For k,ℓ∈Nk,ellinmathbb{N}, we introduce the concepts of 𝑘-ultrahomogeneity and ℓ-tuple regularity for finite groups. Inspired by analogous concepts in graph theory, these form a natural generalization of homogeneity, which was studied by Cherlin and Felgner as well as Li, and automorphism transitivity, which was investigated by Zhang. Additionally, these groups have an interesting algorithmic interpretation. We classify the 𝑘-ultrahomogeneous and ℓ-tuple regular finite groups for k,ℓ≥2k,ellgeq 2. In particular, we show that every 2-tuple regular finite group is ultrahomogeneous.
对于 k , ℓ ∈ N k,ellinmathbb{N}，我们引入了有限群的𝑘-超同质性和ℓ-元组正则性的概念。这些概念受到图论中类似概念的启发，是对 Cherlin 和 Felgner 以及 Li 所研究的同质性和 Zhang 所研究的自动反常性的自然概括。此外，这些群还具有有趣的算法解释。我们对 k , ℓ ≥ 2 k,ellgeq 2 的 𝑘-ultrahomogeneous 和 ℓ-tuple 正则有限群进行了分类。特别是，我们证明了每个 2 元组正则有限群都是超均质的。
{"title":"Tuple regularity and 𝑘-ultrahomogeneity for finite groups","authors":"Sofia Brenner","doi":"10.1515/jgth-2023-0106","DOIUrl":"https://doi.org/10.1515/jgth-2023-0106","url":null,"abstract":"For <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">ℓ</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:mi mathvariant=\"double-struck\">N</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0106_ineq_0001.png\"/> <jats:tex-math>k,ellinmathbb{N}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, we introduce the concepts of 𝑘-ultrahomogeneity and ℓ-tuple regularity for finite groups. Inspired by analogous concepts in graph theory, these form a natural generalization of homogeneity, which was studied by Cherlin and Felgner as well as Li, and automorphism transitivity, which was investigated by Zhang. Additionally, these groups have an interesting algorithmic interpretation. We classify the 𝑘-ultrahomogeneous and ℓ-tuple regular finite groups for <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">ℓ</m:mi> </m:mrow> <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-2023-0106_ineq_0002.png\"/> <jats:tex-math>k,ellgeq 2</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In particular, we show that every 2-tuple regular finite group is ultrahomogeneous.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140830770","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, k(G)k(G) the number of conjugacy classes of 𝐺, and 𝐵 a nilpotent subgroup of 𝐺. In this paper, we prove that |BOπ(G)/Oπ(G)|≤|G|/k(G)lvert BO_{pi}(G)/O_{pi}(G)rvertleqlvert Grvert/k(G) if 𝐺 is solvable and that 157|BOπ(G)/Oπ(G)|≤|G|/k(G)frac{15}{7}lvert BO_{pi}(G)/O_{pi}(G)rvertleqlvert Grvert/k(G)
设𝐺 是一个有限群，k ( G ) k(G) 是𝐺 的共轭类数，𝐵 是𝐺 的一个无穷子群。在本文中我们证明，如果𝐺 是可解的，则 | B O π ( G ) / O π ( G ) | ≤ | G | / k ( G ) lvert BO_{pi}(G)/O_{pi}(G)rvertleqlvert Grvert/k(G) ，而且 15 7 | B O π ( G ) / O π ( G ) | ≤ | G | / k ( G ) frac{15}{7}lvert BO_{pi}(G)/O_{pi}(G)rvertleqlvert Grvert/k(G) if 𝐺 is nonsolvable、其中 π = π ( B ) pi=pi(B) 是 | B |lvert Brvert 的素除数集。这两个边界都是可能的最佳边界。
Suppose 𝐺 is a residually finite group of finite upper rank admitting an automorphism 𝜑 with finite Reidemeister number R(φ)R(varphi) (the number of 𝜑-twisted conjugacy classes). We prove that such a 𝐺 is soluble-by-finite (in other words, any residually finite group of finite upper rank that is not soluble-by-finite has the R∞R_{infty} property). This reduction is the first step in the proof of the second main theorem of the paper: suppose 𝐺 is a residually finite group of finite Prüfer rank and 𝜑 is its automorphism. Then R(φ)R(varphi) (if it is finite) is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations of 𝐺, which are fixed points of the dual map φ̂:[ρ]↦[ρ∘φ]hat{varphi}colon[rho]mapsto[rhocircvarphi] (i.e. we prove the TBFT𝑓, the finite version of the conjecture about the twisted Burnside–Frobenius theorem, for such groups).
