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 w(G)w(G) 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.
众所周知，研究一个词组中的言语亚群是获取词组结构信息的有效工具。因此，对自由组中的词语进行分类的条件至关重要。研究最多的问题之一是确定哪些词是简明词，如果在每个𝑤取值有限的群𝐺中，言语子群 w ( G ) w(G) 都是有限的，则称𝑤为简明词。本文的目的是介绍一些结果，其中引入了词的层次结构，概括了简洁词的概念。
{"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
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
让 σ = { σ i ∣ i ∈ I } is some partition of the set of all primes and 𝐺 a finite group.那么，如果𝐺对所有 i∈I iin I 都有一个 Hall σ i sigma_{i} -子群，则称𝐺为𝜎-full；如果𝐺对某个 𝑖来说是一个 σ i sigma_{i} -群，则称𝐺为𝜎-primary。此外，如果𝐺 的每个主因子都是𝜎-初等群，则𝐺 是𝜎-可溶的；如果𝐺 是𝜎-初等群的直积，则𝐺 是𝜎-无穷群。我们用 G N σ G^{mathfrak{N}_{sigma} 来表示𝐺的𝜎-零potent 残差，它是𝐺的所有正则子群𝑁 与𝜎-零potent G / N G/N 的交集。如果𝐺是满的，并且𝐺与𝐺的所有霍尔σ i sigma_{i} -子群𝐻（即 A H = H AH=HA ）对于所有𝑖都是包络的，那么𝐺的子群𝐴在𝐺中就被称为是可𝜎包络的。如果存在一个子群链 A = A 0 ≤ A 1 ≤ ⋯ ≤ A n = G A=A_{0}leq A_{1}leq A_{1}cdotsleq A_{n}=G，则𝐴在𝐺中是𝜎-次正态的，这样，要么 A i - 1 ⊴ A i A_{i-1}trianglelefteq A_{i} 或 A i / ( A i - 1 ) A i A_{i}/(A_{i-1})_{A_{i}} 对于所有 i = 1 、..., n i=1,ldots,n 。我们证明，如果𝐺是一个𝜎可溶群、那么当且仅当 G N σ ∩ A G = G N σ ∩ A G G^{mathfrak{N}_{sigma}}cap A^{G}=G^{mathfrak{N}_{sigma}}cap A_{G} 对于每一个𝜎-的子正常子群𝐴。
{"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}
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.
{"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}
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
对于科赛特群 𝑤 的元素𝑤 来说，有两个重要的属性，即它的长度和它在 Φ（即 𝑤 的根系统）上的作用中作为不相交循环的乘积的表达式。本文研究了𝑤 的这两个特征之间的相互作用，引入了𝑤 的交叉数概念，即 κ ( w ) kappa(w) 。把 w = c 1 ⋯ c r w=c_{1}cdots c_{r} 写成不相交循环的乘积，我们给每个循环 c i c_{i} 关联一个 "交叉数" κ ( c i ) kappa(c_{i}) ，它是 w ⋅ α wcdotalpha 为负数的 c i c_{i} 中𝛼 的正根的个数。让 Seq κ ( w ) {mathrm{Seq}}_{kappa}({w}) 是 κ ( c i ) kappa(c_{i})按递增顺序写成的序列，让 κ ( w ) = max Seq κ ( w ) kappa(w)=max{mathrm{Seq}}_{kappa}({w})。𝑤的长度可以从这个序列中获取，但是 Seq κ ( w ) {mathrm{Seq}}_{kappa}({w}) 提供了更多的信息。对于𝑋 的共轭类，让 κ min ( X ) = min { κ ( w ) ∣ w∈ X }。 kappa_min}(X)=min{kappa(w)mid win X} 并让κ ( W ) kappa(W)成为κ min kappa_{min} 在𝑋的所有共轭类中的最大值。我们把 κ ( w ) kappa(w) 和 κ ( W ) kappa(W) 分别称为𝑤和 𝑤的交叉数。在这里，我们确定了所有有限柯克赛特群和所有普遍柯克赛特群的交叉数。我们还证明，对于有限不可还原考克西特群，如果 𝑢 和 𝑣 是同一共轭类 𝑋 中长度最小的两个元素、则 Seq κ ( u ) = Seq κ ( v ) {mathrm{Seq}}_{kappa}({u})={mathrm{Seq}}_{kappa}({v}) κ min ( X ) = κ ( u ) = κ ( v ) kappa_{min}(X)=kappa(u)=kappa(v) .
