{"title":"On generalized concise words","authors":"Costantino Delizia, Michele Gaeta, Carmine Monetta","doi":"10.1515/jgth-2024-0148","DOIUrl":"https://doi.org/10.1515/jgth-2024-0148","url":null,"abstract":"The study of verbal subgroups within a group is well known for being an effective tool to obtain structural information about a group. Therefore, conditions that allow the classification of words in a free group are of paramount importance. One of the most studied problems is to establish which words are concise, where a word 𝑤 is said to be concise if the verbal subgroup <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>w</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-2024-0148_ineq_0001.png\"/> <jats:tex-math>w(G)</jats:tex-math> </jats:alternatives> </jats:inline-formula> is finite in each group 𝐺 in which 𝑤 takes only a finite number of values. The purpose of this article is to present some results, in which a hierarchy among words is introduced, generalizing the concept of concise word.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217113","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}
Zhigang Wang, A-Ming Liu, Vasily G. Safonov, Alexander N. Skiba
{"title":"On 𝜎-permutable subgroups of 𝜎-soluble finite groups","authors":"Zhigang Wang, A-Ming Liu, Vasily G. Safonov, Alexander N. Skiba","doi":"10.1515/jgth-2024-0012","DOIUrl":"https://doi.org/10.1515/jgth-2024-0012","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>σ</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:msub> <m:mi>σ</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo fence=\"true\" lspace=\"0em\" rspace=\"0em\">∣</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>∈</m:mo> <m:mi>I</m:mi> </m:mrow> <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-2024-0012_ineq_0001.png\"/> <jats:tex-math>sigma={sigma_{i}mid iin I}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be some partition of the set of all primes and 𝐺 a finite group. Then 𝐺 is said to be 𝜎-full if 𝐺 has a Hall <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>σ</m:mi> <m:mi>i</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2024-0012_ineq_0002.png\"/> <jats:tex-math>sigma_{i}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-subgroup for all <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>i</m:mi> <m:mo>∈</m:mo> <m:mi>I</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2024-0012_ineq_0003.png\"/> <jats:tex-math>iin I</jats:tex-math> </jats:alternatives> </jats:inline-formula> and 𝜎-primary if 𝐺 is a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>σ</m:mi> <m:mi>i</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2024-0012_ineq_0002.png\"/> <jats:tex-math>sigma_{i}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-group for some 𝑖. In addition, 𝐺 is 𝜎-soluble if every chief factor of 𝐺 is 𝜎-primary and 𝜎-nilpotent if 𝐺 is a direct product of 𝜎-primary groups. We write <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>G</m:mi> <m:msub> <m:mi mathvariant=\"fraktur\">N</m:mi> <m:mi>σ</m:mi> </m:msub> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2024-0012_ineq_0005.png\"/> <jats:tex-math>G^{mathfrak{N}_{sigma}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> for the 𝜎-nilpotent residual of 𝐺, which is the intersection of all normal subgroups 𝑁 of 𝐺 with 𝜎-nilpotent <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>N</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2024-0012_ineq_0006.png\"/> <jats:tex-math>G/N</jats:tex-math> </jats:alternatives> </jats:inline-formula>. A subgroup 𝐴 of 𝐺 is said to be 𝜎-permutable in 𝐺 p","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217115","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":"The commuting graph of a solvable 𝐴-group","authors":"Rachel Carleton, Mark L. Lewis","doi":"10.1515/jgth-2023-0076","DOIUrl":"https://doi.org/10.1515/jgth-2023-0076","url":null,"abstract":"Let 𝐺 be a finite group. Recall that an 𝐴-group is a group whose Sylow subgroups are all abelian. In this paper, we investigate the upper bound on the diameter of the commuting graph of a solvable 𝐴-group. Assuming that the commuting graph is connected, we show when the derived length of 𝐺 is 2, the diameter of the commuting graph will be at most 4. In the general case, we show that the diameter of the commuting graph will be at most 6. In both cases, examples are provided to show that the upper bound of the commuting graph cannot be improved.