Saul D. Freedman, Veronica Kelsey, Colva M. Roney-Dougal
The relational complexity of a subgroup 𝐺 of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Sym</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2023-0262_ineq_0001.png" /> <jats:tex-math>mathrm{Sym}({Omega})</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a measure of the way in which the orbits of 𝐺 on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="normal">Ω</m:mi> <m:mi>k</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2023-0262_ineq_0002.png" /> <jats:tex-math>Omega^{k}</jats:tex-math> </jats:alternatives> </jats:inline-formula> for various 𝑘 determine the original action of 𝐺. Very few precise values of relational complexity are known. This paper determines the exact relational complexity of all groups lying between <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>PSL</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="double-struck">F</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2023-0262_ineq_0003.png" /> <jats:tex-math>mathrm{PSL}_{n}(mathbb{F})</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>PGL</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="double-struck">F</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2023-0262_ineq_0004.png" /> <jats:tex-math>mathrm{PGL}_{n}(mathbb{F})</jats:tex-math> </jats:alternatives> </jats:inline-formula>, for an arbitrary field 𝔽, acting on the set of 1-dimensional subspaces of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">F</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2023-0262_ineq_0005.png" /> <jats:tex-math>mathbb{F}^{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. We also bound the relational complexity of all groups lying between <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>PSL</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>q<
Sym ( Ω ) mathrm{Sym}({Omega})的子群𝐺的关系复杂度是一个度量，它衡量了不同𝑘的𝐺在Ω k Omega^{k}上的轨道如何决定𝐺的原始作用。关系复杂度的精确值很少为人所知。本文确定了对于任意域𝔽，介于 PSL n ( F ) mathrm{PSL}_{n}(mathbb{F}) 和 PGL n ( F ) mathrm{PGL}_{n}(mathbb{F}) 之间，作用于 F n mathbb{F}^{n} 的一维子空间集合的所有群的精确关系复杂度。我们还约束了介于 PSL n ( q ) mathrm{PSL}_{n}(q) 和 P Γ L n ( q ) mathrm{P}Gammamathrm{L}_{n}(q) 之间的所有群的关系复杂度，并将这些结果推广到 m ≥ 1 mgeq 1 的𝑚 空间上的作用。
{"title":"The relational complexity of linear groups acting on subspaces","authors":"Saul D. Freedman, Veronica Kelsey, Colva M. Roney-Dougal","doi":"10.1515/jgth-2023-0262","DOIUrl":"https://doi.org/10.1515/jgth-2023-0262","url":null,"abstract":"The relational complexity of a subgroup 𝐺 of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>Sym</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi mathvariant=\"normal\">Ω</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0262_ineq_0001.png\" /> <jats:tex-math>mathrm{Sym}({Omega})</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a measure of the way in which the orbits of 𝐺 on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi mathvariant=\"normal\">Ω</m:mi> <m:mi>k</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0262_ineq_0002.png\" /> <jats:tex-math>Omega^{k}</jats:tex-math> </jats:alternatives> </jats:inline-formula> for various 𝑘 determine the original action of 𝐺. Very few precise values of relational complexity are known. This paper determines the exact relational complexity of all groups lying between <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>PSL</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi mathvariant=\"double-struck\">F</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0262_ineq_0003.png\" /> <jats:tex-math>mathrm{PSL}_{n}(mathbb{F})</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>PGL</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi mathvariant=\"double-struck\">F</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0262_ineq_0004.png\" /> <jats:tex-math>mathrm{PGL}_{n}(mathbb{F})</jats:tex-math> </jats:alternatives> </jats:inline-formula>, for an arbitrary field 𝔽, acting on the set of 1-dimensional subspaces of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi mathvariant=\"double-struck\">F</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0262_ineq_0005.png\" /> <jats:tex-math>mathbb{F}^{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. We also bound the relational complexity of all groups lying between <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>PSL</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>q<","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"15 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139757226","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the average case complexity of the Uniform Membership Problem for subgroups of free groups, and we show that it is orders of magnitude smaller than the worst case complexity of the best known algorithms. This applies to subgroups given by a fixed number of generators as well as to subgroups given by an exponential number of generators. The main idea behind this result is to exploit a generic property of tuples of words, called the central tree property. An application is given to the average case complexity of the Relative Primitivity Problem, using Shpilrain’s recent algorithm to decide primitivity, whose average case complexity is a constant depending only on the rank of the ambient free group.
