Pub Date : 2020-05-16DOI: 10.26493/2590-9770.1350.C36
Maria Elisa Fernandes, Claudio Alexandre Piedade
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 ${6,3}$.
{"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}
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.
{"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}
Pub Date : 2020-05-14DOI: 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}
Pub Date : 2020-05-09DOI: 10.33048/SEMI.2020.17.065
M. I. Fraiman
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 ${[rho]mapsto[rhocircvarphi]}$.
{"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}
Pub Date : 2020-04-29DOI: 10.33048/semi.2020.17.085
A. Sozutov
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 $U_3(Q)$. Keywords: A strongly embedded subgroup of a unitary type, subgroups of Borel, Cartan, involution, finite element.
组$G$的适当子组$B$被称为{it强嵌入}子组，如果$2inpi(B)$和$2notinpi(B cap B^g)$对应于任何元素$g in G setminus B $，那么$ N_G(X) leq B$对应于任何2-子组$ X leq B $。如果对于所有$ gin G $子群$ langle a, a^g rangle $都是{it有限}的，则群$G$中的元素$a$称为有限的。证明了特征为$2$的局部有限域$Q$上，阶为$4$且强嵌入子群与$U_3(Q)$的Borel子群同构的群是局部有限的，且与群$U_3(Q)$同构。关键词:酉型的强嵌入子群，Borel子群，Cartan，对合，有限元。
{"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}
Pub Date : 2020-04-23DOI: 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}
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 ${G_{n,r}}$. In 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: $$1 to langle sigma_{n}rangle to mathrm{Aut}(X_n^{mathbb{Z}}, sigma_{n}) to mathcal{L}_{n}.$$ A consequence of this is that the groups $mathrm{Out}(G_{n,r})$ are centreless and have undecidable order problem.
在本文中，我们进一步探讨了移空间的自同构群与Higman-Thompson群的外自同构群之间的联系的本质${G_{n,r}}$。在之前的工作中，作者证明了$n$ -字母上的单侧移位动力系统的自同构群$mathrm{Aut}(X_n^{mathbb{N}}, sigma_{n})$自然嵌入为Higman-Thompson群的外自同构群$mathop{mathrm{Out}}(G_{n,r})$的子群$G_{n, r}$, $1 le r < n$。在本文中，我们证明了(双边)移动动力系统$mathop{mathrm{Aut}}(X_n^{mathbb{Z}}, sigma_{n})$的自同构群商的中心嵌入为$G_{n,r}$的外部自同构群$mathop{mathrm{Out}}(G_{n,r})$的子群$mathcal{L}_{n}$。根据Ryan的结果，我们有如下的中心扩展:$$1 to langle sigma_{n}rangle to mathrm{Aut}(X_n^{mathbb{Z}}, sigma_{n}) to mathcal{L}_{n}.$$这样做的一个结果是，群$mathrm{Out}(G_{n,r})$是无中心的，并且有不可确定的顺序问题。
{"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}
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. In 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 $A_n$, $8leq nleq 12$.
{"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}
Pub Date : 2020-03-28DOI: 10.30504/jims.2020.108338
G. Bergman
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. We 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.
{"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}
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 $D_{p^n}$. Key Words and Phrases: Co-Prime Order graph,finite abelian group,Dihedral group, Laplacian spectrum.
给定有限群的textbf{协素序图}$Theta (G)$是一个简单的无向图，其顶点集是群$G$本身，并且$Theta (G)$中的任意两个顶点x,y相邻当且仅当$gcd(o(x),o(y))=1$或素数。本文给出了有限阿贝尔群或二面体群的协素数阶图中顶点度的精确计算公式$D_n$，并研究了当G为有限阿贝尔p群${mathbb{Z}_p}^t times {mathbb{Z}_q}^s$或二面体群$D_{p^n}$时，协素数阶图$Theta (G)$的拉普拉斯谱。关键词:协素阶图，有限阿贝尔群，二面体群，拉普拉斯谱。
{"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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：