{"title":"Twisted conjugacy in residually finite groups of finite Prüfer rank","authors":"Evgenij Troitsky","doi":"10.1515/jgth-2023-0083","DOIUrl":"https://doi.org/10.1515/jgth-2023-0083","url":null,"abstract":"Suppose 𝐺 is a residually finite group of finite upper rank admitting an automorphism 𝜑 with finite Reidemeister number <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>R</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>φ</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-2023-0083_ineq_0001.png\" /> <jats:tex-math>R(varphi)</jats:tex-math> </jats:alternatives> </jats:inline-formula> (the number of 𝜑-twisted conjugacy classes). We prove that such a 𝐺 is soluble-by-finite (in other words, any residually finite group of finite upper rank that is not soluble-by-finite has the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>R</m:mi> <m:mi mathvariant=\"normal\">∞</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0083_ineq_0002.png\" /> <jats:tex-math>R_{infty}</jats:tex-math> </jats:alternatives> </jats:inline-formula> property). This reduction is the first step in the proof of the second main theorem of the paper: suppose 𝐺 is a residually finite group of finite Prüfer rank and 𝜑 is its automorphism. Then <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>R</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>φ</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-2023-0083_ineq_0001.png\" /> <jats:tex-math>R(varphi)</jats:tex-math> </jats:alternatives> </jats:inline-formula> (if it is finite) is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations of 𝐺, which are fixed points of the dual map <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mover accent=\"true\"> <m:mi>φ</m:mi> <m:mo>̂</m:mo> </m:mover> <m:mo lspace=\"0.278em\" rspace=\"0.278em\">:</m:mo> <m:mrow> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:mi>ρ</m:mi> <m:mo stretchy=\"false\">]</m:mo> </m:mrow> <m:mo stretchy=\"false\">↦</m:mo> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:mrow> <m:mi>ρ</m:mi> <m:mo lspace=\"0.222em\" rspace=\"0.222em\">∘</m:mo> <m:mi>φ</m:mi> </m:mrow> <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-2023-0083_ineq_0004.png\" /> <jats:tex-math>hat{varphi}colon[rho]mapsto[rhocircvarphi]</jats:tex-math> </jats:alternatives> </jats:inline-formula> (i.e. we prove the TBFT<jats:sub>𝑓</jats:sub>, the finite version of the conjecture about the twisted Burnside–Frobenius theorem, for such groups).","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140565553","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}
Suppose that w=w(x1,…,xn)w=w(x_{1},ldots,x_{n}) is a word, i.e. an element of the free group F=⟨x1,…,xn⟩F=langle x_{1},ldots,x_{n}rangle. The verbal subgroup w(G)w(G) of a group 𝐺 is the subgroup generated by the set {w(x1,…,xn):x1,…,xn∈G}{w(x_{1},ldots,x_{n}):x_{1},ldots,x_{n}in G} of all 𝑤-values in 𝐺. Following J. González-Sánchez and B. Klopsch, a group 𝐺 is 𝑤-maximal if
假设 w = w ( x 1 , ... , x n ) w=w(x_{1},ldots,x_{n}) 是一个词，即自由群 F = ⟨ x 1 , ... , x n ⟩ F=langle x_{1},ldots,x_{n}rangle 的一个元素。群𝐺的言语子群 w ( G ) w(G) 是由集合 { w ( x 1 , ... , x n ) : x 1 , ... , x n∈ G } 产生的子群。 {w(x_{1},ldots,x_{n}):x_{1},ldots,x_{n}在 G} 中的𝑤值。按照冈萨雷斯-桑切斯（J. González-Sánchez ）和克劳普施（B. Klopsch）的观点，如果| H : w ( H ) | < | G : w ( G ) | lvert H:w(H)rvert<lvert G:w(G)rvert for every H < G H<G，那么群𝐺是𝑤-最大的。本文给出了关于𝑤-最大群的新结果，并研究了前述不等式不严格的较弱条件。本文给出了一些应用：例如，如果一个有限群有一个大小为 𝑛 的可解（或无势）部分，那么它就有一个大小至少为 𝑛 的可解（或无势）子群。
{"title":"On groups with large verbal quotients","authors":"Francesca Lisi, Luca Sabatini","doi":"10.1515/jgth-2023-0088","DOIUrl":"https://doi.org/10.1515/jgth-2023-0088","url":null,"abstract":"Suppose that <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:mi>w</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msub> <m:mi>x</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>x</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0088_ineq_0001.png\" /> <jats:tex-math>w=w(x_{1},ldots,x_{n})</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a word, i.e. an element of the free group <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:mrow> <m:mo stretchy=\"false\">⟨</m:mo> <m:msub> <m:mi>x</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>x</m:mi> <m:mi>n</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-2023-0088_ineq_0002.png\" /> <jats:tex-math>F=langle x_{1},ldots,x_{n}rangle</jats:tex-math> </jats:alternatives> </jats:inline-formula>. 