{"title":"The central tree property and algorithmic problems on subgroups of free groups","authors":"Mallika Roy, Enric Ventura, Pascal Weil","doi":"10.1515/jgth-2023-0050","DOIUrl":"https://doi.org/10.1515/jgth-2023-0050","url":null,"abstract":"We study the average case complexity of the Uniform Membership Problem for subgroups of free groups, and we show that it is orders of magnitude smaller than the worst case complexity of the best known algorithms. This applies to subgroups given by a fixed number of generators as well as to subgroups given by an exponential number of generators. The main idea behind this result is to exploit a generic property of tuples of words, called the central tree property. An application is given to the average case complexity of the Relative Primitivity Problem, using Shpilrain’s recent algorithm to decide primitivity, whose average case complexity is a constant depending only on the rank of the ambient free group.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"7 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139757282","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In a recent paper by A. A. Klyachko, V. Y. Miroshnichenko, and A. Y. Olshanskii, it is proven that the center of any finite strongly verbally closed group is a direct factor. In this paper, we extend this result to the case of finite normal subgroups of any strongly verbally closed group. It follows that finitely generated nilpotent groups with nonabelian torsion subgroups are not strongly verbally closed.
A. A. Klyachko、V. Y. Miroshnichenko 和 A. Y. Olshanskii 最近的一篇论文证明，任何有限强言闭群的中心都是直接因子。在本文中，我们将这一结果扩展到任何强言闭群的有限正则子群的情况。由此可知，具有非阿贝尔扭转子群的有限生成零能群不是强封闭群。
{"title":"Finite normal subgroups of strongly verbally closed groups","authors":"Filipp D. Denissov","doi":"10.1515/jgth-2023-0015","DOIUrl":"https://doi.org/10.1515/jgth-2023-0015","url":null,"abstract":"In a recent paper by A. A. Klyachko, V. Y. Miroshnichenko, and A. Y. Olshanskii, it is proven that the center of any finite strongly verbally closed group is a direct factor. In this paper, we extend this result to the case of finite normal subgroups of any strongly verbally closed group. It follows that finitely generated nilpotent groups with nonabelian torsion subgroups are not strongly verbally closed.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"220 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139757140","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A Schmidt group is a finite non-nilpotent group such that every proper subgroup is nilpotent. In this paper, we prove that if every Schmidt subgroup of a finite group 𝐺 is subnormal or modular, then G/F(G)G/F(G) is cyclic. Moreover, for a given prime 𝑝, we describe the structure of finite groups with subnormal or modular Schmidt subgroups of order divisible by 𝑝.
施密特群是一个有限非零能群，它的每个适当子群都是零能的。在本文中，我们证明了如果有限群𝐺 的每个施密特子群都是子常群或模群，那么 G / F ( G ) G/F(G) 是循环群。此外，对于给定素数𝑝，我们描述了具有阶可被𝑝整除的亚正态或模态施密特子群的有限群的结构。
{"title":"Groups with subnormal or modular Schmidt 𝑝𝑑-subgroups","authors":"Victor S. Monakhov, Irina L. Sokhor","doi":"10.1515/jgth-2023-0096","DOIUrl":"https://doi.org/10.1515/jgth-2023-0096","url":null,"abstract":"A Schmidt group is a finite non-nilpotent group such that every proper subgroup is nilpotent. In this paper, we prove that if every Schmidt subgroup of a finite group 𝐺 is subnormal or modular, then <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:mi>F</m:mi> </m:mrow> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0096_ineq_0001.png\" /> <jats:tex-math>G/F(G)</jats:tex-math> </jats:alternatives> </jats:inline-formula> is cyclic. Moreover, for a given prime 𝑝, we describe the structure of finite groups with subnormal or modular Schmidt subgroups of order divisible by 𝑝.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"110 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139757568","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is shown that any finite group 𝐴 is realizable as the automizer in a finite perfect group 𝐺 of an abelian subgroup whose conjugates generate 𝐺. The construction uses techniques from fusion systems on arbitrary finite groups, most notably certain realization results for fusion systems of the type studied originally by Park.