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217116","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":"Root cycles in Coxeter groups","authors":"Sarah Hart, Veronica Kelsey, Peter Rowley","doi":"10.1515/jgth-2023-0027","DOIUrl":"https://doi.org/10.1515/jgth-2023-0027","url":null,"abstract":"For an element 𝑤 of a Coxeter group 𝑊, there are two important attributes, namely its length, and its expression as a product of disjoint cycles in its action on Φ, the root system of 𝑊. This paper investigates the interaction between these two features of 𝑤, introducing the notion of the crossing number of 𝑤, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>κ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>w</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-2023-0027_ineq_0001.png\"/> <jats:tex-math>kappa(w)</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Writing <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>w</m:mi> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi>c</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo></m:mo> <m:mi mathvariant=\"normal\">⋯</m:mi> <m:mo></m:mo> <m:msub> <m:mi>c</m:mi> <m:mi>r</m:mi> </m:msub> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0027_ineq_0002.png\"/> <jats:tex-math>w=c_{1}cdots c_{r}</jats:tex-math> </jats:alternatives> </jats:inline-formula> as a product of disjoint cycles, we associate to each cycle <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>c</m:mi> <m:mi>i</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0027_ineq_0003.png\"/> <jats:tex-math>c_{i}</jats:tex-math> </jats:alternatives> </jats:inline-formula> a “crossing number” <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>κ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msub> <m:mi>c</m:mi> <m:mi>i</m:mi> </m:msub> <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-2023-0027_ineq_0004.png\"/> <jats:tex-math>kappa(c_{i})</jats:tex-math> </jats:alternatives> </jats:inline-formula>, which is the number of positive roots 𝛼 in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>c</m:mi> <m:mi>i</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0027_ineq_0003.png\"/> <jats:tex-math>c_{i}</jats:tex-math> </jats:alternatives> </jats:inline-formula> for which <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>w</m:mi> <m:mo lspace=\"0.222em\" rspace=\"0.222em\">⋅</m:mo> <m:mi>α</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0027_ineq_0006.png\"/> <jats:tex-m","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217129","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":"Reflection length at infinity in hyperbolic reflection groups","authors":"Marco Lotz","doi":"10.1515/jgth-2023-0073","DOIUrl":"https://doi.org/10.1515/jgth-2023-0073","url":null,"abstract":"In a discrete group generated by hyperplane reflections in the 𝑛-dimensional hyperbolic space, the reflection length of an element is the minimal number of hyperplane reflections in the group that suffices to factor the element. For a Coxeter group that arises in this way and does not split into a direct product of spherical and affine reflection groups, the reflection length is unbounded. The action of the Coxeter group induces a tessellation of the hyperbolic space. After fixing a fundamental domain, there exists a bijection between the tiles and the group elements. We describe certain points in the visual boundary of the 𝑛-dimensional hyperbolic space for which every neighbourhood contains tiles of every reflection length. To prove this, we show that two disjoint hyperplanes in the 𝑛-dimensional hyperbolic space without common boundary points have a unique common perpendicular.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141781166","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":"Relative homology of arithmetic subgroups of SU(3)","authors":"Claudio Bravo","doi":"10.1515/jgth-2023-0140","DOIUrl":"https://doi.org/10.1515/jgth-2023-0140","url":null,"abstract":"Let 𝑘 be a global field of positive characteristic. Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"script\">G</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mi>SU</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mn>3</m:mn> <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-2023-0140_ineq_0001.png\"/> <jats:tex-math>mathcal{G}=mathrm{SU}(3)</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the non-split group scheme defined from an (isotropic) hermitian form in three variables. In this work, we describe, in terms of the Euler–Poincaré characteristic, the relative homology groups of certain arithmetic subgroups 𝐺 of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"script\">G</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>k</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-2023-0140_ineq_0002.png\"/> <jats:tex-math>mathcal{G}(k)</jats:tex-math> </jats:alternatives> </jats:inline-formula> modulo a representative system 𝔘 of the conjugacy classes of their maximal unipotent subgroups. In other words, we measure how far the homology groups of 𝐺 are from being the coproducts of the corresponding homology groups of the subgroups <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>U</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant=\"fraktur\">U</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0140_ineq_0003.png\"/> <jats:tex-math>Uinmathfrak{U}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141568007","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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