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Fusion systems with J-components over 𝐹2𝑒 with 𝑒 > 1 融合系统与j组件𝐹2𝑒与𝑒> 1
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-08-20 DOI: 10.1515/jgth-2020-0156
M. Aschbacher
{"title":"Fusion systems with J-components over 𝐹2𝑒 with 𝑒 > 1","authors":"M. Aschbacher","doi":"10.1515/jgth-2020-0156","DOIUrl":"https://doi.org/10.1515/jgth-2020-0156","url":null,"abstract":"Abstract Let 𝐾 be a finite simple group of Lie type over a field of even order q > 2 q>2 . If 𝐾 is not F 4 2 ⁢ ( q ) {}^{2}F_{4}(q) , then we determine the fusion systems ℱ of J-component type with a fully centralized involution 𝑗 such that C F ⁢ ( j ) C_{mathcal{F}}(j) has a component realized by 𝐾. The exceptional case is treated in a later paper.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85344581","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}
引用次数: 0
On finite 𝜎-tower groups 有限的𝜎-tower群
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-08-02 DOI: 10.1515/jgth-2022-0058
Jinzhuan Cai, Zhigang Wang, I. N. Safonova, A. Skiba
{"title":"On finite 𝜎-tower groups","authors":"Jinzhuan Cai, Zhigang Wang, I. N. Safonova, A. Skiba","doi":"10.1515/jgth-2022-0058","DOIUrl":"https://doi.org/10.1515/jgth-2022-0058","url":null,"abstract":"Abstract In this paper, 𝐺 is a finite group and 𝜎 a partition of the set of all primes ℙ, that is, σ = { σ i ∣ i ∈ I } sigma={sigma_{i}mid iin I} , where P = ⋃ i ∈ I σ i mathbb{P}=bigcup_{iin I}sigma_{i} and σ i ∩ σ j = ∅ sigma_{i}capsigma_{j}=emptyset for all i ≠ j ineq j . If 𝑛 is an integer, we write σ ⁢ ( n ) = { σ i ∣ σ i ∩ π ⁢ ( n ) ≠ ∅ } sigma(n)={sigma_{i}midsigma_{i}cappi(n)neqemptyset} and σ ⁢ ( G ) = σ ⁢ ( | G | ) sigma(G)=sigma(lvert Grvert) . A group 𝐺 is said to be 𝜎-primary if 𝐺 is a σ i sigma_{i} -group for some i = i ⁢ ( G ) i=i(G) and 𝜎-soluble if every chief factor of 𝐺 is 𝜎-primary. We say that 𝐺 is a 𝜎-tower group if either G = 1 G=1 or 𝐺 has a normal series 1 = G 0 < G 1 < ⋯ < G t - 1 < G t = G 1=G_{0}<G_{1}<cdots<G_{t-1}<G_{t}=G such that G i / G i - 1 G_{i}/G_{i-1} is a σ i sigma_{i} -group, σ i ∈ σ ⁢ ( G ) sigma_{i}insigma(G) , and G / G i G/G_{i} and G i - 1 G_{i-1} are σ i ′ sigma_{i}^{prime} -groups for all i = 1 , … , t i=1,ldots,t . A subgroup 𝐴 of 𝐺 is said to be 𝜎-subnormal in 𝐺 if there is a subgroup chain A = A 0 ≤ A 1 ≤ ⋯ ≤ A t = G A=A_{0}leq A_{1}leqcdotsleq A_{t}=G such that either A i - 1 ⁢ ⊴ ⁢ A i A_{i-1}trianglelefteq A_{i} or A i / ( A i - 1 ) A i A_{i}/(A_{i-1})_{A_{i}} is 𝜎-primary for all i = 1 , … , t i=1,ldots,t . In this paper, answering to Question 4.8 in [A. N. Skiba, On 𝜎-subnormal and 𝜎-permutable subgroups of finite groups, J. Algebra 436 (2015), 1–16], we prove that a 𝜎-soluble group G ≠ 1 Gneq 1 with | σ ⁢ ( G ) | = n lvertsigma(G)rvert=n is a 𝜎-tower group if each of its ( n + 1 ) (n+1) -maximal subgroups is 𝜎-subnormal in 𝐺.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72823234","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}
引用次数: 1
Shift dynamics of the groups of Fibonacci type 斐波那契类型群的移位动力学
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-07-30 DOI: 10.1515/jgth-2022-0003
Kirk McDermott