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-2023-0088_ineq_0003.png\" /> <jats:tex-math>w(G)</jats:tex-math> </jats:alternatives> </jats:inline-formula> of a group 𝐺 is the subgroup generated by the set <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:mi>w</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msub> <m:mi>x</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>x</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo rspace=\"0.278em\" stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo rspace=\"0.278em\">:</m:mo> <m:mrow> <m:mrow> <m:msub> <m:mi>x</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>x</m:mi> <m:mi>n</m:mi> </m:msub> </m:mrow> <m:mo>∈</m:mo> <m:mi>G</m:mi> </m:mrow> <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-2023-0088_ineq_0004.png\" /> <jats:tex-math>{w(x_{1},ldots,x_{n}):x_{1},ldots,x_{n}in G}</jats:tex-math> </jats:alternatives> </jats:inline-formula> of all 𝑤-values in 𝐺. Following J. González-Sánchez and B. Klopsch, a group 𝐺 is 𝑤-maximal if <jats:inline-formu","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140325979","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}
Javier Aramayona, George Domat, Christopher J. Leininger
We classify surface Houghton groups, as well as their pure subgroups, up to isomorphism, commensurability, and quasi-isometry.
我们对表面霍顿群及其纯子群进行了分类，直至同构、可通约和准同构。
{"title":"Isomorphisms and commensurability of surface Houghton groups","authors":"Javier Aramayona, George Domat, Christopher J. Leininger","doi":"10.1515/jgth-2023-0297","DOIUrl":"https://doi.org/10.1515/jgth-2023-0297","url":null,"abstract":"We classify surface Houghton groups, as well as their pure subgroups, up to isomorphism, commensurability, and quasi-isometry.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140196420","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}
The famous Brauer–Fowler theorem states that the order of a finite simple group can be bounded in terms of the order of the centralizer of an involution. Using the classification of finite simple groups, we generalize this theorem and prove that if a simple locally finite group has an involution which commutes with at most 𝑛 involutions, then the group is finite and its order is bounded in terms of 𝑛 only. This answers a question of Strunkov from the Kourovka notebook.
{"title":"A generalization of the Brauer–Fowler theorem","authors":"Saveliy V. Skresanov","doi":"10.1515/jgth-2024-0041","DOIUrl":"https://doi.org/10.1515/jgth-2024-0041","url":null,"abstract":"The famous Brauer–Fowler theorem states that the order of a finite simple group can be bounded in terms of the order of the centralizer of an involution. Using the classification of finite simple groups, we generalize this theorem and prove that if a simple locally finite group has an involution which commutes with at most 𝑛 involutions, then the group is finite and its order is bounded in terms of 𝑛 only. This answers a question of Strunkov from the Kourovka notebook.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140147125","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}
Given a permutation group 𝐺, the derangement graph of 𝐺 is the Cayley graph with connection set the derangements of 𝐺. The group 𝐺 is said to be innately transitive if 𝐺 has a transitive minimal normal subgroup. Clearly, every primitive group is innately transitive. We show that, besides an infinite family of explicit exceptions, there exists a function f:N→Nfcolonmathbb{N}tomathbb{N} such that, if 𝐺 is innately transitive of degree 𝑛 and the derangement graph of 𝐺 has no clique of size 𝑘, then n≤f(k)nleq f(k). Motivation for this work arises from investigations on Erdős–Ko–Rado type theorems for permutation groups.
