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On multidimensional Schur rings of finite groups 有限群的多维Schur环
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2023-02-02 DOI: 10.1515/jgth-2023-0032
Gang Chen, Qingchun Ren, Ilia N. Ponomarenko
{"title":"On multidimensional Schur rings of finite groups","authors":"Gang Chen, Qingchun Ren, Ilia N. Ponomarenko","doi":"10.1515/jgth-2023-0032","DOIUrl":"https://doi.org/10.1515/jgth-2023-0032","url":null,"abstract":"Abstract For any finite group 𝐺 and a positive integer 𝑚, we define and study a Schur ring over the direct power G m G^{m} , which gives an algebraic interpretation of the partition of G m G^{m} obtained by the 𝑚-dimensional Weisfeiler–Leman algorithm. It is proved that this ring determines the group 𝐺 up to isomorphism if m ≥ 3 mgeq 3 , and approaches the Schur ring associated with the group Aut ⁡ ( G ) operatorname{Aut}(G) acting on G m G^{m} naturally if 𝑚 increases. It turns out that the problem of finding this limit ring is polynomial-time equivalent to the group isomorphism problem.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84448814","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
The R ∞ R_{infty} property for nilpotent quotients of Generalized Solvable Baumslag–Solitar groups 广义可解baumslag -孤子群幂零商的R∞R＿ {infty}性质
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2023-01-31 DOI: 10.1515/jgth-2022-0129
Wagner C. Sgobbi, Da Silva, D. Vendrúscolo
{"title":"The R ∞ R_{infty} property for nilpotent quotients of Generalized Solvable Baumslag–Solitar groups","authors":"Wagner C. Sgobbi, Da Silva, D. Vendrúscolo","doi":"10.1515/jgth-2022-0129","DOIUrl":"https://doi.org/10.1515/jgth-2022-0129","url":null,"abstract":"Abstract We say a group 𝐺 has property R ∞ R_{infty} if the number R ⁢ ( φ ) R(varphi) of twisted conjugacy classes is infinite for every automorphism 𝜑 of 𝐺. For such groups, the R ∞ R_{infty} -nilpotency degree is the least integer 𝑐 such that G / γ c + 1 ⁢ ( G ) G/gamma_{c+1}(G) has property R ∞ R_{infty} . In this work, we compute the R ∞ R_{infty} -nilpotency degree of all Generalized Solvable Baumslag–Solitar groups Γ n Gamma_{n} . Moreover, we compute the lower central series of Γ n Gamma_{n} , write the nilpotent quotients Γ n , c = Γ n / γ c + 1 ⁢ ( Γ n ) Gamma_{n,c}=Gamma_{n}/gamma_{c+1}(Gamma_{n}) as semidirect products of finitely generated abelian groups and classify which invertible integer matrices can be extended to automorphisms of Γ n , c Gamma_{n,c} .","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88833587","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
Curvature distribution and hyperbolicity 曲率分布和双曲性
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2023-01-17 DOI: 10.1515/jgth-2022-0106
C. Chalk, M. Edjvet
{"title":"Curvature distribution and hyperbolicity","authors":"C. Chalk, M. Edjvet","doi":"10.1515/jgth-2022-0106","DOIUrl":"https://doi.org/10.1515/jgth-2022-0106","url":null,"abstract":"Abstract We describe a method, based on curvature distribution techniques on van Kampen diagrams, for proving finitely presented groups hyperbolic. We apply our method and show that the generalised Fibonacci group F ⁢ ( r , n ) F(r,n) is hyperbolic when r ≥ 3 rgeq 3 and n ≥ 6 ⁢ r + 1 ngeq 6r+1 and determine which of the groups F ⁢ ( 3 , n ) F(3,n) are hyperbolic.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84051218","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 generations by conjugate elements in almost simple groups with socle 2𝐹4(𝑞2)′ 近似单群共轭元的代论(2)𝐹4(𝑞2)
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-12-28 DOI: 10.1515/jgth-2022-0216
D. Revin, A. Zavarnitsine
