":null,"abstract":"Using a long-standing conjecture from combinatorial group theory, we explore,\u0000from multiple angles, the challenges of finding rare instances carrying\u0000disproportionately high rewards. Based on lessons learned in the mathematical\u0000context defined by the Andrews-Curtis conjecture, we propose algorithmic\u0000improvements that can be relevant in other domains with ultra-sparse reward\u0000problems. Although our case study can be formulated as a game, its shortest\u0000winning sequences are potentially $10^6$ or $10^9$ times longer than those\u0000encountered in chess. In the process of our study, we demonstrate that one of\u0000the potential counterexamples due to Akbulut and Kirby, whose status escaped\u0000direct mathematical methods for 39 years, is stably AC-trivial.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206490","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article investigates the properties of order-divisor graphs associated with finite groups. An order-divisor graph of a finite group is an undirected graph in which the set of vertices includes all elements of the group, and two distinct vertices with different orders are adjacent if the order of one vertex divides the order of the other. We prove some beautiful results in order-divisor graphs of finite groups. The primary focus is on examining the girth, degree of vertices, and size of the order-divisor graph. In particular, we provide a comprehensive description of these parameters for the order-divisor graphs of finite cyclic groups and dihedral groups.
{"title":"Some Properties of Order-Divisor Graphs of Finite Groups","authors":"Shafiq ur Rehman, Raheela Tahir, Farhat Noor","doi":"arxiv-2408.14104","DOIUrl":"https://doi.org/arxiv-2408.14104","url":null,"abstract":"This article investigates the properties of order-divisor graphs associated\u0000with finite groups. An order-divisor graph of a finite group is an undirected\u0000graph in which the set of vertices includes all elements of the group, and two\u0000distinct vertices with different orders are adjacent if the order of one vertex\u0000divides the order of the other. We prove some beautiful results in\u0000order-divisor graphs of finite groups. The primary focus is on examining the\u0000girth, degree of vertices, and size of the order-divisor graph. In particular,\u0000we provide a comprehensive description of these parameters for the\u0000order-divisor graphs of finite cyclic groups and dihedral groups.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206493","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose self-similar contracting groups as a platform for cryptographic schemes based on simultaneous conjugacy search problem (SCSP). The class of these groups contains extraordinary examples like Grigorchuk group, which is known to be non-linear, thus making some of existing attacks against SCSP inapplicable. The groups in this class admit a natural normal form based on the notion of a nucleus portrait, that plays a key role in our approach. While for some groups in the class the conjugacy search problem has been studied, there are many groups for which no algorithms solving it are known. Moreover, there are some self-similar groups with undecidable conjugacy problem. We discuss benefits and drawbacks of using these groups in group-based cryptography and provide computational analysis of variants of the length-based attack on SCSP for some groups in the class, including Grigorchuk group, Basilica group, and others.
{"title":"Contracting Self-similar Groups in Group-Based Cryptography","authors":"Delaram Kahrobaei, Arsalan Akram Malik, Dmytro Savchuk","doi":"arxiv-2408.14355","DOIUrl":"https://doi.org/arxiv-2408.14355","url":null,"abstract":"We propose self-similar contracting groups as a platform for cryptographic\u0000schemes based on simultaneous conjugacy search problem (SCSP). The class of\u0000these groups contains extraordinary examples like Grigorchuk group, which is\u0000known to be non-linear, thus making some of existing attacks against SCSP\u0000inapplicable. The groups in this class admit a natural normal form based on the\u0000notion of a nucleus portrait, that plays a key role in our approach. While for\u0000some groups in the class the conjugacy search problem has been studied, there\u0000are many groups for which no algorithms solving it are known. Moreover, there\u0000are some self-similar groups with undecidable conjugacy problem. We discuss\u0000benefits and drawbacks of using these groups in group-based cryptography and\u0000provide computational analysis of variants of the length-based attack on SCSP\u0000for some groups in the class, including Grigorchuk group, Basilica group, and\u0000others.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A base for a permutation group $G$ acting on a set $Omega$ is a sequence $mathcal{B}$ of points of $Omega$ such that the pointwise stabiliser $G_{mathcal{B}}$ is trivial. Denote the minimum size of a base for $G$ by $b(G)$. There is a natural greedy algorithm for constructing a base of relatively small size; denote by $mathcal{G}(G)$ the maximum size of a base it produces. Motivated by a long-standing conjecture of Cameron, we determine $mathcal{G}(G)$ for every almost simple primitive group $G$ with socle a sporadic simple group, showing that $mathcal{G}(G)=b(G)$.
{"title":"Greedy base sizes for sporadic simple groups","authors":"Coen del Valle","doi":"arxiv-2408.14139","DOIUrl":"https://doi.org/arxiv-2408.14139","url":null,"abstract":"A base for a permutation group $G$ acting on a set $Omega$ is a sequence\u0000$mathcal{B}$ of points of $Omega$ such that the pointwise stabiliser\u0000$G_{mathcal{B}}$ is trivial. Denote the minimum size of a base for $G$ by\u0000$b(G)$. There is a natural greedy algorithm for constructing a base of\u0000relatively small size; denote by $mathcal{G}(G)$ the maximum size of a base it\u0000produces. Motivated by a long-standing conjecture of Cameron, we determine\u0000$mathcal{G}(G)$ for every almost simple primitive group $G$ with socle a\u0000sporadic simple group, showing that $mathcal{G}(G)=b(G)$.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206492","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A word in a group is called a test element if any endomorphism fixing it is necessarily an automorphism. In this note, we give a sufficient condition in geometry to construct test elements for monomorphisms of a free group, by using the Whitehead graph and the action of the free group on its Cayley graph.
