We continue our investigation of binary actions of simple groups. In this paper, we demonstrate a connection between the graph Γ(C)Gamma(mathcal{C}) based on the conjugacy class 𝒞 of the group 𝐺, which was introduced in our previous work, and the notion of a strongly embedded subgroup of 𝐺. We exploit this connection to prove a result concerning the binary actions of finite simple groups that contain a single conjugacy class of involutions.
我们继续研究简单群的二元作用。在本文中，我们证明了基于群𝐺 的共轭类 𝒞 的图Γ ( C ) Gamma(mathcal{C})与𝐺 的强嵌入子群概念之间的联系。我们利用这种联系证明了一个关于有限简单群的二元作用的结果，这些有限简单群包含一个渐开线的共轭类。
{"title":"The binary actions of simple groups with a single conjugacy class of involutions","authors":"Nick Gill, Pierre Guillot","doi":"10.1515/jgth-2024-0066","DOIUrl":"https://doi.org/10.1515/jgth-2024-0066","url":null,"abstract":"We continue our investigation of binary actions of simple groups. In this paper, we demonstrate a connection between the graph <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"normal\">Γ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi mathvariant=\"script\">C</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-2024-0066_ineq_0001.png\"/> <jats:tex-math>Gamma(mathcal{C})</jats:tex-math> </jats:alternatives> </jats:inline-formula> based on the conjugacy class 𝒞 of the group 𝐺, which was introduced in our previous work, and the notion of a strongly embedded subgroup of 𝐺. We exploit this connection to prove a result concerning the binary actions of finite simple groups that contain a single conjugacy class of involutions.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"11 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141512054","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}
Stefanos Aivazidis, Maria Loukaki, Thomas W. Müller
Let 𝐺 be a finite group, and let 𝐻 be a subgroup of 𝐺. We compute the probability, denoted by PG(H)P_{G}(H), that a left transversal of 𝐻 in 𝐺 is also a right transversal, thus a two-sided one. Moreover, we define, and denote by tp(G)operatorname{tp}(G), the common transversal probability of 𝐺 to be the minimum, taken over all subgroups 𝐻 of 𝐺, of PG(H)P_{G}(H). We prove a number of results regarding the invariant tp(G)operatorname{tp}(G), like lower and upper bounds, and possible values it can attain. We also show that tp(G)operatorname{tp}(G) determines structural properties of 𝐺. Finally, several open problems are formulated and discussed.
设𝐺 是一个有限群，又设𝐻 是𝐺 的一个子群。我们用 P G ( H ) P_{G}(H)来计算𝐻 在𝐺 中的左横也是右横的概率，即双面概率。此外，我们定义𝐺 的公共横切概率为 P G ( H ) P_{G}(H) 在𝐺 的所有子群 𝐻 中的最小值，并用 tp ( G ) （operatorname{tp}(G) ）表示。我们证明了一些关于不变式 tp ( G ) （operatorname{tp}(G) ）的结果，如下限和上限，以及它可能达到的值。我们还证明了 tp ( G ) operatorname{tp}(G) 决定了𝐺的结构属性。最后，我们提出并讨论了几个悬而未决的问题。
{"title":"On the common transversal probability","authors":"Stefanos Aivazidis, Maria Loukaki, Thomas W. Müller","doi":"10.1515/jgth-2024-0030","DOIUrl":"https://doi.org/10.1515/jgth-2024-0030","url":null,"abstract":"Let 𝐺 be a finite group, and let 𝐻 be a subgroup of 𝐺. We compute the probability, denoted by <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>P</m:mi> <m:mi>G</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>H</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-2024-0030_ineq_0001.png\"/> <jats:tex-math>P_{G}(H)</jats:tex-math> </jats:alternatives> </jats:inline-formula>, that a left transversal of 𝐻 in 𝐺 is also a right transversal, thus a two-sided one. Moreover, we define, and denote by <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>tp</m:mi> <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-2024-0030_ineq_0002.png\"/> <jats:tex-math>operatorname{tp}(G)</jats:tex-math> </jats:alternatives> </jats:inline-formula>, the common transversal probability of 𝐺 to be the minimum, taken over all subgroups 𝐻 of 𝐺, of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>P</m:mi> <m:mi>G</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>H</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-2024-0030_ineq_0001.png\"/> <jats:tex-math>P_{G}(H)</jats:tex-math> </jats:alternatives> </jats:inline-formula>. We prove a number of results regarding the invariant <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>tp</m:mi> <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-2024-0030_ineq_0002.png\"/> <jats:tex-math>operatorname{tp}(G)</jats:tex-math> </jats:alternatives> </jats:inline-formula>, like lower and upper bounds, and possible values it can attain. We also show that <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>tp</m:mi> <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-2024-0030_ineq_0002.png\"/> <jats:tex-math>operatorname{tp}(G)</jats:tex-math> </jats:alternatives> </jats:inline-formula> determines structural properties of 𝐺. Finally, several open problems are formulated and discussed.