研究表明，任何有限群𝐴都可以实现为一个有限完全群𝐺的自动子群，其共轭子群生成𝐺。这个构造使用了任意有限群上融合系统的技术，其中最著名的是 Park 最初研究的那类融合系统的某些实现结果。
{"title":"Realizing finite groups as automizers","authors":"Sylvia Bayard, Justin Lynd","doi":"10.1515/jgth-2022-0145","DOIUrl":"https://doi.org/10.1515/jgth-2022-0145","url":null,"abstract":"It is shown that any finite group 𝐴 is realizable as the automizer in a finite perfect group 𝐺 of an abelian subgroup whose conjugates generate 𝐺. The construction uses techniques from fusion systems on arbitrary finite groups, most notably certain realization results for fusion systems of the type studied originally by Park.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"69 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139757138","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Hasan Akın, Dikran Dikranjan, Anna Giordano Bruno, Daniele Toller
The aim of this paper is to present one-dimensional finitary linear cellular automata 𝑆 on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="double-struck">Z</m:mi> <m:mi>m</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2023-0092_ineq_0001.png" /> <jats:tex-math>mathbb{Z}_{m}</jats:tex-math> </jats:alternatives> </jats:inline-formula> from an algebraic point of view. Among various other results, we (i) show that the Pontryagin dual <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mover accent="true"> <m:mi>S</m:mi> <m:mo>̂</m:mo> </m:mover> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2023-0092_ineq_0002.png" /> <jats:tex-math>hat{S}</jats:tex-math> </jats:alternatives> </jats:inline-formula> of 𝑆 is a classical one-dimensional linear cellular automaton 𝑇 on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="double-struck">Z</m:mi> <m:mi>m</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2023-0092_ineq_0001.png" /> <jats:tex-math>mathbb{Z}_{m}</jats:tex-math> </jats:alternatives> </jats:inline-formula>; (ii) give several equivalent conditions for 𝑆 to be invertible with inverse a finitary linear cellular automaton; (iii) compute the algebraic entropy of 𝑆, which coincides with the topological entropy of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>T</m:mi> <m:mo>=</m:mo> <m:mover accent="true"> <m:mi>S</m:mi> <m:mo>̂</m:mo> </m:mover> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2023-0092_ineq_0004.png" /> <jats:tex-math>T=hat{S}</jats:tex-math> </jats:alternatives> </jats:inline-formula> by the so-called Bridge Theorem. In order to better understand and describe entropy, we introduce the degree <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>deg</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>S</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2023-0092_ineq_0005.png" /> <jats:tex-math>deg(S)</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>deg</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>T</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2023-0092_ineq_0006.png" /> <jats:tex-mat
本文旨在从代数角度介绍 Z m mathbb{Z}_{m} 上的一维有限线性蜂窝自动机𝑆。在其他各种结果中，我们 (i) 证明了𝑆 的庞特里亚金对偶 S ̂ hat{S} 是 Z m mathbb{Z}_{m} 上的经典一维线性蜂窝自动机 𝑇 ；(iii) 计算𝑆的代数熵，根据所谓的桥定理，代数熵与 T = S ̂ T=hat{S} 的拓扑熵重合。为了更好地理解和描述熵，我们引入了𝑆 和 𝑇 的度 deg ( S ) deg(S) 和 deg ( T ) deg(T)。
{"title":"The algebraic entropy of one-dimensional finitary linear cellular automata","authors":"Hasan Akın, Dikran Dikranjan, Anna Giordano Bruno, Daniele Toller","doi":"10.1515/jgth-2023-0092","DOIUrl":"https://doi.org/10.1515/jgth-2023-0092","url":null,"abstract":"The aim of this paper is to present one-dimensional finitary linear cellular automata 𝑆 on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"double-struck\">Z</m:mi> <m:mi>m</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0092_ineq_0001.png\" /> <jats:tex-math>mathbb{Z}_{m}</jats:tex-math> </jats:alternatives> </jats:inline-formula> from an algebraic point of view. Among various other results, we (i) show that the Pontryagin dual <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mover accent=\"true\"> <m:mi>S</m:mi> <m:mo>̂</m:mo> </m:mover> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0092_ineq_0002.png\" /> <jats:tex-math>hat{S}</jats:tex-math> </jats:alternatives> </jats:inline-formula> of 𝑆 is a classical one-dimensional linear cellular automaton 𝑇 on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"double-struck\">Z</m:mi> <m:mi>m</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0092_ineq_0001.png\" /> <jats:tex-math>mathbb{Z}_{m}</jats:tex-math> </jats:alternatives> </jats:inline-formula>; (ii) give several equivalent conditions for 𝑆 to be invertible with inverse a finitary linear cellular automaton; (iii) compute the algebraic entropy of 𝑆, which coincides with the topological entropy of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>T</m:mi> <m:mo>=</m:mo> <m:mover accent=\"true\"> <m:mi>S</m:mi> <m:mo>̂</m:mo> </m:mover> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0092_ineq_0004.png\" /> <jats:tex-math>T=hat{S}</jats:tex-math> </jats:alternatives> </jats:inline-formula> by the so-called Bridge Theorem. In order to better understand and describe entropy, we introduce the degree <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>deg</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>S</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0092_ineq_0005.png\" /> <jats:tex-math>deg(S)</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>deg</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>T</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0092_ineq_0006.png\" /> <jats:tex-mat","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"14 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139647210","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The three famous problems concerning units, zero-divisors and idempotents in group rings of torsion-free groups, commonly attributed to Kaplansky, have been around for more than 60 years and still remain open in characteristic zero. In this article, we introduce the corresponding problems in the considerably more general context of arbitrary rings graded by torsion-free groups. For natural reasons, we will restrict our attention to rings without non-trivial homogeneous zero-divisors with respect to the given grading. We provide a partial solution to the extended problems by solving them for rings graded by unique product groups. We also show that the extended problems exhibit the same (potential) hierarchy as the classical problems for group rings. Furthermore, a ring which is graded by an arbitrary torsion-free group is shown to be indecomposable, and to have no non-trivial central zero-divisor and no non-homogeneous central unit. We also present generalizations of the classical group ring conjectures.
