{"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}
Javier Aramayona, George Domat, Christopher J. Leininger
{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