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"54 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141512056","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 group and 𝑘 a positive integer. We say that 𝐺 is conjugate separable abelian (CSA) if every maximal abelian subgroup of 𝐺 is malnormal. In this paper, as a natural generalization, we study groups with the property that all maximal nilpotent subgroups of class at most 𝑘 are malnormal, which we refer to as CSN𝑘 groups, and we show that they have many properties in common with the more widely studied CSA groups. In addition, we introduce the class of nilpotency transitive groups of class 𝑘, denoted NT𝑘, and in the presence of a special residuality condition, we prove that the CSN𝑘 and NT𝑘 properties are equivalent.
{"title":"On conjugate separability of nilpotent subgroups","authors":"Mohammad Shahryari","doi":"10.1515/jgth-2024-0023","DOIUrl":"https://doi.org/10.1515/jgth-2024-0023","url":null,"abstract":"Let 𝐺 be a group and 𝑘 a positive integer. We say that 𝐺 is conjugate separable abelian (CSA) if every maximal abelian subgroup of 𝐺 is malnormal. In this paper, as a natural generalization, we study groups with the property that all maximal nilpotent subgroups of class at most 𝑘 are malnormal, which we refer to as CSN<jats:sub>𝑘</jats:sub> groups, and we show that they have many properties in common with the more widely studied CSA groups. In addition, we introduce the class of nilpotency transitive groups of class 𝑘, denoted NT<jats:sub>𝑘</jats:sub>, and in the presence of a special residuality condition, we prove that the CSN<jats:sub>𝑘</jats:sub> and NT<jats:sub>𝑘</jats:sub> properties are equivalent.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"33 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141512055","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 group and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="normal">P</m:mi> <m:mi mathvariant="bold">v</m:mi> </m:msub> <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-2024-0016_ineq_0001.png"/> <jats:tex-math>mathrm{P}_{mathbf{v}}(G)</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the proportion of elements <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:mi>G</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2024-0016_ineq_0002.png"/> <jats:tex-math>gin G</jats:tex-math> </jats:alternatives> </jats:inline-formula> such that <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>χ</m:mi> <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:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2024-0016_ineq_0003.png"/> <jats:tex-math>chi(g)=0</jats:tex-math> </jats:alternatives> </jats:inline-formula> for some irreducible character 𝜒. In a recent paper, we proved that if <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi mathvariant="normal">P</m:mi> <m:mi mathvariant="bold">v</m:mi> </m:msub> <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:mo><</m:mo> <m:mrow> <m:msub> <m:mi mathvariant="normal">P</m:mi> <m:mi mathvariant="bold">v</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>A</m:mi> <m:mn>7</m:mn> </m:msub> <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-2024-0016_ineq_0004.png"/> <jats:tex-math>mathrm{P}_{mathbf{v}}(G)<mathrm{P}_{mathbf{v}}(A_{7})</jats:tex-math> </jats:alternatives> </jats:inline-formula>, then <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi mathvariant="normal">P</m:mi> <m:mi mathvariant="bold">v</m:mi> </m:msub> <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:mo>=</m:mo> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>m</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>/</m:mo> <m:mi>m</m:mi> </m:mrow> </m:mrow> </m:math> <jats