{"title":"On generations by conjugate elements in almost simple groups with socle 2𝐹4(𝑞2)′","authors":"D. Revin, A. Zavarnitsine","doi":"10.1515/jgth-2022-0216","DOIUrl":"https://doi.org/10.1515/jgth-2022-0216","url":null,"abstract":"Abstract We prove that if L = F 4 2 ⁢ ( 2 2 ⁢ n + 1 ) ′ L={}^{2}F_{4}(2^{2n+1})^{prime} and 𝑥 is a nonidentity automorphism of 𝐿, then G = ⟨ L , x ⟩ G=langle L,xrangle has four elements conjugate to 𝑥 that generate 𝐺. This result is used to study the following conjecture about the 𝜋-radical of a finite group. Let 𝜋 be a proper subset of the set of all primes and let 𝑟 be the least prime not belonging to 𝜋. Set m = r m=r if r = 2 r=2 or 3 and m = r − 1 m=r-1 if r ⩾ 5 rgeqslant 5 . Supposedly, an element 𝑥 of a finite group 𝐺 is contained in the 𝜋-radical O π ⁡ ( G ) operatorname{O}_{pi}(G) if and only if every 𝑚 conjugates of 𝑥 generate a 𝜋-subgroup. Based on the results of this and previous papers, the conjecture is confirmed for all finite groups whose every nonabelian composition factor is isomorphic to a sporadic, alternating, linear, unitary simple group, or to one of the groups of type B 2 2 ⁢ ( 2 2 ⁢ n + 1 ) {}^{2}B_{2}(2^{2n+1}) , G 2 2 ⁢ ( 3 2 ⁢ n + 1 ) {}^{2}G_{2}(3^{2n+1}) , F 4 2 ⁢ ( 2 2 ⁢ n + 1 ) ′ {}^{2}F_{4}(2^{2n+1})^{prime} , G 2 ⁢ ( q ) G_{2}(q) , or D 4 3 ⁢ ( q ) {}^{3}D_{4}(q) .","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81422613","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 bound for the orders of centralizers of irreducible subgroups of algebraic groups 代数群的不可约子群的中心器阶的界
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-12-07 DOI: 10.1515/jgth-2022-0111
M. Liebeck
{"title":"A bound for the orders of centralizers of irreducible subgroups of algebraic groups","authors":"M. Liebeck","doi":"10.1515/jgth-2022-0111","DOIUrl":"https://doi.org/10.1515/jgth-2022-0111","url":null,"abstract":"Abstract We prove that if 𝐺 is a connected semisimple algebraic group of rank 𝑟, and 𝐻 is a subgroup of 𝐺 that is contained in no proper parabolic subgroup, then we have | C G ⁢ ( H ) | < c r ⁢ | Z ⁢ ( G ) | lvert C_{G}(H)rvert<c^{r}lvert Z(G)rvert , where 𝑐 is an absolute constant ( c = 16 c=16 if all simple factors of 𝐺 are classical, and c ≤ 197 cleq 197 in general).","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90460034","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
A classification of skew morphisms of dihedral groups 二面体群的偏斜态射分类
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-11-16 DOI: 10.1515/jgth-2022-0085
Kan Hu, I. Kovács, Young Soo Kwon
{"title":"A classification of skew morphisms of dihedral groups","authors":"Kan Hu, I. Kovács, Young Soo Kwon","doi":"10.1515/jgth-2022-0085","DOIUrl":"https://doi.org/10.1515/jgth-2022-0085","url":null,"abstract":"Abstract A skew morphism of a finite group 𝐴 is a permutation 𝜑 of 𝐴 fixing the identity element and for which there is an integer-valued function 𝜋 on 𝐴 such that φ ⁢ ( x ⁢ y ) = φ ⁢ ( x ) ⁢ φ π ⁢ ( x ) ⁢ ( y ) varphi(xy)=varphi(x)varphi^{pi(x)}(y) for all x , y ∈ A x,yin A . In this paper, we restrict ourselves to the case when A = D n A=D_{n} , the dihedral group of order 2 ⁢ n 2n . Wang et al. [Smooth skew morphisms of dihedral groups, Ars Math. Contemp. 16 (2019), 2, 527–547] determined all 𝜑 under the condition that π ( φ ( x ) ) ≡ π ( x ) ( mod | φ | ) ) pi(varphi(x))equivpi(x)pmod{lvertvarphirvert}) holds for every x ∈ D n xin D_{n} , and later Kovács and Kwon [Regular Cayley maps for dihedral groups, J. Combin. Theory Ser. B 148 (2021), 84–124] characterised those 𝜑 such that there exists an inverse-closed ⟨ φ ⟩ langlevarphirangle -orbit, which generates D n D_{n} . We show that these two types of skew morphisms comprise all skew morphisms of D n D_{n} . The result is used to classify the finite groups with a complementary factorisation into a dihedral and a core-free cyclic subgroup. As another application, a formula for the total number of skew morphisms of D p t D_{p^{t}} is also derived for any prime 𝑝.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79876747","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: 5
