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Pro-$p$ groups with few relations and universal Koszulity 亲$p$群体与很少的关系和普遍的Koszulity
arXiv: Group Theory Pub Date : 2020-03-20 DOI: 10.7146/MATH.SCAND.A-123644
C. Quadrelli
{"title":"Pro-$p$ groups with few relations and universal Koszulity","authors":"C. Quadrelli","doi":"10.7146/MATH.SCAND.A-123644","DOIUrl":"https://doi.org/10.7146/MATH.SCAND.A-123644","url":null,"abstract":"Let $p$ be a prime. We show that if a pro-$p$ group with at most 2 defining relations has quadratic $mathbb{F}_p$-cohomology, then such algebra is universally Koszul. This proves the \"Universal Koszulity Conjecture\" formulated by J. Minac et al. in the case of maximal pro-$p$ Galois groups of fields with at most 2 defining relations.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75835263","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: 5
Spectral aspects of commuting conjugacy class graph of finite groups 有限群的交换共轭类图的谱方面
arXiv: Group Theory Pub Date : 2020-03-12 DOI: 10.29252/AS.2021.1979
Parthajit Bhowal, R. K. Nath
{"title":"Spectral aspects of commuting conjugacy class graph of finite groups","authors":"Parthajit Bhowal, R. K. Nath","doi":"10.29252/AS.2021.1979","DOIUrl":"https://doi.org/10.29252/AS.2021.1979","url":null,"abstract":"The commuting conjugacy class graph of a non-abelian group $G$, denoted by $mathcal{CCC}(G)$, is a simple undirected graph whose vertex set is the set of conjugacy classes of the non-central elements of $G$ and two distinct vertices $x^G$ and $y^G$ are adjacent if there exists some elements $x' in x^G$ and $y' in y^G$ such that $x'y' = y'x'$. In this paper we compute various spectra and energies of commuting conjugacy class graph of the groups $D_{2n}, Q_{4m}, U_{(n, m)}, V_{8n}$ and $SD_{8n}$. Our computation shows that $mathcal{CCC}(G)$ is super integral for these groups. We compare various energies and as a consequence it is observed that $mathcal{CCC}(G)$ satisfy E-LE Conjecture of Gutman et al. We also provide negative answer to a question posed by Dutta et al. comparing Laplacian and Signless Laplacian energy. Finally, we conclude this paper by characterizing the above mentioned groups $G$ such that $mathcal{CCC}(G)$ is hyperenergetic, L-hyperenergetic or Q-hyperenergetic.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78554726","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: 4
Unique product groups and congruence subgroups 唯一乘积群和同余子群
arXiv: Group Theory Pub Date : 2020-03-10 DOI: 10.1142/S0219498822500256
William Craig, P. Linnell
{"title":"Unique product groups and congruence subgroups","authors":"William Craig, P. Linnell","doi":"10.1142/S0219498822500256","DOIUrl":"https://doi.org/10.1142/S0219498822500256","url":null,"abstract":"We prove that a uniform pro-p group with no nonabelian free subgroups has a normal series with torsion-free abelian factors. We discuss this in relation to unique product groups. We also consider generalizations of Hantzsche-Wendt groups.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89379167","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: 4
An algebraic characterization of 𝑘–colorability 𝑘-colorability的代数表征
arXiv: Group Theory Pub Date : 2020-03-03 DOI: 10.1090/proc/15391
Ramón Flores, Delaram Kahrobaei, T. Koberda
{"title":"An algebraic characterization of 𝑘–colorability","authors":"Ramón Flores, Delaram Kahrobaei, T. Koberda","doi":"10.1090/proc/15391","DOIUrl":"https://doi.org/10.1090/proc/15391","url":null,"abstract":"We characterize $k$--colorability of a simplicial graph via the intrinsic algebraic structure of the associated right-angled Artin group. As a consequence, we show that a certain problem about the existence of homomorphisms from right-angled Artin groups to products of free groups is NP--complete.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77989519","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
A result on the number of cyclic subgroups of a finite group 关于有限群的循环子群数目的一个结果
arXiv: Group Theory Pub Date : 2020-03-01 DOI: 10.3792/PJAA.96.018
M. Tarnauceanu
{"title":"A result on the number of cyclic subgroups of a finite group","authors":"M. Tarnauceanu","doi":"10.3792/PJAA.96.018","DOIUrl":"https://doi.org/10.3792/PJAA.96.018","url":null,"abstract":"Let $G$ be a finite group, $L_1(G)$ be its poset of cyclic subgroups and consider the quantity $alpha(G)=frac{|L_1(G)|}{|G|}$. The aim of this paper is to study the class $cal{C}$ of finite nilpotent groups having $alpha(G)=frac{3}{4}$. We show that if $G$ belongs to this class, then it is a 2-group satisfying certain conditions. Also, we study the appartenance of some classes of finite groups to $cal{C}$.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72876666","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: 1
