Journal of Group Theory最新文献

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

{"title":"Polynomial maps and polynomial sequences in groups","authors":"Ya-Qing Hu","doi":"10.1515/jgth-2023-0051","DOIUrl":"https://doi.org/10.1515/jgth-2023-0051","url":null,"abstract":"This paper presents a modified version of Leibman’s group-theoretic generalizations of the difference calculus for polynomial maps from nonempty commutative semigroups to groups, and proves that it has many desirable formal properties when the target group is locally nilpotent and also when the semigroup is the set of nonnegative integers. We will apply it to solve Waring’s problem for general discrete Heisenberg groups in a sequel to this paper.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139476592","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":"Orders on free metabelian groups","authors":"Wenhao Wang","doi":"10.1515/jgth-2022-0203","DOIUrl":"https://doi.org/10.1515/jgth-2022-0203","url":null,"abstract":"A bi-order on a group 𝐺 is a total, bi-multiplication invariant order. A subset 𝑆 in an ordered group <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:mo lspace=\"0em\" rspace=\"0em\">⩽</m:mo> <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-2022-0203_ineq_0001.png\" /> <jats:tex-math>(G,leqslant)</jats:tex-math> </jats:alternatives> </jats:inline-formula> is convex if, for all <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>f</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-2022-0203_ineq_0002.png\" /> <jats:tex-math>fleqslant g</jats:tex-math> </jats:alternatives> </jats:inline-formula> in 𝑆, every element <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>h</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-2022-0203_ineq_0003.png\" /> <jats:tex-math>hin G</jats:tex-math> </jats:alternatives> </jats:inline-formula> satisfying <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>f</m:mi> <m:mo>⩽</m:mo> <m:mi>h</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-2022-0203_ineq_0004.png\" /> <jats:tex-math>fleqslant hleqslant g</jats:tex-math> </jats:alternatives> </jats:inline-formula> belongs to 𝑆. In this paper, we show that the derived subgroup of the free metabelian group of rank 2 is convex with respect to any bi-order. Moreover, we study the convex hull of the derived subgroup of a free metabelian group of higher rank. As an application, we prove that the space of bi-orders of a non-abelian free metabelian group of finite rank is homeomorphic to the Cantor set. In addition, we show that no bi-order for these groups can be recognised by a regular language.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138504132","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":"Virtual planar braid groups and permutations","authors":"Tushar Kanta Naik, Neha Nanda, Mahender Singh","doi":"10.1515/jgth-2023-0010","DOIUrl":"https://doi.org/10.1515/jgth-2023-0010","url":null,"abstract":"Twin groups and virtual twin groups are planar analogues of braid groups and virtual braid groups, respectively. These groups play the role of braid groups in the Alexander–Markov correspondence for the theory of stable isotopy classes of immersed circles on orientable surfaces. Motivated by the general idea of Artin and recent work of Bellingeri and Paris [P. Bellingeri and L. Paris, Virtual braids and permutations, <jats:italic>Ann. Inst. Fourier (Grenoble)</jats:italic> 70 (2020), 3, 1341–1362], we obtain a complete description of homomorphisms between virtual twin groups and symmetric groups, which as an application gives us the precise structure of the automorphism group of the virtual twin group <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>VT</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0010_ineq_0001.png\" /> <jats:tex-math>mathrm{VT}_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> on <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:mn>2</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0010_ineq_0002.png\" /> <jats:tex-math>ngeq 2</jats:tex-math> </jats:alternatives> </jats:inline-formula> strands. This is achieved by showing the existence of an irreducible right-angled Coxeter group <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>KT</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0010_ineq_0003.png\" /> <jats:tex-math>mathrm{KT}_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> inside <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>VT</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0010_ineq_0001.png\" /> <jats:tex-math>mathrm{VT}_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. As a by-product, it also follows that the twin group <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"normal\">T</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0010_ineq_0005.png\" /> <jats:tex-math>mathrm{T}_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> embeds inside the virtual twin group <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>VT</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138504131","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":"An exact sequence for the graded Picent","authors":"Andrei Marcus, Virgilius-Aurelian Minuță","doi":"10.1515/jgth-2023-0040","DOIUrl":"https://doi.org/10.1515/jgth-2023-0040","url":null,"abstract":"To a strongly 𝐺-graded algebra 𝐴 with 1-component 𝐵, we associate the group <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>Picent</m:mi> <m:mi>gr</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>A</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-2023-0040_ineq_0001.png\" /> <jats:tex-math>mathrm{Picent}^{mathrm{gr}}(A)</jats:tex-math> </jats:alternatives> </jats:inline-formula> of isomorphism classes of invertible 𝐺-graded <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>A</m:mi> <m:mo>,</m:mo> <m:mi>A</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-2023-0040_ineq_0002.png\" /> <jats:tex-math>(A,A)</jats:tex-math> </jats:alternatives> </jats:inline-formula>-bimodules over the centralizer of 𝐵 in 𝐴. Our main result is a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>Picent</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0040_ineq_0003.png\" /> <jats:tex-math>mathrm{Picent}</jats:tex-math> </jats:alternatives> </jats:inline-formula> version of