{"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}
In this paper, we study nilpotent Q[x]mathbb{Q}[x]-powered groups that satisfy the following property: for some set of primes 𝜔 in Q[x]mathbb{Q}[x], every ω′omega^{prime}-isolated Q[x]mathbb{Q}[x]-subgroup in some family of its Q[x]mathbb{Q}[x]-subgroups is finite 𝜔-type separable.
本文研究满足以下性质的无幂 Q [ x ] mathbb{Q}[x] 有幂群：对于 Q [ x ] mathbb{Q}[x]中的某个prime 細集，其 Q [ x ] mathbb{Q}[x]-子群的某个族中的每Ω ′ omega^{prime} -隔离的 Q [ x ] mathbb{Q}[x]-子群都是有限娀型可分离的。
{"title":"Separability properties of nilpotent ℚ[𝑥]-powered groups II","authors":"Stephen Majewicz, Marcos Zyman","doi":"10.1515/jgth-2023-0288","DOIUrl":"https://doi.org/10.1515/jgth-2023-0288","url":null,"abstract":"In this paper, we study nilpotent <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"double-struck\">Q</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:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0288_ineq_0001.png\"/> <jats:tex-math>mathbb{Q}[x]</jats:tex-math> </jats:alternatives> </jats:inline-formula>-powered groups that satisfy the following property: for some set of primes 𝜔 in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"double-struck\">Q</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:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0288_ineq_0001.png\"/> <jats:tex-math>mathbb{Q}[x]</jats:tex-math> </jats:alternatives> </jats:inline-formula>, every <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>ω</m:mi> <m:mo>′</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0288_ineq_0003.png\"/> <jats:tex-math>omega^{prime}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-isolated <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"double-struck\">Q</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:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0288_ineq_0001.png\"/> <jats:tex-math>mathbb{Q}[x]</jats:tex-math> </jats:alternatives> </jats:inline-formula>-subgroup in some family of its <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"double-struck\">Q</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:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0288_ineq_0001.png\"/> <jats:tex-math>mathbb{Q}[x]</jats:tex-math> </jats:alternatives> </jats:inline-formula>-subgroups is finite 𝜔-type separable.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947942","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 well-known conjecture of Gluck claims that |G:F(G)|≤b(G)2lvert G:mathbf{F}(G)rvertleq b(G)^{2} for all finite solvable groups 𝐺, where F(G)mathbf{F}(G) is the Fitting subgroup and b(G)b(G) is the largest degree of a complex irreducible character of 𝐺. In this paper, we prove that Gluck’s conjecture holds for all wreath product type groups of the form G≀H1≀H2≀⋯≀HrGwr H_{1}wr H_{2}wrcdotswr H_{r}, where 𝐺 is a finite solvable group acting primitively on F
Gluck 的一个著名猜想声称 | G : F ( G ) | ≤ b ( G ) 2 lvert G:mathbf{F}(G)rvertleq b(G)^{2} 适用于所有有限可解群𝐺，其中 F ( G ) mathbf{F}(G) 是 Fitting 子群，而 b ( G ) b(G) 是𝐺 的复不可约特征的最大度数。在本文中，我们将证明格鲁克猜想对于所有形式为 G ≀ H 1 ≀ H 2 ⋯ ≀ H r Gwr H_{1}wr H_{2}wrcdotswr H_{r} 的花环积类型群都成立、其中，𝐺 是有限可解的群，原始地作用于 F ( G ) / Φ ( G ) mathbf{F}(G)/Phi(G) , 每个 H i H_{i} 是有限度的可解原始置换群。
