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":"49 1","pages":""},"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 <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
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":"55 1","pages":""},"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":"159 1","pages":""},"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":"126 1","pages":""},"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":"11 1","pages":""},"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":"126 1","pages":""},"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":"67 1","pages":""},"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":"5 1","pages":""},"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}
In this article, we address the Amit–Ashurst conjecture on lower bounds of a probability distribution associated to a word on a finite nilpotent group. We obtain an extension of a result of [R. D. Camina, A. Iñiguez and A. Thillaisundaram, Word problems for finite nilpotent groups, Arch. Math. (Basel) 115 (2020), 6, 599–609] by providing improved bounds for the case of finite nilpotent groups of class 2 for words in an arbitrary number of variables, and also settle the conjecture in some cases. We achieve this by obtaining words that are identically distributed on a group to a given word. In doing so, we also obtain an improvement of a result of [A. Iñiguez and J. Sangroniz, Words and characters in finite 𝑝-groups, J. Algebra 485 (2017), 230–246] using elementary techniques.
在这篇文章中，我们讨论了阿米特-阿舒斯特猜想中与有限零能群上的一个词相关的概率分布的下界问题。我们得到了[R.D. Camina, A. Iñiguez and A. Thillaisundaram, Word problems for finite nilpotent groups, Arch.Math. (Basel) 115 (2020), 6, 599-609] 为任意变量个数的单词提供了改进的 2 类有限零potent 群的约束，并在某些情况下解决了猜想。我们通过获得与给定词在一个群上同分布的词来实现这一目标。在此过程中，我们还得到了对 [A. Iñiguez and J. M. B.] 结果的改进。Iñiguez and J. Sangroniz, Words and characters in finite 𝑝-groups, J. Algebra 485 (2017), 230-246] 的一个结果的改进。
{"title":"Automorphic word maps and the Amit–Ashurst conjecture","authors":"Harish Kishnani, Amit Kulshrestha","doi":"10.1515/jgth-2023-0151","DOIUrl":"https://doi.org/10.1515/jgth-2023-0151","url":null,"abstract":"In this article, we address the Amit–Ashurst conjecture on lower bounds of a probability distribution associated to a word on a finite nilpotent group. We obtain an extension of a result of [R. D. Camina, A. Iñiguez and A. Thillaisundaram, Word problems for finite nilpotent groups, <jats:italic>Arch. Math. (Basel)</jats:italic> 115 (2020), 6, 599–609] by providing improved bounds for the case of finite nilpotent groups of class 2 for words in an arbitrary number of variables, and also settle the conjecture in some cases. We achieve this by obtaining words that are identically distributed on a group to a given word. In doing so, we also obtain an improvement of a result of [A. Iñiguez and J. Sangroniz, Words and characters in finite 𝑝-groups, <jats:italic>J. Algebra</jats:italic> 485 (2017), 230–246] using elementary techniques.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"69 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139757141","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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