Suppose that $G$ is a groupoid with binary operation $otimes$. The pair $(G,otimes)$ is said to be a gyrogroup if the operation $otimes$ has a left identity, each element $a in G$ has a left inverse and the gyroassociative law and the left loop property are satisfied in $G$. In this paper, a method for constructing new gyrogroups from old ones is presented and the structure of subgyrogroups of these gyrogroups are also given. As a consequence of this work, five $2-$gyrogroups of order $2^n$, $ngeq 3$, are presented. Some open questions are also proposed.
假设$G$是一个具有二进制操作$otimes$的类群。如果运算$otimes$有一个左恒等式，每个元素$a in G$有一个左逆，并且在$G$中满足陀螺结合律和左环性质，则称对$(G,otimes)$为一个陀螺群。本文提出了一种由旧的陀螺群构造新陀螺群的方法，并给出了这些陀螺群的子陀螺群的结构。作为这项工作的结果，提出了五个阶为$2^n$, $ngeq 3$的$2-$陀螺群。还提出了一些悬而未决的问题。
{"title":"Construction of New Gyrogroups and the Structure of their Subgyrogroups","authors":"S. Mahdavi, A. Ashrafi, M. Salahshour","doi":"10.29252/AS.2020.1971","DOIUrl":"https://doi.org/10.29252/AS.2020.1971","url":null,"abstract":"Suppose that $G$ is a groupoid with binary operation $otimes$. The pair $(G,otimes)$ is said to be a gyrogroup if the operation $otimes$ has a left identity, each element $a in G$ has a left inverse and the gyroassociative law and the left loop property are satisfied in $G$. In this paper, a method for constructing new gyrogroups from old ones is presented and the structure of subgyrogroups of these gyrogroups are also given. As a consequence of this work, five $2-$gyrogroups of order $2^n$, $ngeq 3$, are presented. Some open questions are also proposed.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78660450","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}
We show that a group is a GVZ-group if and only if it is a flat group. We show that the nilpotence class of a GVZ-group is bounded by the number of distinct degrees of irreducible characters. We also show that certain CM-groups can be characterized as GVZ-groups whose irreducible character values lie in the prime field.
{"title":"GVZ-groups, Flat groups, and CM-Groups","authors":"Shawn T. Burkett, M. Lewis","doi":"10.5802/CRMATH.185","DOIUrl":"https://doi.org/10.5802/CRMATH.185","url":null,"abstract":"We show that a group is a GVZ-group if and only if it is a flat group. We show that the nilpotence class of a GVZ-group is bounded by the number of distinct degrees of irreducible characters. We also show that certain CM-groups can be characterized as GVZ-groups whose irreducible character values lie in the prime field.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83920269","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 paper, we prove a series of results on group embeddings in groups with a small number of generators. We show that each finitely generated group $G$ lying in a variety ${mathcal M}$ can be embedded in a $4$-generated group $H in {mathcal M}{mathcal A}$ (${mathcal A}$ means the variety of abelian groups). If $G$ is a finite group, then $H$ can also be found as a finite group. It follows, that any finitely generated (finite) solvable group $G$ of the derived length $l$ can be embedded in a $4$-generated (finite) solvable group $H$ of length $l+1$. Thus, we answer the question of V. H. Mikaelian and this http URL. Olshanskii. It is also shown that any countable group $Gin {mathcal M}$, such that the abelianization $G_{ab}$ is a free abelian group, is embeddable in a $2$-generated group $Hin {mathcal M}{mathcal A}$.
{"title":"Embedding theorems for solvable groups","authors":"V. Roman’kov","doi":"10.1090/PROC/15562","DOIUrl":"https://doi.org/10.1090/PROC/15562","url":null,"abstract":"In this paper, we prove a series of results on group embeddings in groups with a small number of generators. We show that each finitely generated group $G$ lying in a variety ${mathcal M}$ can be embedded in a $4$-generated group $H in {mathcal M}{mathcal A}$ (${mathcal A}$ means the variety of abelian groups). If $G$ is a finite group, then $H$ can also be found as a finite group. It follows, that any finitely generated (finite) solvable group $G$ of the derived length $l$ can be embedded in a $4$-generated (finite) solvable group $H$ of length $l+1$. Thus, we answer the question of V. H. Mikaelian and this http URL. Olshanskii. It is also shown that any countable group $Gin {mathcal M}$, such that the abelianization $G_{ab}$ is a free abelian group, is embeddable in a $2$-generated group $Hin {mathcal M}{mathcal A}$.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75479948","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}
There are only 10 Euclidean forms, that is flat closed three dimensional manifolds: six are orientable $mathcal{G}_1,dots,mathcal{G}_6$ and four are non-orientable $mathcal{B}_1,dots,mathcal{B}_4$. In the present paper we investigate the manifold $mathcal{G}_6$, also known as Hantzsche-Wendt manifold; this is the unique Euclidean $3$-form with finite first homology group $H_1(mathcal{G}_6) = mathbb{Z}^2_4$. The aim of this paper is to describe all types of $n$-fold coverings over $mathcal{G}_{6}$ and calculate the numbers of non-equivalent coverings of each type. We classify subgroups in the fundamental group $pi_1(mathcal{G}_{6})$ up to isomorphism. Given index $n$, we calculate the numbers of subgroups and the numbers of conjugacy classes of subgroups for each isomorphism type and provide the Dirichlet generating series for the above sequences.