给定一个置换群𝐺，𝐺的derangement图就是连接集为𝐺的derangements的Cayley图。如果𝐺有一个传递的最小正子群，则称𝐺群为先天传递群。显然，每个基元群都是先天传递群。我们证明，除了无穷系列的明确例外之外，还存在一个函数 f : N → N fcolonmathbb{N}tomathbb{N} ，使得如果𝐺 是阶为 𝑛 的先天传递性的，并且𝐺 的错乱图没有大小为 𝑘 的簇，那么 n ≤ f ( k ) nleq f(k) 。这项工作的动机来自于对置换群的厄尔多斯-柯-拉多类型定理的研究。
{"title":"Cliques in derangement graphs for innately transitive groups","authors":"Marco Fusari, Andrea Previtali, Pablo Spiga","doi":"10.1515/jgth-2023-0284","DOIUrl":"https://doi.org/10.1515/jgth-2023-0284","url":null,"abstract":"Given a permutation group 𝐺, the derangement graph of 𝐺 is the Cayley graph with connection set the derangements of 𝐺. The group 𝐺 is said to be innately transitive if 𝐺 has a transitive minimal normal subgroup. Clearly, every primitive group is innately transitive. We show that, besides an infinite family of explicit exceptions, there exists a function <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 lspace=\"0.278em\" rspace=\"0.278em\">:</m:mo> <m:mrow> <m:mi mathvariant=\"double-struck\">N</m:mi> <m:mo stretchy=\"false\">→</m:mo> <m:mi mathvariant=\"double-struck\">N</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0284_ineq_0001.png\" /> <jats:tex-math>fcolonmathbb{N}tomathbb{N}</jats:tex-math> </jats:alternatives> </jats:inline-formula> such that, if 𝐺 is innately transitive of degree 𝑛 and the derangement graph of 𝐺 has no clique of size 𝑘, then <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>n</m:mi> <m:mo>≤</m:mo> <m:mrow> <m:mi>f</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:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0284_ineq_0002.png\" /> <jats:tex-math>nleq f(k)</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Motivation for this work arises from investigations on Erdős–Ko–Rado type theorems for permutation groups.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140147133","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}
We consider a particular class of Garside groups, which we call circular groups. We mainly prove that roots are unique up to conjugacy in circular groups. This allows us to completely classify these groups up to isomorphism. As a consequence, we obtain the uniqueness of roots up to conjugacy in complex braid groups of rank 2. We also consider a generalization of circular groups, called hosohedral-type groups. These groups are defined using circular groups, and a procedure called the Δ-product, which we study in generality. We also study the uniqueness of roots up to conjugacy in hosohedral-type groups.
{"title":"Uniqueness of roots up to conjugacy in circular and hosohedral-type Garside groups","authors":"Owen Garnier","doi":"10.1515/jgth-2023-0268","DOIUrl":"https://doi.org/10.1515/jgth-2023-0268","url":null,"abstract":"We consider a particular class of Garside groups, which we call circular groups. We mainly prove that roots are unique up to conjugacy in circular groups. This allows us to completely classify these groups up to isomorphism. As a consequence, we obtain the uniqueness of roots up to conjugacy in complex braid groups of rank 2. We also consider a generalization of circular groups, called hosohedral-type groups. These groups are defined using circular groups, and a procedure called the Δ-product, which we study in generality. We also study the uniqueness of roots up to conjugacy in hosohedral-type groups.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140070093","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}
We use a constructive method to obtain all but finitely many 𝑝-local representation zeta functions of a family of finitely generated nilpotent groups MnM_{n} with maximal nilpotency class. For representation dimensions coprime to all primes p<np<n, we construct all irreducible representations of MnM_{n} by defining a standard form for the matrices of these representations.