{"title":"Units, zero-divisors and idempotents in rings graded by torsion-free groups","authors":"Johan Öinert","doi":"10.1515/jgth-2023-0110","DOIUrl":"https://doi.org/10.1515/jgth-2023-0110","url":null,"abstract":"The three famous problems concerning units, zero-divisors and idempotents in group rings of torsion-free groups, commonly attributed to Kaplansky, have been around for more than 60 years and still remain open in characteristic zero. In this article, we introduce the corresponding problems in the considerably more general context of arbitrary rings graded by torsion-free groups. For natural reasons, we will restrict our attention to rings without non-trivial homogeneous zero-divisors with respect to the given grading. We provide a partial solution to the extended problems by solving them for rings graded by unique product groups. We also show that the extended problems exhibit the same (potential) hierarchy as the classical problems for group rings. Furthermore, a ring which is graded by an arbitrary torsion-free group is shown to be indecomposable, and to have no non-trivial central zero-divisor and no non-homogeneous central unit. We also present generalizations of the classical group ring conjectures.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"98 2 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139554853","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let Ω be a set equipped with an equivalence relation ∼sim; we refer to the equivalence classes as blocks of Ω. A permutation group G≤Sym(Ω)Gleqmathrm{Sym}(Omega) is 𝑘-by-block-transitive if ∼sim is 𝐺-invariant, with at least 𝑘 blocks, and 𝐺 is transitive on the set of 𝑘-tuples of points such that no two entries lie in the same block. The action is block-faithful if the action on the set of blocks is faithful. In this article, we classify the finite block-faithful 2-by-block-transitive actions. We also show that, for k≥3kgeq 3, there are no finite block-faithful 𝑘-by-block-transitive actions with nontrivial blocks.
{"title":"Multiple transitivity except for a system of imprimitivity","authors":"Colin D. Reid","doi":"10.1515/jgth-2023-0062","DOIUrl":"https://doi.org/10.1515/jgth-2023-0062","url":null,"abstract":"Let Ω be a set equipped with an equivalence relation <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mo>∼</m:mo> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0062_ineq_0001.png\" /> <jats:tex-math>sim</jats:tex-math> </jats:alternatives> </jats:inline-formula>; we refer to the equivalence classes as blocks of Ω. A permutation group <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>G</m:mi> <m:mo>≤</m:mo> <m:mrow> <m:mi>Sym</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi mathvariant=\"normal\">Ω</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0062_ineq_0002.png\" /> <jats:tex-math>Gleqmathrm{Sym}(Omega)</jats:tex-math> </jats:alternatives> </jats:inline-formula> is <jats:italic>𝑘-by-block-transitive</jats:italic> if <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mo>∼</m:mo> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0062_ineq_0001.png\" /> <jats:tex-math>sim</jats:tex-math> </jats:alternatives> </jats:inline-formula> is 𝐺-invariant, with at least 𝑘 blocks, and 𝐺 is transitive on the set of 𝑘-tuples of points such that no two entries lie in the same block. The action is <jats:italic>block-faithful</jats:italic> if the action on the set of blocks is faithful. In this article, we classify the finite block-faithful 2-by-block-transitive actions. We also show that, for <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>k</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0062_ineq_0004.png\" /> <jats:tex-math>kgeq 3</jats:tex-math> </jats:alternatives> </jats:inline-formula>, there are no finite block-faithful 𝑘-by-block-transitive actions with nontrivial blocks.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"48 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139496858","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let 𝐺 be a finite reductive group such that the derived subgroup of the underlying algebraic group is a product of quasi-simple groups of type 𝖠. In this paper, we give an explicit description of the action of automorphisms of 𝐺 on the set of its irreducible complex characters. This generalizes a recent result of M. Cabanes and B. Späth [Equivariant character correspondences and inductive McKay condition for type 𝖠, J. Reine Angew. Math. 728 (2017), 153–194] and provides a useful tool for investigating the local sides of the local-global conjectures as one usually needs to deal with Levi subgroups. As an application we obtain a generalization of the stabilizer condition in the so-called inductive McKay condition [B. Späth, Inductive McKay condition in defining characteristic, Bull. Lond. Math. Soc. 44 (2012), 3, 426–438; Theorem 2.12] for irreducible characters of 𝐺. Moreover, a criterion is given to explicitly determine whether an irreducible character is a constituent of a given generalized Gelfand–Graev character of 𝐺.