{"title":"On the coverings of Hantzsche-Wendt manifold","authors":"G. Chelnokov, A. Mednykh","doi":"10.2748/tmj.20210308","DOIUrl":"https://doi.org/10.2748/tmj.20210308","url":null,"abstract":"There are only 10 Euclidean forms, that is flat closed three dimensional manifolds: six are orientable $mathcal{G}_1,dots,mathcal{G}_6$ and four are non-orientable $mathcal{B}_1,dots,mathcal{B}_4$. In the present paper we investigate the manifold $mathcal{G}_6$, also known as Hantzsche-Wendt manifold; this is the unique Euclidean $3$-form with finite first homology group $H_1(mathcal{G}_6) = mathbb{Z}^2_4$. \u0000The aim of this paper is to describe all types of $n$-fold coverings over $mathcal{G}_{6}$ and calculate the numbers of non-equivalent coverings of each type. We classify subgroups in the fundamental group $pi_1(mathcal{G}_{6})$ up to isomorphism. Given index $n$, we calculate the numbers of subgroups and the numbers of conjugacy classes of subgroups for each isomorphism type and provide the Dirichlet generating series for the above sequences.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74755710","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 study connections between components of the Cayley graph $mathrm{Cay}(G,A)$, where $A$ is an arbitrary subset of a group $G$, and cosets of the subgroup of $G$ generated by $A$. In particular, we show how to construct generating sets of $G$ if $mathrm{Cay}(G,A)$ has finitely many components. Furthermore, we provide an algorithm for finding minimal generating sets of finite groups using their Cayley graphs.
{"title":"An algorithm for finding minimal generating sets of finite groups.","authors":"Tanakorn Udomworarat, T. Suksumran","doi":"10.29252/AS.2021.2029","DOIUrl":"https://doi.org/10.29252/AS.2021.2029","url":null,"abstract":"In this article, we study connections between components of the Cayley graph $mathrm{Cay}(G,A)$, where $A$ is an arbitrary subset of a group $G$, and cosets of the subgroup of $G$ generated by $A$. In particular, we show how to construct generating sets of $G$ if $mathrm{Cay}(G,A)$ has finitely many components. Furthermore, we provide an algorithm for finding minimal generating sets of finite groups using their Cayley graphs.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85072077","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 short note, we show that a group acting geometrically on a CAT(0) cube complex with virtually abelian hyperplane-stabilisers must decompose virtually as a free product of free abelian groups and surface groups.
{"title":"CAT(0) cube complexes with flat hyperplanes","authors":"A. Genevois","doi":"10.1090/PROC/15490","DOIUrl":"https://doi.org/10.1090/PROC/15490","url":null,"abstract":"In this short note, we show that a group acting geometrically on a CAT(0) cube complex with virtually abelian hyperplane-stabilisers must decompose virtually as a free product of free abelian groups and surface groups.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91380601","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}
Surface groups are determined among limit groups by their profinite completions. As a corollary, the set of surface words in a free group is closed in the profinite topology.
平面群是由极限群的无限完井来确定的。作为推论，自由群中的面词集在无限拓扑中是封闭的。
{"title":"On the profinite rigidity of surface groups and surface words","authors":"H. Wilton","doi":"10.5802/CRMATH.121","DOIUrl":"https://doi.org/10.5802/CRMATH.121","url":null,"abstract":"Surface groups are determined among limit groups by their profinite completions. As a corollary, the set of surface words in a free group is closed in the profinite topology.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76162234","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 : 2020-08-16DOI: 10.22108/IJGT.2020.123980.1638
M. Vaughan-Lee
There is a long-standing conjecture attributed to I Schur that if $G$ is a finite group with Schur multiplier $M(G)$ then the exponent of $M(G)$ divides the exponent of $G$. It is easy to see that this conjecture holds for exponent 2 and exponent 3, but it has been known since 1974 that the conjecture fails for exponent 4. In this note I give an example of a group $G$ with exponent 5 with Schur multiplier $M(G)$ of exponent 25, and an example of a group $A$ of exponent 9 with Schur multiplier $M(A)$ of exponent 27.
