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The synthetic presentation of the main research directions in groupoid theory 类群理论主要研究方向的综合介绍
arXiv - MATH - Group Theory Pub Date : 2024-08-01 DOI: arxiv-2408.00562
Gheorghe Ivan
{"title":"The synthetic presentation of the main research directions in groupoid theory","authors":"Gheorghe Ivan","doi":"arxiv-2408.00562","DOIUrl":"https://doi.org/arxiv-2408.00562","url":null,"abstract":"The purpose of this paper is to present a systematic exposition of the main\u0000results obtained in the studies carried out in groupoid theory. Key words and\u0000phrases: groupoid, topological groupoid, Lie groupoid, group-groupoid, vector\u0000space-groupoid.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141884921","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}
引用次数: 0
A Zero-Knowledge Proof of Knowledge for Subgroup Distance Problem 子群距离问题的零知识证明
arXiv - MATH - Group Theory Pub Date : 2024-08-01 DOI: arxiv-2408.00395
Cansu Betin Onur
{"title":"A Zero-Knowledge Proof of Knowledge for Subgroup Distance Problem","authors":"Cansu Betin Onur","doi":"arxiv-2408.00395","DOIUrl":"https://doi.org/arxiv-2408.00395","url":null,"abstract":"In this study, we introduce a novel zero-knowledge identification scheme\u0000based on the hardness of the subgroup distance problem in the Hamming metric.\u0000The proposed protocol, named Subgroup Distance Zero Knowledge Proof (SDZKP),\u0000employs a cryptographically secure pseudorandom number generator to mask\u0000secrets and utilizes a Stern-type algorithm to ensure robust security\u0000properties.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141884891","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}
引用次数: 0
Embedding Partial HNN Extensions In Ascending HNN Extensions 将部分 HNN 扩展嵌入升序 HNN 扩展中
arXiv - MATH - Group Theory Pub Date : 2024-08-01 DOI: arxiv-2408.00453
Hip Kuen Chong, Daniel T. Wise
{"title":"Embedding Partial HNN Extensions In Ascending HNN Extensions","authors":"Hip Kuen Chong, Daniel T. Wise","doi":"arxiv-2408.00453","DOIUrl":"https://doi.org/arxiv-2408.00453","url":null,"abstract":"We show that any partial ascending HNN extension of a free group embeds in an\u0000actual ascending HNN extension of a free group. Moreover, we can ensure that it\u0000embeds as the parabolic subgroup of a relatively hyperbolic group.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141884865","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}
引用次数: 0
On the Nilpotent Graph of a finite Group 论有限群的无势图
arXiv - MATH - Group Theory Pub Date : 2024-08-01 DOI: arxiv-2408.00910
Jaime Torres, Ismael Gutierrez, E. J. Garcia-Claro
{"title":"On the Nilpotent Graph of a finite Group","authors":"Jaime Torres, Ismael Gutierrez, E. J. Garcia-Claro","doi":"arxiv-2408.00910","DOIUrl":"https://doi.org/arxiv-2408.00910","url":null,"abstract":"If G is a non-nilpotent group and nil(G) = {g in G : <g, h> is nilpotent for\u0000all hin G}, the nilpotent graph of G is the graph with set of vertices\u0000G-nil(G) in which two distinct vertices are related if they generate a\u0000nilpotent subgroup of G. Several properties of the nilpotent graph associated\u0000with a finite non-nilpotent group G are studied in this work. Lower bounds for\u0000the clique number and the number of connected components of the nilpotent graph\u0000of G are presented in terms of the size of its Fitting subgroup and the number\u0000of its strongly self-centralizing subgroups, respectively. It is proved the\u0000nilpotent graph of the symmetric group of degree n is disconnected if and only\u0000if n or n-1 is a prime number, and no finite non-nilpotent group has a\u0000self-complementary nilpotent graph. For the dihedral group Dn, it is determined\u0000the number of connected components of its nilpotent graph is one more than n\u0000when n is odd; or one more than the 2'-part of n when n is even. In addition, a\u0000formula for the number of connected components of the nilpotent graph of\u0000PSL(2,q), where q is a prime power, is provided. Finally, necessary and\u0000sufficient conditions for specific subsets of a group, containing connected\u0000components of its nilpotent graph, to contain one of its Sylow p-subgroups are\u0000studied; and it is shown the nilpotent graph of a finite non-nilpotent group G\u0000with nil(G) of even order is non-Eulerian.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141968899","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}
引用次数: 0
Counting pseudo-Anosovs as weakly contracting isometries 将伪阿诺索夫算作弱收缩等轴线
arXiv - MATH - Group Theory Pub Date : 2024-08-01 DOI: arxiv-2408.00603
Inhyeok Choi
{"title":"Counting pseudo-Anosovs as weakly contracting isometries","authors":"Inhyeok Choi","doi":"arxiv-2408.00603","DOIUrl":"https://doi.org/arxiv-2408.00603","url":null,"abstract":"Let $S$ be a finite generating set of the mapping class group of a\u0000finite-type hyperbolic surface. We show that mapping classes supported on a\u0000fixed subsurface are not generic in the word metric with respect to $S$. We\u0000also show that pseudo-Anosov mapping classes are generic in the word metric\u0000with respect to $S'$, where $S'$ is $S$ plus a single mapping class. We also\u0000observe the analogous results for well-behaved hierarchically hyperbolic groups\u0000and groups quasi-isometric to them. This gives a version of quasi-isometry\u0000invariant theory of counting group elements in groups.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141884867","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}