{"title":"On Gluck’s conjecture for wreath product type groups","authors":"Hangyang Meng, Xiuyun Guo","doi":"10.1515/jgth-2024-0042","DOIUrl":"https://doi.org/10.1515/jgth-2024-0042","url":null,"abstract":"A well-known conjecture of Gluck claims that <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mo stretchy=\"false\">|</m:mo> <m:mi>G</m:mi> <m:mo lspace=\"0.278em\" rspace=\"0.278em\">:</m:mo> <m:mi mathvariant=\"bold\">F</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo stretchy=\"false\">|</m:mo> </m:mrow> <m:mo>≤</m:mo> <m:mi>b</m:mi> <m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2024-0042_ineq_0001.png\"/> <jats:tex-math>lvert G:mathbf{F}(G)rvertleq b(G)^{2}</jats:tex-math> </jats:alternatives> </jats:inline-formula> for all finite solvable groups 𝐺, where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"bold\">F</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-2024-0042_ineq_0002.png\"/> <jats:tex-math>mathbf{F}(G)</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the Fitting subgroup and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>b</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-2024-0042_ineq_0003.png\"/> <jats:tex-math>b(G)</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the largest degree of a complex irreducible character of 𝐺. In this paper, we prove that Gluck’s conjecture holds for all wreath product type groups of the form <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 lspace=\"0.222em\" rspace=\"0.222em\">≀</m:mo> <m:msub> <m:mi>H</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo lspace=\"0.222em\" rspace=\"0.222em\">≀</m:mo> <m:msub> <m:mi>H</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo lspace=\"0.222em\" rspace=\"0.222em\">≀</m:mo> <m:mi mathvariant=\"normal\">⋯</m:mi> <m:mo lspace=\"0.222em\" rspace=\"0.222em\">≀</m:mo> <m:msub> <m:mi>H</m:mi> <m:mi>r</m:mi> </m:msub> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2024-0042_ineq_0004.png\"/> <jats:tex-math>Gwr H_{1}wr H_{2}wrcdotswr H_{r}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where 𝐺 is a finite solvable group acting primitively on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mrow> <m:mi mathvariant=\"bold\">F</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stre","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947943","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}
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.
{"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}
Let 𝑘 be a global field of positive characteristic. Let G=SU(3)mathcal{G}=mathrm{SU}(3) 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 G(k)mathcal{G}(k) 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 U∈UUinmathfrak{U}.
让 𝑘 是一个正特征的全局域。让 G = SU ( 3 ) （mathcal{G}=mathrm{SU}(3) 是由三变量（各向同性）赫米特形式定义的非分裂群方案。在这项工作中，我们用欧拉-庞加莱特征来描述 G ( k ) mathcal{G}(k)的某些算术子群𝐺 modulo a representative system 𝔘 of the conjugacy classes of their maximal unipotent subgroups 的相对同调群。换句话说，我们测量的是𝐺 的同调群距离子群 U∈Uinmathfrak{U} 的相应同调群的共轭类有多远。
{"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}
For n≥2ngeq 2, let G1=A1∗⋯∗AnG_{1}=A_{1}astdotsast A_{n} and G2=B1∗⋯∗BnG_{2}=B_{1}astdotsast B_{n} where the AiA_{i}’s and BiB_{i}’s are non-elementary relatively hyperbolic groups. Suppose that, for 1≤i≤n1leq ileq n, the Bowditch boundary of
对于 n ≥ 2 ngeq 2 、让 G 1 = A 1 ∗ ⋯ ∗ A n G_{1}=A_{1}astdotsast A_{n} 和 G 2 = B 1 ∗ ⋯ ∗ B n G_{2}=B_{1}astdotsast B_{n} 其中 A i A_{i} 's 和 B i B_{i} 's 是非元素相对双曲群。假设对于 1 ≤ i ≤ n 1leq ileq n ，A i A_{i} 的鲍迪奇边界与 B i B_{i} 的鲍迪奇边界同构。我们证明 G 1 G_{1} 的鲍迪奇边界与 G 2 G_{2} 的鲍迪奇边界同构。我们将这一结果推广到具有有限边群的相对双曲群图。这扩展了马丁-西里ą托克斯基在相对双曲群背景下的工作。