我们用一种构造方法来获得具有最大无幂级数的有限生成无幂群 M n M_{n} 族的所有𝑝局部表示 zeta 函数。对于与所有素数 p < n p<n 共价的表示维数，我们通过定义这些表示的矩阵的标准形式来构造 M n M_{n} 的所有不可还原表示。
{"title":"Representation zeta function of a family of maximal class groups: Non-exceptional primes","authors":"Shannon Ezzat","doi":"10.1515/jgth-2022-0213","DOIUrl":"https://doi.org/10.1515/jgth-2022-0213","url":null,"abstract":"We use a constructive method to obtain all but finitely many 𝑝-local representation zeta functions of a family of finitely generated nilpotent groups <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>M</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-2022-0213_ineq_0001.png\" /> <jats:tex-math>M_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with maximal nilpotency class. For representation dimensions coprime to all primes <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>p</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-2022-0213_ineq_0002.png\" /> <jats:tex-math>p<n</jats:tex-math> </jats:alternatives> </jats:inline-formula>, we construct all irreducible representations of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>M</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-2022-0213_ineq_0001.png\" /> <jats:tex-math>M_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> by defining a standard form for the matrices of these representations.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140070166","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 5-group of maximal class with major centralizer G1=CG(G2/G4)G_{1}=C_{G}({G_{2}}/{G_{4}}). In this paper, we prove that the irreducible character degrees of a 5-group 𝐺 of maximal class are almost determined by the irreducible character degrees of the major centralizer G1G_{1} and show that the set of irreducible character degrees of a 5-group of maximal class is either {1,5,53}{1,5,5^{3}} or {1,5,…,5k}{1,5,ldots,5^{k}} with k≥1kgeq 1.
设𝐺是最大类的 5 群，其主要中心子 G 1 = C G ( G 2 / G 4 ) G_{1}=C_{G}({G_{2}}/{G_{4}}) 。本文证明了最大类 5 群𝐺 的不可还原特征度几乎由主要中心子 G 1 G_{1} 的不可还原特征度决定，并证明了最大类 5 群的不可还原特征度集合要么是 { 1 , 5 , 5 3 }，要么是 { 1 , 5 , 5 3 }。 {1,5,5^{3}} 或者 { 1 , 5 , ... , 5 k }。 k ≥ 1 kgeq 1 。
{"title":"Character degrees of 5-groups of maximal class","authors":"Lijuan He, Heng Lv, Dongfang Yang","doi":"10.1515/jgth-2023-0103","DOIUrl":"https://doi.org/10.1515/jgth-2023-0103","url":null,"abstract":"Let 𝐺 be a 5-group of maximal class with major centralizer <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>G</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi>C</m:mi> <m:mi>G</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:msub> <m:mi>G</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>/</m:mo> <m:msub> <m:mi>G</m:mi> <m:mn>4</m:mn> </m:msub> </m:mrow> <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-2023-0103_ineq_0001.png\" /> <jats:tex-math>G_{1}=C_{G}({G_{2}}/{G_{4}})</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this paper, we prove that the irreducible character degrees of a 5-group 𝐺 of maximal class are almost determined by the irreducible character degrees of the major centralizer <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>G</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0103_ineq_0002.png\" /> <jats:tex-math>G_{1}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and show that the set of irreducible character degrees of a 5-group of maximal class is either <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:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>5</m:mn> <m:mo>,</m:mo> <m:msup> <m:mn>5</m:mn> <m:mn>3</m:mn> </m:msup> <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-2023-0103_ineq_0003.png\" /> <jats:tex-math>{1,5,5^{3}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> or <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:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>5</m:mn> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">…</m:mi> <m:mo>,</m:mo> <m:msup> <m:mn>5</m:mn> <m:mi>k</m:mi> </m:msup> <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-2023-0103_ineq_0004.png\" /> <jats:tex-math>{1,5,ldots,5^{k}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>k</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0103_ineq_0005.png\" /> <jats:tex-math>kgeq 1</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140047328","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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