The purpose of this paper is to present a systematic exposition of the main results obtained in the studies carried out in groupoid theory. Key words and phrases: groupoid, topological groupoid, Lie groupoid, group-groupoid, vector space-groupoid.
{"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":"75 1","pages":""},"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}
In this study, we introduce a novel zero-knowledge identification scheme based on the hardness of the subgroup distance problem in the Hamming metric. The proposed protocol, named Subgroup Distance Zero Knowledge Proof (SDZKP), employs a cryptographically secure pseudorandom number generator to mask secrets and utilizes a Stern-type algorithm to ensure robust security properties.
{"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":"8 1","pages":""},"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}
We show that any partial ascending HNN extension of a free group embeds in an actual ascending HNN extension of a free group. Moreover, we can ensure that it embeds as the parabolic subgroup of a relatively hyperbolic group.
{"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":"217 1","pages":""},"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}
Jaime Torres, Ismael Gutierrez, E. J. Garcia-Claro
If G is a non-nilpotent group and nil(G) = {g in G : is nilpotent for all hin G}, the nilpotent graph of G is the graph with set of vertices G-nil(G) in which two distinct vertices are related if they generate a nilpotent subgroup of G. Several properties of the nilpotent graph associated with a finite non-nilpotent group G are studied in this work. Lower bounds for the clique number and the number of connected components of the nilpotent graph of G are presented in terms of the size of its Fitting subgroup and the number of its strongly self-centralizing subgroups, respectively. It is proved the nilpotent graph of the symmetric group of degree n is disconnected if and only if n or n-1 is a prime number, and no finite non-nilpotent group has a self-complementary nilpotent graph. For the dihedral group Dn, it is determined the number of connected components of its nilpotent graph is one more than n when n is odd; or one more than the 2'-part of n when n is even. In addition, a formula for the number of connected components of the nilpotent graph of PSL(2,q), where q is a prime power, is provided. Finally, necessary and sufficient conditions for specific subsets of a group, containing connected components of its nilpotent graph, to contain one of its Sylow p-subgroups are studied; and it is shown the nilpotent graph of a finite non-nilpotent group G with nil(G) of even order is non-Eulerian.
如果 G 是一个非零potent 群，并且 nil(G) = {g in G : is nilpotent forall hin G}，那么 G 的零potent 图就是具有顶点集 G-nil(G) 的图，其中两个不同的顶点如果生成 G 的一个零potent 子群，那么它们就是相关的。根据 G 的 Fitting 子群的大小和强自中心化子群的数量，分别给出了 G 的无穷图的小群数和连通成分数的下限。证明了当且仅当 n 或 n-1 是素数时，阶数为 n 的对称群的无穷图是断开的，并且没有有限非无穷群具有自补无穷图。对于二面体群 Dn，可以确定当 n 为奇数时，其无穷图的连通成分数比 n 多一个；当 n 为偶数时，其无穷图的连通成分数比 n 的 2'- 部分多一个。此外，还提供了 PSL(2,q)（其中 q 是质数幂）无勢图的连通部分数公式。最后，研究了一个群的特定子集（包含其无穷图的连通成分）包含其一个 Sylow p 子群的必要条件和充分条件；并证明了具有偶数阶 nil(G) 的有限非无穷群 G 的无穷图是非欧拉图。
{"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":"11 1","pages":""},"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}
Let $S$ be a finite generating set of the mapping class group of a finite-type hyperbolic surface. We show that mapping classes supported on a fixed subsurface are not generic in the word metric with respect to $S$. We also show that pseudo-Anosov mapping classes are generic in the word metric with respect to $S'$, where $S'$ is $S$ plus a single mapping class. We also observe the analogous results for well-behaved hierarchically hyperbolic groups and groups quasi-isometric to them. This gives a version of quasi-isometry invariant theory of counting group elements in groups.
{"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":"48 1","pages":""},"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}
In this paper, we study matings of (anti-)polynomials and Fuchsian, reflection groups as Schwarz reflections, B-involutions or as (anti-)holomorphic correspondences, as well as their parameter spaces. We prove the existence of matings of generic (anti-)polynomials, such as periodically repelling, or geometrically finite (anti-)polynomials, with circle maps arising from the corresponding groups. These matings emerge naturally as degenerate (anti-)polynomial-like maps, and we show that the corresponding parameter space slices for such matings bear strong resemblance with parameter spaces of polynomial maps. Furthermore, we provide algebraic descriptions for these matings, and construct algebraic correspondences that combine generic (anti-)polynomials and genus zero orbifolds in a common dynamical plane, providing a new concrete evidence to Fatou's vision of a unified theory of groups and maps.
{"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":"4 1","pages":""},"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}
We compute the essential dimension of finite groups of order $leqslant 63$.
我们计算了阶为 $leqslant 63$ 的有限群的基本维数。
{"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":"296 1","pages":""},"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}
Thomas Breuer, Frank Calegari, Silvio Dolfi, Gabriel Navarro, Pham Huu Tiep
We determine the finite groups whose real irreducible representations have different degrees.
我们确定了实不可还原表示具有不同度的有限群。
{"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":"99 1","pages":""},"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}
The braid group $B_4$ naturally acts on the rational projective plane $mathbb{P}^2(mathbb{Q})$, this action corresponds to the classical integral reduced Burau representation of $B_4$. The first result of this paper is a classification of the orbits of this action. The Burau representation then defines an action of $B_4$ on $mathbb{P}^2(mathbb{Z}(q))$, where $q$ is a formal parameter and $mathbb{Z}(q)$ is the field of rational functions in $q$ with integer coefficients. We study orbits of the $B_4$-action on $mathbb{P}^2(mathbb{Z}(q))$, and show existence of embeddings of the $q$-deformed projective line $mathbb{P}^1(mathbb{Z}(q))$ that precisely correspond to the notion of $q$-rationals due to Morier-Genoud and Ovsienko.
{"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":"78 1","pages":""},"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}
Let $G$ be a finite permutation group acting on a set $Omega$. An ordered sequence $(omega_1,ldots,omega_ell)$ of elements of $Omega$ is an irredundant base for $G$ if the pointwise stabilizer of the sequence is trivial and no point is fixed by the stabilizer of its predecessors. We show that any interval of natural numbers can be realized as the set of cardinalities of irredundant bases for some finite primitive group.
{"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":"39 1","pages":""},"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}
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