让𝐺 是一个有限群，并让 P v ( G ) mathrm{P}_{mathbf{v}}(G) 是元素 g ∈ G gin G 中的比例，使得对于某个不可还原字符𝜒，χ ( g ) = 0 chi(g)=0 。在最近的一篇论文中，我们证明了如果 P v ( G ) < P v ( A 7 ) mathrm{P}_{mathbf{v}}(G)<；mathrm{P}_{mathbf{v}}(A_{7}) , then P v ( G ) = ( m - 1 ) / m mathrm{P}_{mathbf{v}}(G)=(m-1)/m for some 1 ≤ m ≤ 6 1leq mleq 6 .这里我们将所有有限群𝐺进行分类，使得 P v ( G ) = ( m - 1 ) / m mathrm{P}_{mathbf{v}}(G)=(m-1)/m 且 m = 1 , 2 , ... , 6 m=1,2,ldots,6 。
{"title":"Finite groups with a small proportion of vanishing elements","authors":"Dongfang Yang, Yu Zeng, Silvio Dolfi","doi":"10.1515/jgth-2024-0016","DOIUrl":"https://doi.org/10.1515/jgth-2024-0016","url":null,"abstract":"Let 𝐺 be a finite group and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi mathvariant=\"normal\">P</m:mi> <m:mi mathvariant=\"bold\">v</m:mi> </m:msub> <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-2024-0016_ineq_0001.png\"/> <jats:tex-math>mathrm{P}_{mathbf{v}}(G)</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the proportion of elements <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:mi>G</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2024-0016_ineq_0002.png\"/> <jats:tex-math>gin G</jats:tex-math> </jats:alternatives> </jats:inline-formula> such that <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>χ</m:mi> <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:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2024-0016_ineq_0003.png\"/> <jats:tex-math>chi(g)=0</jats:tex-math> </jats:alternatives> </jats:inline-formula> for some irreducible character 𝜒. In a recent paper, we proved that if <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:msub> <m:mi mathvariant=\"normal\">P</m:mi> <m:mi mathvariant=\"bold\">v</m:mi> </m:msub> <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:mo><</m:mo> <m:mrow> <m:msub> <m:mi mathvariant=\"normal\">P</m:mi> <m:mi mathvariant=\"bold\">v</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msub> <m:mi>A</m:mi> <m:mn>7</m:mn> </m:msub> <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-2024-0016_ineq_0004.png\"/> <jats:tex-math>mathrm{P}_{mathbf{v}}(G)<mathrm{P}_{mathbf{v}}(A_{7})</jats:tex-math> </jats:alternatives> </jats:inline-formula>, then <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:msub> <m:mi mathvariant=\"normal\">P</m:mi> <m:mi mathvariant=\"bold\">v</m:mi> </m:msub> <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:mo>=</m:mo> <m:mrow> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>m</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>/</m:mo> <m:mi>m</m:mi> </m:mrow> </m:mrow> </m:math> <jats","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"62 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141530295","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}
Michael G. Cowling, Karl H. Hofmann, Sidney A. Morris
The aim of this note is to insert in the literature some easy but apparently not widely known facts about morphisms of locally compact groups, all of which are concerned with the openness of the morphism.
{"title":"Open mappings of locally compact groups","authors":"Michael G. Cowling, Karl H. Hofmann, Sidney A. Morris","doi":"10.1515/jgth-2024-0017","DOIUrl":"https://doi.org/10.1515/jgth-2024-0017","url":null,"abstract":"The aim of this note is to insert in the literature some easy but apparently not widely known facts about morphisms of locally compact groups, all of which are concerned with the openness of the morphism.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"26 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141149712","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 say that a group 𝐺 is of profinite type if it can be realized as a Galois group of some field extension. Using Krull’s theory, this is equivalent to 𝐺 admitting a profinite topology. We also say that a group of profinite type is profinitely rigid if it admits a unique profinite topology. In this paper, we study when abelian groups and some group extensions are of profinite type or profinitely rigid. We also discuss the connection between the properties of profinite type and profinite rigidity to the injectivity and surjectivity of the cohomology comparison maps, which were studied by Sury and other authors.