有一个由I Schur提出的长期猜想，如果$G$是具有Schur乘子$M(G)$的有限群，则$M(G)$的指数除以$G$的指数。很容易看出，这个猜想对指数2和指数3成立，但自1974年以来，人们已经知道，这个猜想对指数4不成立。在本文中，我给出了一个指数为5的群$G$和指数为25的舒尔乘法器$M(G)$的例子，以及指数为9的群$ a $和指数为27的舒尔乘法器$M(a)$的例子。
{"title":"Schur's exponent conjecture -- counterexamples of exponent 5 and exponent 9.","authors":"M. Vaughan-Lee","doi":"10.22108/IJGT.2020.123980.1638","DOIUrl":"https://doi.org/10.22108/IJGT.2020.123980.1638","url":null,"abstract":"There is a long-standing conjecture attributed to I Schur that if $G$ is a finite group with Schur multiplier $M(G)$ then the exponent of $M(G)$ divides the exponent of $G$. It is easy to see that this conjecture holds for exponent 2 and exponent 3, but it has been known since 1974 that the conjecture fails for exponent 4. In this note I give an example of a group $G$ with exponent 5 with Schur multiplier $M(G)$ of exponent 25, and an example of a group $A$ of exponent 9 with Schur multiplier $M(A)$ of exponent 27.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72942623","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 paper we define a way to get a bounded invertible automaton starting from a finite graph. It turns out that the corresponding automaton group is regular weakly branch over its commutator subgroup, contains a free semigroup on two elements and is amenable of exponential growth. We also highlight a connection between our construction and the right-angled Artin groups. We then study the Schreier graphs associated with the self-similar action of these automaton groups on the regular rooted tree. We explicitly determine their diameter and their automorphism group in the case where the initial graph is a path. Moreover, we show that the case of cycles gives rise to Schreier graphs whose automorphism group is isomorphic to the dihedral group. It is remarkable that our construction recovers some classical examples of automaton groups like the Adding machine and the Tangled odometer.
{"title":"Graph automaton groups","authors":"Matteo Cavaleri, D. D’Angeli, A. Donno, E. Rodaro","doi":"10.32037/agta-2021-005","DOIUrl":"https://doi.org/10.32037/agta-2021-005","url":null,"abstract":"In this paper we define a way to get a bounded invertible automaton starting from a finite graph. It turns out that the corresponding automaton group is regular weakly branch over its commutator subgroup, contains a free semigroup on two elements and is amenable of exponential growth. We also highlight a connection between our construction and the right-angled Artin groups. We then study the Schreier graphs associated with the self-similar action of these automaton groups on the regular rooted tree. We explicitly determine their diameter and their automorphism group in the case where the initial graph is a path. Moreover, we show that the case of cycles gives rise to Schreier graphs whose automorphism group is isomorphic to the dihedral group. It is remarkable that our construction recovers some classical examples of automaton groups like the Adding machine and the Tangled odometer.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88010036","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 finite group, and $alpha$ a nontrivial character of $G$. The McKay graph $mathcal{M}(G,alpha)$ has the irreducible characters of $G$ as vertices, with an edge from $chi_1$ to $chi_2$ if $chi_2$ is a constituent of $alphachi_1$. We study the diameters of McKay graphs for finite simple groups $G$. For alternating groups, we prove a conjecture made in [LST]: there is an absolute constant $C$ such that $hbox{diam},{mathcal M}(G,alpha) le Cfrac{log |mathsf{A}_n|}{log alpha(1)}$ for all nontrivial irreducible characters $alpha$ of $mathsf{A}_n$. Also for classsical groups of symplectic or orthogonal type of rank $r$, we establish a linear upper bound $Cr$ on the diameters of all nontrivial McKay graphs.
设$G$是一个有限群，$alpha$是$G$的一个非平凡特征。McKay图$mathcal{M}(G,alpha)$以$G$的不可约特征为顶点，如果$chi_2$是$alphachi_1$的一个组成部分，则有一条从$chi_1$到$chi_2$的边。我们研究了有限简单群的McKay图的直径$G$。对于交替群，我们证明了[LST]中的一个猜想:存在一个绝对常数$C$，使得$hbox{diam},{mathcal M}(G,alpha) le Cfrac{log |mathsf{A}_n|}{log alpha(1)}$对于$mathsf{A}_n$的所有非平凡不可约字符$alpha$。对于秩为$r$的辛型或正交型的经典群，我们在所有非平凡McKay图的直径上建立了一个线性上界$Cr$。
{"title":"McKay graphs for alternating and classical groups","authors":"M. Liebeck, A. Shalev, P. Tiep","doi":"10.1090/TRAN/8395","DOIUrl":"https://doi.org/10.1090/TRAN/8395","url":null,"abstract":"Let $G$ be a finite group, and $alpha$ a nontrivial character of $G$. The McKay graph $mathcal{M}(G,alpha)$ has the irreducible characters of $G$ as vertices, with an edge from $chi_1$ to $chi_2$ if $chi_2$ is a constituent of $alphachi_1$. We study the diameters of McKay graphs for finite simple groups $G$. For alternating groups, we prove a conjecture made in [LST]: there is an absolute constant $C$ such that $hbox{diam},{mathcal M}(G,alpha) le Cfrac{log |mathsf{A}_n|}{log alpha(1)}$ for all nontrivial irreducible characters $alpha$ of $mathsf{A}_n$. Also for classsical groups of symplectic or orthogonal type of rank $r$, we establish a linear upper bound $Cr$ on the diameters of all nontrivial McKay graphs.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91326769","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}
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