the Beattie–del Río exact sequence, involving Dade’s 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:mo stretchy=\"false\">[</m:mo> <m:mi>B</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-2023-0040_ineq_0004.png\" /> <jats:tex-math>G[B]</jats:tex-math> </jats:alternatives> </jats:inline-formula>, which relates <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>Picent</m:mi> <m:mi>gr</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>A</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-2023-0040_ineq_0001.png\" /> <jats:tex-math>mathrm{Picent}^{mathrm{gr}}(A)</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>Picent</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>B</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-2023-0040_ineq_0006.png\" /> <jats:tex-math>mathrm{Picent}(B)</jats:tex-math> </jats:alternatives> </jats:inline-formu","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138504129","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":"Invariance of the Schur multiplier, the Bogomolov multiplier and the minimal number of generators under a variant of isoclinism","authors":"Ammu E. Antony, Sathasivam Kalithasan, Viji Z. Thomas","doi":"10.1515/jgth-2023-0066","DOIUrl":"https://doi.org/10.1515/jgth-2023-0066","url":null,"abstract":"We introduce the 𝑞-Bogomolov multiplier as a generalization of the Bogomolov multiplier, and we prove that it is invariant under 𝑞-isoclinism. We prove that the 𝑞-Schur multiplier is invariant under 𝑞-exterior isoclinism, and as an easy consequence, we prove that the Schur multiplier is invariant under exterior isoclinism. We also prove that if 𝐺 and 𝐻 are 𝑝-groups with <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:msup> <m:mi>Z</m:mi> <m:mo>∧</m:mo> </m:msup> </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:mo>≅</m:mo> <m:mrow> <m:mrow> <m:mi>H</m:mi> <m:mo>/</m:mo> <m:msup> <m:mi>Z</m:mi> <m:mo>∧</m:mo> </m:msup> </m:mrow> <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:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0066_ineq_0001.png\" /> <jats:tex-math>G/Z^{wedge}(G)cong H/Z^{wedge}(H)</jats:tex-math> </jats:alternatives> </jats:inline-formula>, then the cardinalities of the minimal number of generators of 𝐺 and 𝐻 are the same. Moreover, we prove some structural results about non-abelian 𝑞-tensor square of groups.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138504128","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 modular 𝜎-subnormal subgroups","authors":"A-Ming Liu, Mingzhu Chen, Inna N. Safonova, Alexander N. Skiba","doi":"10.1515/jgth-2023-0064","DOIUrl":"https://doi.org/10.1515/jgth-2023-0064","url":null,"abstract":"Let 𝜎 be a partition of the set of prime numbers. In this paper, we describe the finite groups for which every 𝜎-subnormal subgroup is modular.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138516835","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":"Structure of the Macdonald groups in one parameter","authors":"Alexander Montoya Ocampo, Fernando Szechtman","doi":"10.1515/jgth-2023-0036","DOIUrl":"https://doi.org/10.1515/jgth-2023-0036","url":null,"abstract":"Abstract Consider the Macdonald groups <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:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>α</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy=\"false\">⟨</m:mo> <m:mrow> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>B</m:mi> </m:mrow> <m:mo fence=\"true\" lspace=\"0em\" rspace=\"0em\">∣</m:mo> <m:mrow> <m:mrow> <m:msup> <m:mi>A</m:mi> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>B</m:mi> <m:mo stretchy=\"false\">]</m:mo> </m:mrow> </m:msup> <m:mo>=</m:mo> <m:msup> <m:mi>A</m:mi> <m:mi>α</m:mi> </m:msup> </m:mrow> <m:mo rspace=\"0.337em\">,</m:mo> <m:mrow> <m:msup> <m:mi>B</m:mi> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:mi>B</m:mi> <m:mo>,</m:mo> <m:mi>A</m:mi> <m:mo stretchy=\"false\">]</m:mo> </m:mrow> </m:msup> <m:mo>=</m:mo> <m:msup> <m:mi>B</m:mi> <m:mi>α</m:mi> </m:msup> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">⟩</m:mo> </m:mrow> </m:mrow> </m:math> G(alpha)=langle A,Bmid A^{[A,B]}=A^{alpha},,B^{[B,A]}=B^{alpha}rangle , <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>α</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant=\"double-struck\">Z</m:mi> </m:mrow> </m:math> alphainmathbb{Z} . We fill a gap in Macdonald’s proof that <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:mo stretchy=\"false\">(</m:mo> <m:mi>α</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> G(alpha) is always nilpotent, and proceed to determine the order, upper and lower central series, nilpotency class, and exponent of <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:mo stretchy=\"false\">(</m:mo> <m:mi>α</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> G(alpha) .","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135340461","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 Reidemeister spectrum of direct products of nilpotent groups","authors":"Pieter Senden","doi":"10.1515/jgth-2022-0159","DOIUrl":"https://doi.org/10.1515/jgth-2022-0159","url":null,"abstract":"Abstract We investigate the Reidemeister spectrum of direct products of nilpotent groups. More specifically, we prove that the Reidemeister spectra of the individual factors yield complete information for the Reidemeister spectrum of the direct product if all groups are finitely generated torsion-free nilpotent and have a directly indecomposable rational Malcev completion. We show this by determining the complete automorphism group of the direct product.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135875707","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":"Narrow normal subgroups of Coxeter groups and of automorphism groups of Coxeter groups","authors":"Luis Paris, Olga Varghese","doi":"10.1515/jgth-2022-0202","DOIUrl":"https://doi.org/10.1515/jgth-2022-0202","url":null,"abstract":"Abstract By definition, a group is called narrow if it does not contain a copy of a non-abelian free group. We describe the structure of finite and narrow normal subgroups in Coxeter groups and their automorphism groups.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135666885","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}