{"title":"Groups of profinite type and profinite rigidity","authors":"Tamar Bar-On, Nikolay Nikolov","doi":"10.1515/jgth-2023-0228","DOIUrl":"https://doi.org/10.1515/jgth-2023-0228","url":null,"abstract":"We say that a group 𝐺 is of <jats:italic>profinite type</jats:italic> if it can be realized as a Galois group of some field extension. Using Krull’s theory, this is equivalent to 𝐺 admitting a profinite topology. We also say that a group of profinite type is <jats:italic>profinitely rigid</jats:italic> if it admits a unique profinite topology. In this paper, we study when abelian groups and some group extensions are of profinite type or profinitely rigid. We also discuss the connection between the properties of profinite type and profinite rigidity to the injectivity and surjectivity of the cohomology comparison maps, which were studied by Sury and other authors.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"47 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141149845","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 simple group of Lie type and let 𝑇 be a maximal torus of 𝐺. It is well known that if the defining field of 𝐺 is large enough, then the normaliser of 𝑇 in 𝐺 is equal to the algebraic normaliser N(G,T)N(G,T). We identify explicitly all the cases when NG(T)N_{G}(T) is not equal to N(G,T)N(G,T).
设𝐺是一个有限李型简单群，设𝑇是𝐺的最大环。众所周知，如果𝐺 的定义域足够大，那么𝐺 中𝑇 的归一化等于代数归一化 N ( G , T ) N(G,T)。我们明确指出 N G ( T ) N_{G}(T) 不等于 N ( G , T ) N(G,T) 的所有情况。
{"title":"On algebraic normalisers of maximal tori in simple groups of Lie type","authors":"Anton A. Baykalov","doi":"10.1515/jgth-2023-0070","DOIUrl":"https://doi.org/10.1515/jgth-2023-0070","url":null,"abstract":"Let 𝐺 be a finite simple group of Lie type and let 𝑇 be a maximal torus of 𝐺. It is well known that if the defining field of 𝐺 is large enough, then the normaliser of 𝑇 in 𝐺 is equal to the algebraic normaliser <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>N</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>G</m:mi> <m:mo>,</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-2023-0070_ineq_0001.png\"/> <jats:tex-math>N(G,T)</jats:tex-math> </jats:alternatives> </jats:inline-formula>. We identify explicitly all the cases when <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>N</m:mi> <m:mi>G</m:mi> </m:msub> <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-2023-0070_ineq_0002.png\"/> <jats:tex-math>N_{G}(T)</jats:tex-math> </jats:alternatives> </jats:inline-formula> is not equal to <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>N</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>G</m:mi> <m:mo>,</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-2023-0070_ineq_0001.png\"/> <jats:tex-math>N(G,T)</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"13 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141060885","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 skew-morphism of a finite group 𝐺 is a permutation 𝜎 on 𝐺 fixing the identity element, and for which there exists an integer-valued function 𝜋 on 𝐺 such that <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>σ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo></m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>σ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo></m:mo> <m:msup> <m:mi>σ</m:mi> <m:mrow> <m:mi>π</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>y</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-2022-0092_ineq_0001.png"/> <jats:tex-math>sigma(xy)=sigma(x)sigma^{pi(x)}(y)</jats:tex-math> </jats:alternatives> </jats:inline-formula> for all <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:mi>G</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2022-0092_ineq_0002.png"/> <jats:tex-math>x,yin G</jats:tex-math> </jats:alternatives> </jats:inline-formula>. It is known that, for a given skew-morphism 𝜎 of 𝐺, the product of the left regular representation of 𝐺 with <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>σ</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-2022-0092_ineq_0003.png"/> <jats:tex-math>langlesigmarangle</jats:tex-math> </jats:alternatives> </jats:inline-formula> forms