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.
{"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}
Jinzhuan Cai, Zhigang Wang, I. N. Safonova, A. Skiba
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}
{"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}
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.
研究了Johnson和Mawdesley引入的Fibonacci型群G= gn¹(x 0¹x m¹x k -1) G=G_{n}(x_{0}x_{m}x_{k}^{-1})的位移动力学。主要结果涉及到𝐺的移位自同构的阶数以及确定它是否为外自同构，并且我们发现当且仅当𝐺不完全时才存在外自同构。Bogley的结果给出了非球面表示决定了在𝐺的非恒等元素上有Z n mathbb{Z}_{n}自由移位的群，由此得出当𝐺是非平凡时，移位是一个𝑛阶的外自同构。因此，本文的重点是非球面的情况，包括斐波那契群和西拉德斯基群。除了少数例外，移位自同构的不动点和自由问题都得到了解决，在某些情况下使用计算和拓扑方法。
{"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}
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.
{"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}
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.
摘要:我们把有限群𝐺的元素阶的集合称为𝐺的谱。对于简单群PSL n n (q) mathm {PSL}_{n}(q)， PSU n n (q) mathm {PSU}_{n}(q)， e6 (q)， e6 2 (q) {}^{2}E_{6}(q)，我们用对角自同构描述了这些群的扩展谱。
{"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}
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.
摘要设𝐺为散在的简单Mathieu群M 11 M_{11}， M 12 M_{12}， M 22 M_{22}， M 23 M_{23}或M 24 M_{24}中的一个，设𝑘为素数特征的代数闭域𝑝，分𝐺的阶。本文描述了k ^ gkg的𝑝-blocks的第一Hochschild上同调群的一些李代数结构。特别是,我们计算的维数HH 1(B) mathrm {HH} ^ {1} (B)𝑝-blocks𝐵kG公斤,在几乎所有的情况下,我们决定HH 1(B) mathrm {HH} ^ {1} (B)是一个可解李代数。
{"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}
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.
{"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}
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.
摘要我们得到了Higman-Thompson群F n∧T n∧V n F_{n}子集T_{n}子集V_{n}的不可数不等式和不可约表示族。这是通过考虑由Cuntz代数表示产生的Higman-Thompson群V n V_{n}的一系列表示来实现的，每个表示作用于希尔伯特空间，该空间建立在点x∈[0,1)xin[0,1)的轨道上，在动力系统Φ (x)=n (x) Phi(x)=nxpmod{1}下。通过V n V_{n}对orb (x) 算子名{orb}(x)的作用来检索每一个这样的表示，并利用群的性质研究了它们对V n V_{n}的子群F n F_{n}和T n T_{n}的限制。
{"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}
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.
摘要设𝐺是定义在有限域F q mathbb{F} ＿q{上的一个线性代数群。我们给出了𝐺的等同性与有理点(G≠(F q n)) n≥1 (G(}mathbb{F} ＿ {smash{q^{n}}})){＿ngeq 1的有限群之间的几个联系}。我们证明了一个同形形φ: G '→Gphicolon G^ {prime}to G / F q mathbb{F} ＿q{在无穷多个𝑛中产生一个固定指标的子群G(F q n) G(}mathbb{F} ＿ {smash{q^{n}}})。反之，我们证明了如果𝐺是约化的，那么对于无穷多个𝑛，固定指标𝑘的子群H {n H＿n}的存在意味着存在一个𝑘阶的同工。特别地，我们证明了无限序列{hn＿h＿n}被有限个同基因所覆盖。这一结果适用于经典群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{，如果𝑘是特征的素数，则可以推广到非约化群。作为一种特殊情况，我们看到，如果𝐺是单连通的，那么G¹(F q n) G(}mathbb{F} ＿ {smash{q^{n}}})的固有子群的最小指标发散到无穷大。通过改变特征𝑝，对序列(G¹(F p)) p (G(mathbb{F} ＿p{))}＿p{也得到了类似的结果。}
{"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}
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.
{"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}
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