A note on stable commutator length in braided Ptolemy-Thompson groups 编织Ptolemy-Thompson群中稳定换向子长度的一个注记
arXiv: Group Theory Pub Date : 2020-02-28 DOI: 10.2996/kmj44206
Shuhei Maruyama
{"title":"A note on stable commutator length in braided Ptolemy-Thompson groups","authors":"Shuhei Maruyama","doi":"10.2996/kmj44206","DOIUrl":"https://doi.org/10.2996/kmj44206","url":null,"abstract":"In this note, we show that the sets of all stable commutator lengths in the braided Ptolemy-Thompson groups are equal to non-negative rational numbers.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81989859","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
A mixed version for a Fuchs’ Lemma Fuchs引理的混合版本
arXiv: Group Theory Pub Date : 2020-02-24 DOI: 10.4171/rsmup/56
Simion Breaz
{"title":"A mixed version for a Fuchs’ Lemma","authors":"Simion Breaz","doi":"10.4171/rsmup/56","DOIUrl":"https://doi.org/10.4171/rsmup/56","url":null,"abstract":"We prove a version for mixed groups for a Fuchs' result about connections between the cancellation property of a group and the unit lifting property of its (Walk-)endomorphism rings.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73410493","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
The mapping class group of the Cantor tree has only geometric normal subgroups 康托树的映射类组只有几何正规子群
arXiv: Group Theory Pub Date : 2020-02-17 DOI: 10.1090/proc/15559
A. McLeay
{"title":"The mapping class group of the Cantor tree has only geometric normal subgroups","authors":"A. McLeay","doi":"10.1090/proc/15559","DOIUrl":"https://doi.org/10.1090/proc/15559","url":null,"abstract":"A normal subgroup of the (extended) mapping class group of a surface is said to be geometric if its automorphism group is the mapping class group. We prove that in the case of the Cantor tree surface, every normal subgroup is geometric. We note that there is no non-trivial finite-type mapping class group for which this statement is true. We study a generalisation of the curve graph, proving that its automorphism group is again the mapping class group. This strategy is adapted from that of Brendle-Margalit and the author for certain normal subgroups in the finite-type setting.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73042717","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
Conjugacy classes of p-elements and normalp-complements p元素与正规补的共轭类
arXiv: Group Theory Pub Date : 2020-02-10 DOI: 10.2140/pjm.2020.308.207
H. Tong-Viet
{"title":"Conjugacy classes of p-elements and normal\u0000p-complements","authors":"H. Tong-Viet","doi":"10.2140/pjm.2020.308.207","DOIUrl":"https://doi.org/10.2140/pjm.2020.308.207","url":null,"abstract":"In this paper, we study the structure of finite groups with a large number of conjugacy classes of $p$-elements for some prime $p$. As consequences, we obtain some new criteria for the existence of normal $p$-complements in finite groups.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87239300","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
Connectivity of generating graphs of nilpotent groups 幂零群生成图的连通性
arXiv: Group Theory Pub Date : 2020-02-09 DOI: 10.5802/alco.132
Scott Harper, A. Lucchini
{"title":"Connectivity of generating graphs of nilpotent groups","authors":"Scott Harper, A. Lucchini","doi":"10.5802/alco.132","DOIUrl":"https://doi.org/10.5802/alco.132","url":null,"abstract":"Let $G$ be $2$-generated group. The generating graph of $Gamma(G)$ is the graph whose vertices are the elements of $G$ and where two vertices $g$ and $h$ are adjacent if $G=langle g,hrangle$. This graph encodes the combinatorial structure of the distribution of generating pairs across $G$. In this paper we study several natural graph theoretic properties related to the connectedness of $Gamma(G)$ in the case where $G$ is a finite nilpotent group. For example, we prove that if $G$ is nilpotent, then the graph obtained from $Gamma(G)$ by removing its isolated vertices is maximally connected and, if $|G| geq 3$, also Hamiltonian. We pose several questions.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87878949","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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