Diagonal embeddings of finite alternating groups 有限交替群的对角嵌入
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-11-15 DOI: 10.1515/jgth-2022-0059
S. Thomas
{"title":"Diagonal embeddings of finite alternating groups","authors":"S. Thomas","doi":"10.1515/jgth-2022-0059","DOIUrl":"https://doi.org/10.1515/jgth-2022-0059","url":null,"abstract":"Abstract We will present an alternative approach to Zalesskiĭ’s theorem on diagonal embeddings of finite alternating groups.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80924387","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
Conjugacy classes of maximal cyclic subgroups of metacyclic 𝑝-groups 亚环的极大环子群的共轭类𝑝-groups
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-10-15 DOI: 10.1515/jgth-2022-0103
M. Bianchi, R. Camina, M. Lewis
{"title":"Conjugacy classes of maximal cyclic subgroups of metacyclic 𝑝-groups","authors":"M. Bianchi, R. Camina, M. Lewis","doi":"10.1515/jgth-2022-0103","DOIUrl":"https://doi.org/10.1515/jgth-2022-0103","url":null,"abstract":"Abstract In this paper, we set η ⁢ ( G ) eta(G) to be the number of conjugacy classes of maximal cyclic subgroups of a finite group 𝐺. We compute η ⁢ ( G ) eta(G) for all metacyclic 𝑝-groups. We show that if 𝐺 is a metacyclic 𝑝-group of order p n p^{n} that is not dihedral, generalized quaternion, or semi-dihedral, then η ⁢ ( G ) ≥ n - 2 eta(G)geq n-2 , and we determine when equality holds.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72541660","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
The average character degree of finite groups and Gluck’s conjecture 有限群的平均特征度与Gluck猜想
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-09-19 DOI: 10.1515/jgth-2022-0120
Alexander Moret'o
{"title":"The average character degree of finite groups and Gluck’s conjecture","authors":"Alexander Moret'o","doi":"10.1515/jgth-2022-0120","DOIUrl":"https://doi.org/10.1515/jgth-2022-0120","url":null,"abstract":"Abstract We prove that the order of a finite group 𝐺 with trivial solvable radical is bounded above in terms of acd ⁡ ( G ) operatorname{acd}(G) , the average degree of the irreducible characters. It is not true that the index of the Fitting subgroup is bounded above in terms of acd ⁡ ( G ) operatorname{acd}(G) , but we show that, in certain cases, it is bounded in terms of the degrees of the irreducible characters of 𝐺 that lie over a linear character of the Fitting subgroup. This leads us to propose a refined version of Gluck’s conjecture.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74759409","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
Character degrees of normally monomial 𝑝-groups of maximal class 极大类正常单项𝑝-groups的特征度
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2022-09-12 DOI: 10.1515/jgth-2021-0212
Dongfang Yang, Heng Lv
{"title":"Character degrees of normally monomial 𝑝-groups of maximal class","authors":"Dongfang Yang, Heng Lv","doi":"10.1515/jgth-2021-0212","DOIUrl":"https://doi.org/10.1515/jgth-2021-0212","url":null,"abstract":"Abstract A finite group 𝐺 is normally monomial if all its irreducible characters are induced from linear characters of normal subgroups of 𝐺. In this paper, we study the largest irreducible character degree and the maximal abelian normal subgroup of normally monomial 𝑝-groups of maximal class in terms of 𝑝. In particular, we determine all possible irreducible character degree sets of normally monomial 5-groups of maximal class.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91007576","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
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