{"title":"Shift dynamics of the groups of Fibonacci type","authors":"Kirk McDermott","doi":"10.1515/jgth-2022-0003","DOIUrl":"https://doi.org/10.1515/jgth-2022-0003","url":null,"abstract":"Abstract We study the shift dynamics of the groups G = G n ⁢ ( x 0 ⁢ x m ⁢ x k - 1 ) G=G_{n}(x_{0}x_{m}x_{k}^{-1}) of Fibonacci type introduced by Johnson and Mawdesley. The main result concerns the order of the shift automorphism of 𝐺 and determining whether it is an outer automorphism, and we find the latter occurs if and only if 𝐺 is not perfect. A result of Bogley provides that the aspherical presentations determine groups admitting a free shift action by Z n mathbb{Z}_{n} on the nonidentity elements of 𝐺, from which it follows that the shift is an outer automorphism of order 𝑛 when 𝐺 is nontrivial. The focus of this paper is therefore on the non-aspherical cases, which include for example the Fibonacci and Sieradski groups. With few exceptions, the fixed-point and freeness problems for the shift automorphism are solved, in some cases using computational and topological methods.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79790453","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}
引用次数: 0
M, B and Co1 are recognisable by their prime graphs M B和Co1可以通过它们的素数图来识别
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-07-28 DOI: 10.1515/jgth-2021-0119
Melissa Lee, Tomasz Popiel
{"title":"M, B and Co1 are recognisable by their prime graphs","authors":"Melissa Lee, Tomasz Popiel","doi":"10.1515/jgth-2021-0119","DOIUrl":"https://doi.org/10.1515/jgth-2021-0119","url":null,"abstract":"Abstract The prime graph, or Gruenberg–Kegel graph, of a finite group 𝐺 is the graph Γ ⁢ ( G ) Gamma(G) whose vertices are the prime divisors of | G | lvert Grvert and whose edges are the pairs { p , q } {p,q} for which 𝐺 contains an element of order p ⁢ q pq . A finite group 𝐺 is recognisable by its prime graph if every finite group 𝐻 with Γ ⁢ ( H ) = Γ ⁢ ( G ) Gamma(H)=Gamma(G) is isomorphic to 𝐺. By a result of Cameron and Maslova, every such group must be almost simple, so one natural case to investigate is that in which 𝐺 is one of the 26 sporadic simple groups. Existing work of various authors answers the question of recognisability by prime graph for all but three of these groups, namely the Monster, M mathrm{M} , the Baby Monster, B mathrm{B} , and the first Conway group, Co 1 mathrm{Co}_{1} . We prove that these three groups are recognisable by their prime graphs.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83454055","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}
引用次数: 2
Orders of inner-diagonal automorphisms of some simple groups of Lie type 李型单群的内对角自同构的阶
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-07-27 DOI: 10.1515/jgth-2021-0192
A. Buturlakin, M. A. Grechkoseeva
{"title":"Orders of inner-diagonal automorphisms of some simple groups of Lie type","authors":"A. Buturlakin, M. A. Grechkoseeva","doi":"10.1515/jgth-2021-0192","DOIUrl":"https://doi.org/10.1515/jgth-2021-0192","url":null,"abstract":"Abstract We refer to the set of element orders of a finite group 𝐺 as the spectrum of 𝐺. For the simple groups PSL n ⁢ ( q ) mathrm{PSL}_{n}(q) , PSU n ⁢ ( q ) mathrm{PSU}_{n}(q) , E 6 ⁢ ( q ) E_{6}(q) , and E 6 2 ⁢ ( q ) {}^{2}E_{6}(q) , we describe the spectra of extensions of these groups by diagonal automorphisms.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83701344","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}
引用次数: 2
The Lie algebra structure of the degree one Hochschild cohomology of the blocks of the sporadic Mathieu groups 零星Mathieu群块的1次Hochschild上同调的李代数结构