We describe the structure of virtually solvable normal subgroups in the automorphism group of a right-angled Artin group Aut(AΓ)mathrm{Aut}(A_{Gamma}). In particular, we prove that a finite normal subgroup in Aut(AΓ)mathrm{Aut}(A_{Gamma}) has at most order two and if Γ is not a clique, then any finite normal subgroup in Aut(AΓ)mathrm{Aut}(A_{Gamma}) is trivial. This property has implications for automatic continuity and C∗C^{ast}-algebras: every algebraic epimorphism φ:L↠Aut(AΓ)varphicolon Ltwoheadrightarrowmathrm{Aut}(A_{Gamma})
我们描述了直角阿汀群 Aut ( A Γ ) mathrm{Aut}(A_{Gamma}) 自变群中的可解正则子群的结构。我们特别证明了 Aut ( A Γ ) mathrm{Aut}(A_{Gamma})中的有限正则子群最多有二阶，并且如果 Γ 不是一个簇，那么 Aut ( A Γ ) mathrm{Aut}(A_{Gamma})中的任何有限正则子群都是微不足道的。这一性质对自动连续性和 C ∗ C^{ast} - 算法都有影响：每个代数外显 φ ： L ↠ Aut ( A Γ ) varphicolon Ltwoheadrightarrowmathrm{Aut}(A_{Gamma}) from a locally compact Hausdorff group 𝐿 is continuous if and only if A Γ A_{Gamma} is not isomorphic to Z n mathbb{Z}^{n} for any n ≥ 1 ngeq 1.此外，如果 Γ 不是连接且至少包含两个顶点，那么可逆元素集在 Aut ( A Γ ) 的还原群 C∗ C^{ast} -代数中是密集的。我们对 Aut ( G Γ ) mathrm{Aut}(G_{Gamma})也得到了类似的结果，其中 G Γ G_{Gamma} 是循环群的图积。此外，我们用定义图形 Γ 来描述 Aut ( G Γ ) mathrm{Aut}(G_{Gamma})的中心。
{"title":"On normal subgroups in automorphism groups","authors":"Philip Möller, Olga Varghese","doi":"10.1515/jgth-2023-0089","DOIUrl":"https://doi.org/10.1515/jgth-2023-0089","url":null,"abstract":"We describe the structure of virtually solvable normal subgroups in the automorphism group of a right-angled Artin group <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>Aut</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msub> <m:mi>A</m:mi> <m:mi mathvariant=\"normal\">Γ</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-2023-0089_ineq_0001.png\"/> <jats:tex-math>mathrm{Aut}(A_{Gamma})</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In particular, we prove that a finite normal subgroup in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>Aut</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msub> <m:mi>A</m:mi> <m:mi mathvariant=\"normal\">Γ</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-2023-0089_ineq_0001.png\"/> <jats:tex-math>mathrm{Aut}(A_{Gamma})</jats:tex-math> </jats:alternatives> </jats:inline-formula> has at most order two and if Γ is not a clique, then any finite normal subgroup in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>Aut</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msub> <m:mi>A</m:mi> <m:mi mathvariant=\"normal\">Γ</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-2023-0089_ineq_0001.png\"/> <jats:tex-math>mathrm{Aut}(A_{Gamma})</jats:tex-math> </jats:alternatives> </jats:inline-formula> is trivial. This property has implications for automatic continuity and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>C</m:mi> <m:mo>∗</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0089_ineq_0004.png\"/> <jats:tex-math>C^{ast}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-algebras: every algebraic epimorphism <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>φ</m:mi> <m:mo lspace=\"0.278em\" rspace=\"0.278em\">:</m:mo> <m:mrow> <m:mi>L</m:mi> <m:mo stretchy=\"false\">↠</m:mo> <m:mrow> <m:mi>Aut</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msub> <m:mi>A</m:mi> <m:mi mathvariant=\"normal\">Γ</m:mi> </m:msub> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0089_ineq_0005.png\"/> <jats:tex-math>varphicolon Ltwoheadrightarrowmathrm{Aut}(A_{Gamma})</jats:tex-math> </jats:alternatives> </jats:inlin","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141530296","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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