引用次数: 0
A general dynamical theory of Schwarz reflections, B-involutions, and algebraic correspondences 施瓦茨反射、B-旋转和代数对应的一般动力学理论
arXiv - MATH - Group Theory Pub Date : 2024-08-01 DOI: arxiv-2408.00204
Yusheng Luo, Mikhail Lyubich, Sabyasachi Mukherjee
{"title":"A general dynamical theory of Schwarz reflections, B-involutions, and algebraic correspondences","authors":"Yusheng Luo, Mikhail Lyubich, Sabyasachi Mukherjee","doi":"arxiv-2408.00204","DOIUrl":"https://doi.org/arxiv-2408.00204","url":null,"abstract":"In this paper, we study matings of (anti-)polynomials and Fuchsian,\u0000reflection groups as Schwarz reflections, B-involutions or as\u0000(anti-)holomorphic correspondences, as well as their parameter spaces. We prove\u0000the existence of matings of generic (anti-)polynomials, such as periodically\u0000repelling, or geometrically finite (anti-)polynomials, with circle maps arising\u0000from the corresponding groups. These matings emerge naturally as degenerate\u0000(anti-)polynomial-like maps, and we show that the corresponding parameter space\u0000slices for such matings bear strong resemblance with parameter spaces of\u0000polynomial maps. Furthermore, we provide algebraic descriptions for these\u0000matings, and construct algebraic correspondences that combine generic\u0000(anti-)polynomials and genus zero orbifolds in a common dynamical plane,\u0000providing a new concrete evidence to Fatou's vision of a unified theory of\u0000groups and maps.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141884868","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}
引用次数: 0
Essential Dimension of Small Finite Groups 小型有限群的基本维度
arXiv - MATH - Group Theory Pub Date : 2024-07-31 DOI: arxiv-2407.21449
Dilpreet Kaur, Zinovy Reichstein
{"title":"Essential Dimension of Small Finite Groups","authors":"Dilpreet Kaur, Zinovy Reichstein","doi":"arxiv-2407.21449","DOIUrl":"https://doi.org/arxiv-2407.21449","url":null,"abstract":"We compute the essential dimension of finite groups of order $leqslant 63$.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872688","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}
引用次数: 0
Finite groups whose real irreducible representations have unique dimensions 实不可还原表示具有唯一维数的有限群
arXiv - MATH - Group Theory Pub Date : 2024-07-30 DOI: arxiv-2407.20854
Thomas Breuer, Frank Calegari, Silvio Dolfi, Gabriel Navarro, Pham Huu Tiep
{"title":"Finite groups whose real irreducible representations have unique dimensions","authors":"Thomas Breuer, Frank Calegari, Silvio Dolfi, Gabriel Navarro, Pham Huu Tiep","doi":"arxiv-2407.20854","DOIUrl":"https://doi.org/arxiv-2407.20854","url":null,"abstract":"We determine the finite groups whose real irreducible representations have\u0000different degrees.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141862895","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}
引用次数: 0
Burau representation of $B_4$ and quantization of the rational projective plane B_4$ 的布劳表示和有理投影面的量子化
arXiv - MATH - Group Theory Pub Date : 2024-07-30 DOI: arxiv-2407.20645
Perrine Jouteur
{"title":"Burau representation of $B_4$ and quantization of the rational projective plane","authors":"Perrine Jouteur","doi":"arxiv-2407.20645","DOIUrl":"https://doi.org/arxiv-2407.20645","url":null,"abstract":"The braid group $B_4$ naturally acts on the rational projective plane\u0000$mathbb{P}^2(mathbb{Q})$, this action corresponds to the classical integral\u0000reduced Burau representation of $B_4$. The first result of this paper is a\u0000classification of the orbits of this action. The Burau representation then\u0000defines an action of $B_4$ on $mathbb{P}^2(mathbb{Z}(q))$, where $q$ is a\u0000formal parameter and $mathbb{Z}(q)$ is the field of rational functions in $q$\u0000with integer coefficients. We study orbits of the $B_4$-action on\u0000$mathbb{P}^2(mathbb{Z}(q))$, and show existence of embeddings of the\u0000$q$-deformed projective line $mathbb{P}^1(mathbb{Z}(q))$ that precisely\u0000correspond to the notion of $q$-rationals due to Morier-Genoud and Ovsienko.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872690","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}
引用次数: 0
Cardinalities of irredundant bases of finite primitive groups 有限基元群非冗余基的枢轴性
arXiv - MATH - Group Theory Pub Date : 2024-07-30 DOI: arxiv-2407.20849
Fabio Mastrogiacomo
{"title":"Cardinalities of irredundant bases of finite primitive groups","authors":"Fabio Mastrogiacomo","doi":"arxiv-2407.20849","DOIUrl":"https://doi.org/arxiv-2407.20849","url":null,"abstract":"Let $G$ be a finite permutation group acting on a set $Omega$. An ordered\u0000sequence $(omega_1,ldots,omega_ell)$ of elements of $Omega$ is an\u0000irredundant base for $G$ if the pointwise stabilizer of the sequence is trivial\u0000and no point is fixed by the stabilizer of its predecessors. We show that any\u0000interval of natural numbers can be realized as the set of cardinalities of\u0000irredundant bases for some finite primitive group.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141862896","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}
引用次数: 0
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