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-07-22 DOI: 10.1515/jgth-2021-0176
William Murphy
{"title":"The Lie algebra structure of the degree one Hochschild cohomology of the blocks of the sporadic Mathieu groups","authors":"William Murphy","doi":"10.1515/jgth-2021-0176","DOIUrl":"https://doi.org/10.1515/jgth-2021-0176","url":null,"abstract":"Abstract Let 𝐺 be one of the sporadic simple Mathieu groups M 11 M_{11} , M 12 M_{12} , M 22 M_{22} , M 23 M_{23} or M 24 M_{24} , and suppose 𝑘 is an algebraically closed field of prime characteristic 𝑝, dividing the order of 𝐺. In this paper, we describe some of the Lie algebra structure of the first Hochschild cohomology groups of the 𝑝-blocks of k ⁢ G kG . In particular, we calculate the dimension of HH 1 ⁢ ( B ) mathrm{HH}^{1}(B) for the 𝑝-blocks 𝐵 of k ⁢ G kG , and in almost all cases, we determine whether HH 1 ⁢ ( B ) mathrm{HH}^{1}(B) is a solvable Lie algebra.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83529184","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}
引用次数: 2
On the proper enhanced power graphs of finite nilpotent groups 有限幂零群的适当增强幂图
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-07-14 DOI: 10.1515/jgth-2022-0057
S. Bera, Hiranya Kishore Dey
{"title":"On the proper enhanced power graphs of finite nilpotent groups","authors":"S. Bera, Hiranya Kishore Dey","doi":"10.1515/jgth-2022-0057","DOIUrl":"https://doi.org/10.1515/jgth-2022-0057","url":null,"abstract":"Abstract For a group 𝐺, the enhanced power graph of 𝐺 is a graph with vertex set 𝐺 in which two distinct vertices x , y x,y are adjacent if and only if there exists an element 𝑤 in 𝐺 such that both 𝑥 and 𝑦 are powers of 𝑤. The proper enhanced power graph is the induced subgraph of the enhanced power graph on the set G ∖ S Gsetminus S , where 𝑆 is the set of dominating vertices of the enhanced power graph. In this paper, we at first classify all nilpotent groups 𝐺 such that the proper enhanced power graphs are connected and calculate their diameter. We also explicitly calculate the domination number of the proper enhanced power graphs of finite nilpotent groups. Finally, we determine the multiplicity of the Laplacian spectral radius of the enhanced power graphs of nilpotent groups.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89270876","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}
引用次数: 4
On a family of representations of the Higman–Thompson groups 关于Higman-Thompson群的一族表示
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-07-06 DOI: 10.1515/jgth-2021-0190
Andr'e Guimaraes, P. R. Pinto
{"title":"On a family of representations of the Higman–Thompson groups","authors":"Andr'e Guimaraes, P. R. Pinto","doi":"10.1515/jgth-2021-0190","DOIUrl":"https://doi.org/10.1515/jgth-2021-0190","url":null,"abstract":"Abstract We obtain an uncountable family of inequivalent and irreducible representations of the Higman–Thompson groups F n ⊂ T n ⊂ V n F_{n}subset T_{n}subset V_{n} . This is accomplished by considering a family of representations of the Higman–Thompson groups V n V_{n} that arise from representations of Cuntz algebras, each one acting on a Hilbert space built upon the orbit of a point x ∈ [ 0 , 1 ) xin[0,1) under the dynamical system Φ ⁢ ( x ) = n ⁢ x ( mod 1 ) Phi(x)=nxpmod{1} . Every such representation is retrieved through the action of V n V_{n} on orb ⁡ ( x ) operatorname{orb}(x) , and their restrictions to the subgroups F n F_{n} and T n T_{n} of V n V_{n} are studied using properties of the groups.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89855506","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}
