Journal of Group Theory最新文献

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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>&lt;</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)&lt;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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"title":"Groups whose proper subgroups of infinite rank have a permutability transitive relation","authors":"Adolfo Ballester Bolinches, Maria De Falco, Francesco de Giovanni, Carmela Musella","doi":"10.1515/jgth-2023-0296","DOIUrl":"https://doi.org/10.1515/jgth-2023-0296","url":null,"abstract":"\u0000 <jats:p>Let 𝐺 be a group.\u0000A subgroup 𝐻 of 𝐺 is called permutable if <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mrow>\u0000 <m:mi>H</m:mi>\u0000 <m:mo>⁢</m:mo>\u0000 <m:mi>X</m:mi>\u0000 </m:mrow>\u0000 <m:mo>=</m:mo>\u0000 <m:mrow>\u0000 <m:mi>X</m:mi>\u0000 <m:mo>⁢</m:mo>\u0000 <m:mi>H</m:mi>\u0000 </m:mrow>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0296_ineq_0001.png\"/>\u0000 <jats:tex-math>HX=XH</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> for all subgroups 𝑋 of 𝐺.\u0000Permutability is not in general a transitive relation, and 𝐺 is called a <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mi>PT</m:mi>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0296_ineq_0002.png\"/>\u0000 <jats:tex-math>mathrm{PT}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula>-group if, whenever 𝐾 is a permutable subgroup of 𝐺 and 𝐻 is a permutable subgroup of 𝐾, we always have that 𝐻 is permutable in 𝐺. The property <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mi>PT</m:mi>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0296_ineq_0002.png\"/>\u0000 <jats:tex-math>mathrm{PT}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> is not inherited by subgroups, and 𝐺 is called a <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mover accent=\"true\">\u0000 <m:mi>PT</m:mi>\u0000 <m:mo>̄</m:mo>\u0000 </m:mover>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0296_ineq_0004.png\"/>\u0000 <jats:tex-math>overline{mathrm{PT}}</jats:tex-ma","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140962131","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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}