{"title":"A note on test elements for monomorphisms of free groups","authors":"Dongxiao Zhao, Qiang Zhang","doi":"arxiv-2408.13449","DOIUrl":"https://doi.org/arxiv-2408.13449","url":null,"abstract":"A word in a group is called a test element if any endomorphism fixing it is\u0000necessarily an automorphism. In this note, we give a sufficient condition in\u0000geometry to construct test elements for monomorphisms of a free group, by using\u0000the Whitehead graph and the action of the free group on its Cayley graph.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206494","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a character triple $(G,N,theta)$, which means that $G$ is a finite group with $N vartriangleleft G$ and $thetain{rm Irr}(N)$ is $G$-invariant, we introduce the notion of a $pi$-quasi extension of $theta$ to $G$ where $pi$ is the set of primes dividing the order of the cohomology element $[theta]_{G/N}in H^2(G/N,mathbb{C}^times)$ associated with the character triple, and then establish the uniqueness of such an extension in the normalized case. As an application, we use the $pi$-quasi extension of $theta$ to construct a bijection from the set of $pi$-defect zero characters of $G/N$ onto the set of relative $pi$-defect zero characters of $G$ over $theta$. Our results generalize the related theorems of M. Murai and of G. Navarro.
给定一个特征三元组 $(G,N,theta)$，这意味着 $G$ 是一个有限群，有 $N vartriangleleft G$，并且 $thetain{rm Irr}(N)$是 $G$ 不变的、我们引入了$theta$到$G$的$pi$-准扩展的概念，其中$pi$是除以H^2(G/N,mathbb{C}^times)$中与特征三元组相关的同调元素$[theta]_{G/N}的阶的素集，然后建立了这种扩展在规范化情况下的唯一性。作为应用，我们使用$theta$的$pi$-准扩展来构造一个从$G/N$的$pi$-缺陷零字符集到$G$在$theta$上的相对$pi$-缺陷零字符集的双射。我们的结果概括了 M. Murai 和 G. Navarro 的相关定理。
{"title":"Character triples and relative defect zero characters","authors":"Junwei Zhang, Lizhong Wang, Ping Jin","doi":"arxiv-2408.13436","DOIUrl":"https://doi.org/arxiv-2408.13436","url":null,"abstract":"Given a character triple $(G,N,theta)$, which means that $G$ is a finite\u0000group with $N vartriangleleft G$ and $thetain{rm Irr}(N)$ is $G$-invariant,\u0000we introduce the notion of a $pi$-quasi extension of $theta$ to $G$ where\u0000$pi$ is the set of primes dividing the order of the cohomology element\u0000$[theta]_{G/N}in H^2(G/N,mathbb{C}^times)$ associated with the character\u0000triple, and then establish the uniqueness of such an extension in the\u0000normalized case. As an application, we use the $pi$-quasi extension of\u0000$theta$ to construct a bijection from the set of $pi$-defect zero characters\u0000of $G/N$ onto the set of relative $pi$-defect zero characters of $G$ over\u0000$theta$. Our results generalize the related theorems of M. Murai and of G.\u0000Navarro.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Daniele D'Angeli, Francesco Matucci, Davide Perego, Emanuele Rodaro
We define a new class of groups arising from context-free inverse graphs. We provide closure properties, prove that their co-word problems are context-free, study the torsion elements, and realize them as subgroups of the asynchronous rational group. Furthermore, we use a generalized version of the free product of graphs and prove that such a product is context-free inverse closed. We also exhibit an example of a group in our class that is not residually finite and one that is not poly-context-free. These properties make them interesting candidates to disprove both the Lehnert conjecture (which characterizes co-context-free groups as all subgroups of Thompson's group V) and the Brough conjecture (which characterizes finitely generated poly-context-free groups as virtual finitely generated subgroups of direct products of free groups).
我们定义了一类由无上下文逆图产生的新群。我们提供了封闭性质，证明了它们的共词问题是无上下文的，研究了扭转元素，并将它们实现为异步有理群的子群。此外，我们还使用了图的自由积的广义版本，并证明这种积是无上下文逆封闭的。我们还举例说明了我们这一类中的一个群不是残差有限群，也不是无多上下文群。这些性质使它们成为推翻莱纳特猜想（该猜想将无上下文群表征为汤普森群 V 的所有子群）和布拉夫猜想（该猜想将有限生成的无多上下文群表征为自由群直接积的虚拟有限生成子群）的有趣候选者。
{"title":"Context-free graphs and their transition groups","authors":"Daniele D'Angeli, Francesco Matucci, Davide Perego, Emanuele Rodaro","doi":"arxiv-2408.13070","DOIUrl":"https://doi.org/arxiv-2408.13070","url":null,"abstract":"We define a new class of groups arising from context-free inverse graphs. We\u0000provide closure properties, prove that their co-word problems are context-free,\u0000study the torsion elements, and realize them as subgroups of the asynchronous\u0000rational group. Furthermore, we use a generalized version of the free product\u0000of graphs and prove that such a product is context-free inverse closed. We also\u0000exhibit an example of a group in our class that is not residually finite and\u0000one that is not poly-context-free. These properties make them interesting\u0000candidates to disprove both the Lehnert conjecture (which characterizes\u0000co-context-free groups as all subgroups of Thompson's group V) and the Brough\u0000conjecture (which characterizes finitely generated poly-context-free groups as\u0000virtual finitely generated subgroups of direct products of free groups).","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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