引用次数: 2
Algebraic groups over finite fields: Connections between subgroups and isogenies 有限域上的代数群:子群与同生之间的联系
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-06-29 DOI: 10.1515/jgth-2022-0110
Davide Sclosa
{"title":"Algebraic groups over finite fields: Connections between subgroups and isogenies","authors":"Davide Sclosa","doi":"10.1515/jgth-2022-0110","DOIUrl":"https://doi.org/10.1515/jgth-2022-0110","url":null,"abstract":"Abstract Let 𝐺 be a linear algebraic group defined over a finite field F q mathbb{F}_{q} . We present several connections between the isogenies of 𝐺 and the finite groups of rational points ( G ⁢ ( F q n ) ) n ≥ 1 (G(mathbb{F}_{smash{q^{n}}}))_{ngeq 1} . We show that an isogeny ϕ : G ′ → G phicolon G^{prime}to G over F q mathbb{F}_{q} gives rise to a subgroup of fixed index in G ⁢ ( F q n ) G(mathbb{F}_{smash{q^{n}}}) for infinitely many 𝑛. Conversely, we show that if 𝐺 is reductive, the existence of a subgroup H n H_{n} of fixed index 𝑘 for infinitely many 𝑛 implies the existence of an isogeny of order 𝑘. In particular, we show that the infinite sequence H n H_{n} is covered by a finite number of isogenies. This result applies to classical groups GL m mathrm{GL}_{m} , SL m mathrm{SL}_{m} , SO m mathrm{SO}_{m} , SU m mathrm{SU}_{m} , Sp 2 ⁢ m mathrm{Sp}_{2m} and can be extended to non-reductive groups if 𝑘 is prime to the characteristic. As a special case, we see that if 𝐺 is simply connected, the minimal indices of proper subgroups of G ⁢ ( F q n ) G(mathbb{F}_{smash{q^{n}}}) diverge to infinity. Similar results are investigated regarding the sequence ( G ⁢ ( F p ) ) p (G(mathbb{F}_{p}))_{p} by varying the characteristic 𝑝.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73801934","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}
引用次数: 0
A classification of finite primitive IBIS groups with alternating socle 具有交替社会的有限原始IBIS群的分类
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-06-03 DOI: 10.1515/jgth-2022-0099
Melissa Lee, Pablo Spiga
{"title":"A classification of finite primitive IBIS groups with alternating socle","authors":"Melissa Lee, Pablo Spiga","doi":"10.1515/jgth-2022-0099","DOIUrl":"https://doi.org/10.1515/jgth-2022-0099","url":null,"abstract":"Abstract Let 𝐺 be a finite permutation group on Ω. An ordered sequence ( ω 1 , … , ω ℓ ) (omega_{1},ldots,omega_{ell}) of elements of Ω is an irredundant base for 𝐺 if the pointwise stabilizer is trivial and no point is fixed by the stabilizer of its predecessors. If all irredundant bases of 𝐺 have the same cardinality, 𝐺 is said to be an IBIS group. Lucchini, Morigi and Moscatiello have proved a theorem reducing the problem of classifying finite primitive IBIS groups 𝐺 to the case that the socle of 𝐺 is either abelian or non-abelian simple. In this paper, we classify the finite primitive IBIS groups having socle an alternating group. Moreover, we propose a conjecture aiming to give a classification of all almost simple primitive IBIS groups.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79095642","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}
引用次数: 2
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