a permutation group on 𝐺, called a skew-product group of 𝐺. In this paper, we study the skew-product groups 𝑋 of elementary abelian 𝑝-groups <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi mathvariant="double-struck">Z</m:mi> <m:mi>p</m:mi> <m:mi>n</m:mi> </m:msubsup> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jgth-2022-0092_ineq_0004.png"/> <jats:tex-math>mathbb{Z}_{p}^{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. We prove that 𝑋 has a normal Sylow 𝑝-subgroup and determine the structure of 𝑋. In particular, we prove that <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi mathvariant="double-struck">Z</m:mi> <m:mi>p</m:mi> <m:mi
有限群𝐺的偏斜形变是在𝐺上固定同一元素的置换𝜎、对于所有的 x , y ∈ G x,yin G 而言，在𝐺上存在一个整数值函数 𝜋 ，使得 σ ( x y ) = σ ( x ) σ π ( x ) ( y ) sigma(xy)=sigma(x)sigma^{pi(x)}(y) 。众所周知，对于𝐺的给定偏斜变形𝜎，𝐺的左正则表达与⟨ σ ⟩ langlesigmarangle 的乘积形成了一个关于𝐺的置换群，称为𝐺的偏斜-乘积群。本文研究基本无边𝑝群 Z p n mathbb{Z}_{p}^{n} 的偏积群𝑋。我们证明了𝑋 有一个正常的 Sylow 𝑝 子群，并确定了 𝑋 的结构。特别是我们证明，如果 p = 2 p=2 并且 Z p n ⊲ X mathbb{Z}_{p}^{n}lhd X 或 ( Z p n ) X ≅ Z p n - 1 (mathbb{Z}_{p}^{n})_{X}congmathbb{Z}_{p}^{n-1} 如果𝑝是奇素数。作为应用，对于 n ≤ 3 nleq 3 ，我们证明𝑋 与仿射群 AGL ( n , p ) 的一个子群 mathrm{AGL}(n,p) 同构，并列举了 Z p n mathbb{Z}_{p}^{n} 的偏斜变形数。
{"title":"Skew-morphisms of elementary abelian 𝑝-groups","authors":"Shaofei Du, Wenjuan Luo, Hao Yu, Junyang Zhang","doi":"10.1515/jgth-2022-0092","DOIUrl":"https://doi.org/10.1515/jgth-2022-0092","url":null,"abstract":"A skew-morphism of a finite group 𝐺 is a permutation 𝜎 on 𝐺 fixing the identity element, and for which there exists an integer-valued function 𝜋 on 𝐺 such that <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>σ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo></m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>σ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo></m:mo> <m:msup> <m:mi>σ</m:mi> <m:mrow> <m:mi>π</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>y</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-2022-0092_ineq_0001.png\"/> <jats:tex-math>sigma(xy)=sigma(x)sigma^{pi(x)}(y)</jats:tex-math> </jats:alternatives> </jats:inline-formula> for all <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:mi>G</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2022-0092_ineq_0002.png\"/> <jats:tex-math>x,yin G</jats:tex-math> </jats:alternatives> </jats:inline-formula>. It is known that, for a given skew-morphism 𝜎 of 𝐺, the product of the left regular representation of 𝐺 with <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>σ</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-2022-0092_ineq_0003.png\"/> <jats:tex-math>langlesigmarangle</jats:tex-math> </jats:alternatives> </jats:inline-formula> forms a permutation group on 𝐺, called a skew-product group of 𝐺. In this paper, we study the skew-product groups 𝑋 of elementary abelian 𝑝-groups <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mi mathvariant=\"double-struck\">Z</m:mi> <m:mi>p</m:mi> <m:mi>n</m:mi> </m:msubsup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2022-0092_ineq_0004.png\"/> <jats:tex-math>mathbb{Z}_{p}^{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. We prove that 𝑋 has a normal Sylow 𝑝-subgroup and determine the structure of 𝑋. In particular, we prove that <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msubsup> <m:mi mathvariant=\"double-struck\">Z</m:mi> <m:mi>p</m:mi> <m:mi","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"44 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140884033","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}
For k,ℓ∈Nk,ellinmathbb{N}, we introduce the concepts of 𝑘-ultrahomogeneity and ℓ-tuple regularity for finite groups. Inspired by analogous concepts in graph theory, these form a natural generalization of homogeneity, which was studied by Cherlin and Felgner as well as Li, and automorphism transitivity, which was investigated by Zhang. Additionally, these groups have an interesting algorithmic interpretation. We classify the 𝑘-ultrahomogeneous and ℓ-tuple regular finite groups for k,ℓ≥2k,ellgeq 2. In particular, we show that every 2-tuple regular finite group is ultrahomogeneous.
