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The degrees of toroidal regular proper hypermaps 环面正则固有超映射的度
arXiv: Group Theory Pub Date : 2020-05-16 DOI: 10.26493/2590-9770.1350.C36
Maria Elisa Fernandes, Claudio Alexandre Piedade
{"title":"The degrees of toroidal regular proper hypermaps","authors":"Maria Elisa Fernandes, Claudio Alexandre Piedade","doi":"10.26493/2590-9770.1350.C36","DOIUrl":"https://doi.org/10.26493/2590-9770.1350.C36","url":null,"abstract":"Recently the classification of all possible faithful transitive permutation representations of the group of symmetries of a regular toroidal map was accomplished. In this paper we complete this investigation on a surface of genus 1 considering the group of a regular toroidal hypermap of type $(3,3,3)$ that is a subgroup of index $2$ of the group of symmetries of a toroidal map of type ${6,3}$.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89620047","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Acylindrical hyperbolicity and existential closedness 非圆柱形双曲线和存在的封闭性
arXiv: Group Theory Pub Date : 2020-05-14 DOI: 10.1090/proc/15409
Simon Andr'e
{"title":"Acylindrical hyperbolicity and existential closedness","authors":"Simon Andr'e","doi":"10.1090/proc/15409","DOIUrl":"https://doi.org/10.1090/proc/15409","url":null,"abstract":"Let $G$ be a finitely presented group, and let $H$ be a subgroup of $G$. We prove that if $H$ is acylindrically hyperbolic and existentially closed in $G$, then $G$ is acylindrically hyperbolic. As a corollary, any finitely presented group which is existentially equivalent to the mapping class group of a surface of finite type, to $mathrm{Out}(F_n)$ or $mathrm{Aut}(F_n)$ for $ngeq 2$ or to the Higman group, is acylindrically hyperbolic.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78864278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
A model theoretic solution to a problem of László Fuchs László Fuchs问题的模型理论解
arXiv: Group Theory Pub Date : 2020-05-14 DOI: 10.1016/J.JALGEBRA.2020.09.029
Marcos Mazari-Armida
{"title":"A model theoretic solution to a problem of László Fuchs","authors":"Marcos Mazari-Armida","doi":"10.1016/J.JALGEBRA.2020.09.029","DOIUrl":"https://doi.org/10.1016/J.JALGEBRA.2020.09.029","url":null,"abstract":"","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73898039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 10
Twisted Burnside–Frobenius theorem and $R_infty$-property for lamplighter-type groups lamplighter型群的扭曲Burnside-Frobenius定理和$ r_inty $-性质
arXiv: Group Theory Pub Date : 2020-05-09 DOI: 10.33048/SEMI.2020.17.065
M. I. Fraiman
{"title":"Twisted Burnside–Frobenius theorem and $R_infty$-property for lamplighter-type groups","authors":"M. I. Fraiman","doi":"10.33048/SEMI.2020.17.065","DOIUrl":"https://doi.org/10.33048/SEMI.2020.17.065","url":null,"abstract":"We prove that the restricted wreath product ${mathbb{Z}_n mathbin{mathrm{wr}} mathbb{Z}^k}$ has the $R_infty$-property, i. e. every its automorphism $varphi$ has infinite Reidemeister number $R(varphi)$, in exactly two cases: (1) for any $k$ and even $n$; (2) for odd $k$ and $n$ divisible by 3. In the remaining cases there are automorphisms with finite Reidemeister number, for which we prove the finite-dimensional twisted Burnside--Frobenius theorem (TBFT): $R(varphi)$ is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations fixed by the action ${[rho]mapsto[rhocircvarphi]}$.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76614780","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
On groups with a strongly embedded unitary subgroup 在具有强内嵌酉子群的群上
arXiv: Group Theory Pub Date : 2020-04-29 DOI: 10.33048/semi.2020.17.085
A. Sozutov
{"title":"On groups with a strongly embedded unitary subgroup","authors":"A. Sozutov","doi":"10.33048/semi.2020.17.085","DOIUrl":"https://doi.org/10.33048/semi.2020.17.085","url":null,"abstract":"The proper subgroup $B$ of the group $G$ is called {it strongly embedded}, if $2inpi(B)$ and $2notinpi(B cap B^g)$ for any element $g in G setminus B $ and, therefore, $ N_G(X) leq B$ for any 2-subgroup $ X leq B $. An element $a$ of a group $G$ is called {it finite} if for all $ gin G $ the subgroups $ langle a, a^g rangle $ are finite. In the paper, it is proved that the group with finite element of order $4$ and strongly embedded subgroup isomorphic to the Borel subgroup of $U_3(Q)$ over a locally finite field $Q$ of characteristic $2$ is locally finite and isomorphic to the group $U_3(Q)$. \u0000Keywords: A strongly embedded subgroup of a unitary type, subgroups of Borel, Cartan, involution, finite element.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76342097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Spread of Almost Simple Classical Groups 几乎简单古典群的传播
arXiv: Group Theory Pub Date : 2020-04-23 DOI: 10.1007/978-3-030-74100-6
Scott Harper
{"title":"The Spread of Almost Simple Classical Groups","authors":"Scott Harper","doi":"10.1007/978-3-030-74100-6","DOIUrl":"https://doi.org/10.1007/978-3-030-74100-6","url":null,"abstract":"","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74775442","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
Automorphisms of shift spaces and the Higman-Thompson groups: the two-sided case 平移空间的自同构与Higman-Thompson群:双面情况
arXiv: Group Theory Pub Date : 2020-04-17 DOI: 10.19086/da.28243
C. Bleak, P. Cameron, F. Olukoya
