We show that if 𝐺 is an admissible group acting geometrically on a CAT(0)operatorname{CAT}(0) space 𝑋, then 𝐺 is a hierarchically hyperbolic space and its 𝜅-Morse boundary (∂κG,ν)(partial_{kappa}G,nu) is a model for the Poisson boundary of (G,μ)(G,mu), where 𝜈 is the hitting measure associated to the random walk driven by 𝜇.
{"title":"Sublinearly Morse boundary of CAT(0) admissible groups","authors":"Hoang Thanh Nguyen, Yulan Qing","doi":"10.1515/jgth-2023-0145","DOIUrl":"https://doi.org/10.1515/jgth-2023-0145","url":null,"abstract":"We show that if 𝐺 is an admissible group acting geometrically on a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>CAT</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mn>0</m:mn> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0145_ineq_0001.png\" /> <jats:tex-math>operatorname{CAT}(0)</jats:tex-math> </jats:alternatives> </jats:inline-formula> space 𝑋, then 𝐺 is a hierarchically hyperbolic space and its 𝜅-Morse boundary <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:msub> <m:mo lspace=\"0em\" rspace=\"0em\">∂</m:mo> <m:mi>κ</m:mi> </m:msub> <m:mi>G</m:mi> </m:mrow> <m:mo>,</m:mo> <m:mi>ν</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0145_ineq_0002.png\" /> <jats:tex-math>(partial_{kappa}G,nu)</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a model for the Poisson boundary of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>G</m:mi> <m:mo>,</m:mo> <m:mi>μ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0145_ineq_0003.png\" /> <jats:tex-math>(G,mu)</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where 𝜈 is the hitting measure associated to the random walk driven by 𝜇.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":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}
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.
{"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}
A bi-order on a group 𝐺 is a total, bi-multiplication invariant order. A subset 𝑆 in an ordered group (G,⩽)(G,leqslant) is convex if, for all f⩽gfleqslant g in 𝑆, every element h∈Ghin G satisfying f⩽h⩽gfleqslant hleqslant g 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.
群𝐺上的双阶是一个总的双乘法不变阶。一个有序群(G，≤)(G, leqslant)中的子集𝑆是凸的，如果对于𝑆中的所有f≤G≤leqslant G，每个元素h∈G h in G满足f≤h≤G≤leqslant h leqslant G属于𝑆。在本文中，我们证明了秩为2的自由亚丫群的派生子群对于任意双阶是凸的。此外，我们还研究了一类高秩自由亚元群的派生子群的凸包。作为一个应用，证明了有限秩非阿贝尔自由亚贝尔群的双阶空间与康托尔集是同胚的。此外，我们证明了这些组的双序不能被常规语言识别。
{"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}
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, Ann. Inst. Fourier (Grenoble) 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 VTnmathrm{VT}_{n} on n≥2ngeq 2 strands. This is achieved by showing the existence of an irreducible right-angled Coxeter group KTnmathrm{KT}_{n} inside VTnmathrm{VT}_{n}. As a by-product, it also follows that the twin group Tnmathrm{T}_{n} embeds inside the virtual twin group VTn
孪生群和虚拟孪生群分别是编织群和虚拟编织群的平面类似物。这些群在可定向表面上沉圆稳定同位素类理论的Alexander-Markov对应中起辫群的作用。受马丁的总体思想和贝林盖里和帕里斯最近的工作的启发[P。《虚拟辫子和排列》，安。Inst. Fourier (Grenoble) 70(2020)， 3,1341 - 1362]，我们得到了虚拟双胞胎群与对称群之间同态的完整描述，作为应用，我们得到了虚拟双胞胎群VT n mathrm{VT} ＿n在n≥2 n {}geq 2链上的自同态群的精确结构。这是通过证明在VT n mathrm{VT} ＿n内存在一个不可约的直角Coxeter群KT {n}mathrm{KT} ＿n{来实现的。作为副产品，还可以得出双胞胎组T n }mathrm{T} ＿n{嵌入到虚拟双胞胎组VT n }mathrm{VT} ＿n{中，这与编织组的类似结果类似。}
{"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}
To a strongly 𝐺-graded algebra 𝐴 with 1-component 𝐵, we associate the group Picentgr(A)mathrm{Picent}^{mathrm{gr}}(A) of isomorphism classes of invertible 𝐺-graded (A,A)(A,A)-bimodules over the centralizer of 𝐵 in 𝐴. Our main result is a Picentmathrm{Picent} version of the Beattie–del Río exact sequence, involving Dade’s group G[B]G[B], which relates Picentgr(A)mathrm{Picent}^{mathrm{gr}}(A), Picent(B)mathrm{Picent}(B)
对于一个具有1组分变量的强𝐺-graded代数变量变量，我们将可逆的𝐺-graded (a, a) (a, a) -双模的同构类的群Picent gr²(a) mathm {Picent}^{ mathm {gr}}(a)关联到变量的中心化算子上。我们的主要结果是Beattie-del Río精确序列的一个Picent mathm {Picent}版本，涉及Dade的群G≠[B] G[B]，它涉及到Picent gr (a) mathm {Picent}^{ mathm {gr}}(a)， Picent≠(B) mathm {Picent}(B)和群上同调。
{"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}
Ammu E. Antony, Sathasivam Kalithasan, Viji Z. Thomas
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 G/Z∧(G)≅H/Z∧(H)G/Z^{wedge}(G)cong H/Z^{wedge}(H), 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.
引入了𝑞-Bogomolov乘子作为Bogomolov乘子的推广，并证明了它在𝑞-isoclinism下是不变的。我们证明了𝑞-Schur乘子在𝑞-exterior等斜下是不变的，作为一个简单的结果，我们证明了Schur乘子在外等斜下是不变的。我们还证明了如果𝐺和𝐻是𝑝-groups with G/Z∧∧(G) = H/Z∧∧(H) G/Z^{wedge}(G)cong H/Z^{wedge}(H)，则𝐺和𝐻的最小生成器数的基数是相同的。此外，我们还证明了群的非阿贝尔𝑞-tensor平方的一些结构结果。
{"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}
A-Ming Liu, Mingzhu Chen, Inna N. Safonova, Alexander N. Skiba
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.
设为质数集的一个划分。在本文中，我们描述了每个𝜎-subnormal子群都是模的有限群。
{"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}
Abstract Consider the Macdonald groups G(α)=⟨A,B∣A[A,B]=Aα,B[B,A]=Bα⟩ G(alpha)=langle A,Bmid A^{[A,B]}=A^{alpha},,B^{[B,A]}=B^{alpha}rangle , α∈Z alphainmathbb{Z} . We fill a gap in Macdonald’s proof that G(α) G(alpha) is always nilpotent, and proceed to determine the order, upper and lower central series, nilpotency class, and exponent of G(α) G(alpha) .
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.
{"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}
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.
{"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}
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