设𝐺是一个有限还原群，其底层代数群的导出子群是𝖠型准简单群的乘积。在本文中，我们给出了𝐺 的自变量对其不可还原复字符集的作用的明确描述。这概括了 M. Cabanes 和 B. Späth 最近的一个结果 [Equivariant character correspondences and inductive McKay condition for type 𝖠, J. Reine Angew.Math.728 (2017), 153-194] 并为研究局部-全局猜想的局部边提供了有用的工具，因为人们通常需要处理 Levi 子群。作为应用，我们在所谓的归纳麦凯条件中得到了稳定器条件的广义化[B. Späth, Inductive McKay condition]。Späth, Inductive McKay condition in defining characteristic, Bull.Lond.Math.44 (2012), 3, 426-438; Theorem 2.12]。此外，还给出了明确判定一个不可约字符是否为𝐺的给定广义格尔芬-格拉夫字符的一个成分的标准。
{"title":"Action of automorphisms on irreducible characters of finite reductive groups of type 𝖠","authors":"Farrokh Shirjian, Ali Iranmanesh, Farideh Shafiei","doi":"10.1515/jgth-2022-0034","DOIUrl":"https://doi.org/10.1515/jgth-2022-0034","url":null,"abstract":"Let 𝐺 be a finite reductive group such that the derived subgroup of the underlying algebraic group is a product of quasi-simple groups of type 𝖠. In this paper, we give an explicit description of the action of automorphisms of 𝐺 on the set of its irreducible complex characters. This generalizes a recent result of M. Cabanes and B. Späth [Equivariant character correspondences and inductive McKay condition for type 𝖠, <jats:italic>J. Reine Angew. Math.</jats:italic> 728 (2017), 153–194] and provides a useful tool for investigating the local sides of the local-global conjectures as one usually needs to deal with Levi subgroups. As an application we obtain a generalization of the stabilizer condition in the so-called inductive McKay condition [B. Späth, Inductive McKay condition in defining characteristic, <jats:italic>Bull. Lond. Math. Soc.</jats:italic> 44 (2012), 3, 426–438; Theorem 2.12] for irreducible characters of 𝐺. Moreover, a criterion is given to explicitly determine whether an irreducible character is a constituent of a given generalized Gelfand–Graev character of 𝐺.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"195 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139476967","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that if 𝐺 is an admissible group acting geometrically on a CAT(0)operatorname{CAT}(0) space 𝑋, then 𝐺 is a hierarchically hyperbolic space and its 𝜅-Morse boundary (∂κG,ν)(partial_{kappa}G,nu) is a model for the Poisson boundary of (G,μ)(G,mu), where 𝜈 is the hitting measure associated to the random walk driven by 𝜇.
{"title":"Sublinearly Morse boundary of CAT(0) admissible groups","authors":"Hoang Thanh Nguyen, Yulan Qing","doi":"10.1515/jgth-2023-0145","DOIUrl":"https://doi.org/10.1515/jgth-2023-0145","url":null,"abstract":"We show that if 𝐺 is an admissible group acting geometrically on a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>CAT</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mn>0</m:mn> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0145_ineq_0001.png\" /> <jats:tex-math>operatorname{CAT}(0)</jats:tex-math> </jats:alternatives> </jats:inline-formula> space 𝑋, then 𝐺 is a hierarchically hyperbolic space and its 𝜅-Morse boundary <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:msub> <m:mo lspace=\"0em\" rspace=\"0em\">∂</m:mo> <m:mi>κ</m:mi> </m:msub> <m:mi>G</m:mi> </m:mrow> <m:mo>,</m:mo> <m:mi>ν</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0145_ineq_0002.png\" /> <jats:tex-math>(partial_{kappa}G,nu)</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a model for the Poisson boundary of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>G</m:mi> <m:mo>,</m:mo> <m:mi>μ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0145_ineq_0003.png\" /> <jats:tex-math>(G,mu)</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where 𝜈 is the hitting measure associated to the random walk driven by 𝜇.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"34 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139476598","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：