{"title":"Automorphisms of shift spaces and the Higman-Thompson groups: the two-sided case","authors":"C. Bleak, P. Cameron, F. Olukoya","doi":"10.19086/da.28243","DOIUrl":"https://doi.org/10.19086/da.28243","url":null,"abstract":"In this article, we further explore the nature of a connection between groups of automorphisms of shift spaces and the groups of outer automorphisms of the Higman-Thompson groups ${G_{n,r}}$. \u0000In previous work, the authors show that the group $mathrm{Aut}(X_n^{mathbb{N}}, sigma_{n})$ of automorphisms of the one-sided shift dynamical system over an $n$-letter alphabet naturally embeds as a subgroup of the group $mathop{mathrm{Out}}(G_{n,r})$ of outer-automorphisms of the Higman-Thompson group $G_{n, r}$, $1 le r < n$. In the current article we show that the quotient of the group of automorphisms of the (two-sided) shift dynamical system $mathop{mathrm{Aut}}(X_n^{mathbb{Z}}, sigma_{n})$ by its centre embeds as a subgroup $mathcal{L}_{n}$ of the outer automorphism group $mathop{mathrm{Out}}(G_{n,r})$ of $G_{n,r}$. It follows by a result of Ryan that we have the following central extension: $$1 to langle sigma_{n}rangle to mathrm{Aut}(X_n^{mathbb{Z}}, sigma_{n}) to mathcal{L}_{n}.$$ \u0000A consequence of this is that the groups $mathrm{Out}(G_{n,r})$ are centreless and have undecidable order problem.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90226388","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
$2A$-Majorana Representations of $A_{12}$ $2A$-Majorana $A_{12}$
arXiv: Group Theory Pub Date : 2020-04-08 DOI: 10.1090/tran/8669
Clara Franchi, A. Ivanov, Mario Mainardis
{"title":"$2A$-Majorana Representations of $A_{12}$","authors":"Clara Franchi, A. Ivanov, Mario Mainardis","doi":"10.1090/tran/8669","DOIUrl":"https://doi.org/10.1090/tran/8669","url":null,"abstract":"Majorana representations have been introduced by Ivanov in order to provide an axiomatic framework for studying the actions on the Griess algebra of the Monster and of its subgroups generated by Fischer involutions. A crucial step in this programme is to obtain an explicit description of the Majorana representations of $A_{12}$, for this might eventually lead to a new and independent construction of the Monster group. \u0000In this paper we prove that $A_{12}$ has a unique Majorana representation on the set of its involutions of type $2^2$ and $2^6$ (that is the involutions that fall into the class of Fischer involutions when $A_{12}$ is embedded in the Monster) and we determine the degree and the decomposition into irreducibles of such representation. As a consequence we get that Majorana algebras affording a $2A$-representation of $A_{12}$ and of the Harada-Norton sporadic simple group satisfy the Straight Flush Conjecture. As a by-product we also determine the degree and the decomposition into irreducibles of the Majorana representation induced on the $A_8$ subgroup of $A_{12}$. We finally state a conjecture about Majorana representations of the alternating groups $A_n$, $8leq nleq 12$.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73506213","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A note on factorizations of finite groups 有限群的分解问题
arXiv: Group Theory Pub Date : 2020-03-28 DOI: 10.30504/jims.2020.108338
G. Bergman
{"title":"A note on factorizations of finite groups","authors":"G. Bergman","doi":"10.30504/jims.2020.108338","DOIUrl":"https://doi.org/10.30504/jims.2020.108338","url":null,"abstract":"In Question 19.35 of the Kourovka Notebook, M. H. Hooshmand asks whether, given a finite group $G$ and a factorization $mathrm{card}(G)= n_1ldots n_k$, one can always find subsets $A_1,ldots,A_k$ of $G$ with $mathrm{card}(A_i)=n_i$ such that $G=A_1ldots A_k;$ equivalently, such that the group multiplication map $A_1timesldotstimes A_kto G$ is a bijection. \u0000We show that for $G$ the alternating group on 4 elements, $k=3$, and $(n_1,n_2,n_3) = (2,3,2)$, the answer is negative. We then generalize some of the tools used in our proof, and note an open question.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75727136","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
Co-prime order graphs of finite Abelian groups and dihedral groups 有限阿贝尔群和二面体群的协素阶图
arXiv: Group Theory Pub Date : 2020-03-22 DOI: 10.22436/jmcs.023.03.03
Amit Sehgal, Manjeet, Dalip Singh
{"title":"Co-prime order graphs of finite Abelian groups and dihedral groups","authors":"Amit Sehgal, Manjeet, Dalip Singh","doi":"10.22436/jmcs.023.03.03","DOIUrl":"https://doi.org/10.22436/jmcs.023.03.03","url":null,"abstract":"The textbf{Co-Prime Order Graph} $Theta (G)$ of a given finite group is a simple undirected graph whose vertex set is the group $G$ itself, and any two vertexes x,y in $Theta (G)$ are adjacent if and only if $gcd(o(x),o(y))=1$ or prime. In this paper, we find a precise formula to count the degree of a vertex in the Co-Prime Order graph of a finite abelian group or Dihedral group $D_n$.We also investigate the Laplacian spectrum of the Co-Prime Order Graph $Theta (G)$ when G is finite abelian p-group, ${mathbb{Z}_p}^t times {mathbb{Z}_q}^s$ or Dihedral group $D_{p^n}$. \u0000Key Words and Phrases: Co-Prime Order graph,finite abelian group,Dihedral group, Laplacian spectrum.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77821365","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 8
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