对于 k , ℓ ∈ N k,ellinmathbb{N}，我们引入了有限群的𝑘-超同质性和ℓ-元组正则性的概念。这些概念受到图论中类似概念的启发，是对 Cherlin 和 Felgner 以及 Li 所研究的同质性和 Zhang 所研究的自动反常性的自然概括。此外，这些群还具有有趣的算法解释。我们对 k , ℓ ≥ 2 k,ellgeq 2 的 𝑘-ultrahomogeneous 和 ℓ-tuple 正则有限群进行了分类。特别是，我们证明了每个 2 元组正则有限群都是超均质的。
{"title":"Tuple regularity and 𝑘-ultrahomogeneity for finite groups","authors":"Sofia Brenner","doi":"10.1515/jgth-2023-0106","DOIUrl":"https://doi.org/10.1515/jgth-2023-0106","url":null,"abstract":"For <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">ℓ</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:mi mathvariant=\"double-struck\">N</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0106_ineq_0001.png\"/> <jats:tex-math>k,ellinmathbb{N}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, we introduce the concepts of 𝑘-ultrahomogeneity and ℓ-tuple regularity for finite groups. Inspired by analogous concepts in graph theory, these form a natural generalization of homogeneity, which was studied by Cherlin and Felgner as well as Li, and automorphism transitivity, which was investigated by Zhang. Additionally, these groups have an interesting algorithmic interpretation. We classify the 𝑘-ultrahomogeneous and ℓ-tuple regular finite groups for <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">ℓ</m:mi> </m:mrow> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0106_ineq_0002.png\"/> <jats:tex-math>k,ellgeq 2</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In particular, we show that every 2-tuple regular finite group is ultrahomogeneous.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"23 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140830770","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}
Anakin Dey, Kolton O’Neal, Duc Van Khanh Tran, Camron Upshur, Yong Yang
Let 𝐺 be a finite solvable permutation group acting faithfully and primitively on a finite set Ω. Let G0G_{0} be the stabilizer of a point 𝛼 in Ω. The rank of 𝐺 is defined as the number of orbits of G0G_{0} in Ω, including the trivial orbit {α}{alpha}. In this paper, we completely classify the cases where 𝐺 has rank 5 and 6, continuing the previous works on classifying groups of rank 4 or lower.
{"title":"Classifying primitive solvable permutation groups of rank 5 and 6","authors":"Anakin Dey, Kolton O’Neal, Duc Van Khanh Tran, Camron Upshur, Yong Yang","doi":"10.1515/jgth-2023-0205","DOIUrl":"https://doi.org/10.1515/jgth-2023-0205","url":null,"abstract":"Let 𝐺 be a finite solvable permutation group acting faithfully and primitively on a finite set Ω. Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>G</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0205_ineq_0001.png\"/> <jats:tex-math>G_{0}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the stabilizer of a point 𝛼 in Ω. The rank of 𝐺 is defined as the number of orbits of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>G</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0205_ineq_0001.png\"/> <jats:tex-math>G_{0}</jats:tex-math> </jats:alternatives> </jats:inline-formula> in Ω, including the trivial orbit <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>α</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-2023-0205_ineq_0003.png\"/> <jats:tex-math>{alpha}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this paper, we completely classify the cases where 𝐺 has rank 5 and 6, continuing the previous works on classifying groups of rank 4 or lower.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"50 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140831007","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}
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