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Some open questions in the theory of generalized permutable subgroups

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Science China Mathematics 2009年第10期52卷 2132-2144页

作者： GUO WenBin XIE FengYan LI BaoJun School of Mathematical Sciences Xuzhou Normal UniversityXuzhou 221116China School of Mathematics Chengdu University of Information TechnologyChengdu 610225China

A subgroup H of a group G is said to be weakly s-supplemented in G if H has a supplement T in G such that H ∩ T HsG, where HsG is the largest s-permutable subgroup of G contained in H. This paper constructs an example to show that the open questions 6.3 and 6.4 in J Algebra, 315: 192–209 (2007) have negative solutions, and shows that in many cases Question 6.4 is positive. A series of known results are unified and generalized.

关键词： finite group formations weakly s-supplemented subgroup maximal subgroup minimal subgroup 20D10 20D20

On an open problem of Guo-Skiba

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Frontiers of Mathematics in China 2016年第6期11卷 1603-1612页

作者： Zhenfeng WU Wenbin GUO Baojun LI Department of Mathematics University of Science and Technology of China Hefei 230026 China College of Applied Mathematics Chengdu University of information Technology Chengdu 610225 China

Let G be a finite group, and let A be a proper subgroup of G. Then any chief factor H/AG of G is called a G-boundary factor of A. For any G- boundary factor H/AG of A, the subgroup （A ∩ H）/AG of G/AG is called a G-trace of A. In this paper, we prove that G is p-soluble if and only if every maximal chain of G of length 2 contains a proper subgroup M of G such that either some G-trace of M is subnormal or every G-boundary factor of M is a p′- group. This result give a positive answer to a recent open problem of Guo and Skiba. We also give some new characterizations of p-hypercyclically embedded subgroups.

关键词： finite group p-hypercyclically embedded subgroup G-boundary factor G-trace of subgroup meet-irreducible subgroup

Recognizing the Automorphism groups of Mathieu groups Through Their Orders and Large Degrees of Their Irreducible Characters＊

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Chinese Annals of Mathematics,Series B 2016年第4期37卷 495-502页

作者： Yanxiong YAN Liangcai ZHANG Haijing XU Guiyun CHEN School of Mathematics and Statistics Southwest University Chongqing 400715 China College of Mathematics and Statistics Chongqing University Chongqing 401331 China

It is a well-known fact that characters of a finite group can give important information about the structure of the group. It was also proved by the third author that a finite simple group can be uniquely determined by its character table. Here the authors attempt to investigate how to characterize a finite almost-simple group by using less information of its character table, and successfully characterize the automorphism groups of Mathieu groups by their orders their character tables. and at most two irreducible character degrees of

关键词： finite group Character degrees Irreducible characters Simple groups Mathieu groups

A Characterization of PSL(3. q) for q=2~m

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Acta Mathematica Sinica,English Series 2002年第3期18卷 463-472页

作者： A.IRANMANESH S.H.ALAVI B.KHOSRAVI Department of Mathematics Tarbiat Modarres University P. O. Bax: 14115-137 Tehran. Iran Institute of Studies in Theoretical Physics and Mathematics Tehran Iran

The order components of a finite group are introduced in [12]. In [9], it is proved that the group PSL(3, q), where q is an odd prime power, is uniquely determined by its order components. In this paper, we show that ... 详细信息

关键词： finite group Simple group Prime graph Order component

On subgroup functors of finite soluble groups

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Science China Mathematics 2017年第3期60卷 439-448页

作者： BALLESTER-BOLINCHES Adolfo COSME-LLOPEZ Enric KAMORNIKOV Sergey Fedorovich Departament d'Algebra Universitat de Valencia Burjassot 46100 Spain Gomel Branch of International University "MITSO" Gomel 246029 Belarus

The principal aim of this paper is to study the regular and transitive subgroup functors in the universe of all finite soluble *** prove that they form a complemented and non-modular lattice containing two relevant **... 详细信息

关键词： finite group soluble group lattices of subgroups subgroup functors formations

The Hecke Algebra of a Frobenius P-Category

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Algebra Colloquium 2014年第1期21卷 1-52页

作者： Lluis Puig CNRS Institut de Mathematiques de Jussieu 6 Av Bizet 94340 Joinville-le-Pont France

We introduce a new avatar of a Frobenius P-category F under the form of a suitable subring HF of the double Burnside ring of P -- called the Hecke algebra of F- where we are able to formulate：（i） the generalization to a Frobenius P-category of the Alperin Fusion Theorem, （ii） the ＂canonical decomposition＂ of the morphisms in the exterior quotient of a Frobenius P-category restricted to the selfcentralizing objects as developed in chapter 6 of [4], and （iii） the ＂basic P × P-sets＂ in chapter 21 of [4] with its generalization by Kari Ragnarsson and Radu Stancu to the virtual P ×P-sets in [6]. We also explain the relationship with the usual Hecke algebra of a finite group.

关键词： finite group Frobenius category Hecke algebra Burnside ring

A New Characterization of L_2(r) by Their Sylow Numbers

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Acta Mathematica Sinica,English Series 2015年第10期31卷 1593-1598页

作者： Alireza Khalili ASBOEI Department of Mathematics Farhangian University Department of Mathematics Buinzahra Branch Islamic Azad University

Let G be a finite centerless group, let π（G） be the set of prime divisors of the order of G, and let np（G） be the number of Sylow p-subgroups of G, that is, np（G） = ｜Sylp（G）｜. Set NS（G）：= ｜np（G）｜ p ∈ π（G）}. In this paper, we are investigating whether L2（r） is determined up to isomorphism by NS（L2（r）） when r is prime.

关键词： finite group simple group Sylow subgroup

NON-ISOMORPHIC groupS WITH ISOMORPHICSPECTRAL TABLES AND BURNSIDE MATRICES

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Chinese Annals of Mathematics,Series B 1994年第3期15卷 273-282页

作者： W. KIMMERLE K. W. ROGGENKAMP(Mathematisches institut B, University of Stuttgalt, Pfthenwaldring 57, 70550 Stuttgart, Germany.) University of Stuttgart 德国

It was shown by Formanek and Sibley that the group determined characterizes a finite groupG up to isomorphism. Hoehnke and Johnson (independelltly the suthors--using an argumentof Manslield) showed the corresponding result for k-characters, k = 1, 2, 3. The notion of kcharacters dates back to nobenius. They are determined by the group doterminaDt and maybe derived from the character table CT(G) provided one knows additionally the functionswhere C(C) = {Cg, g E G} denotes the set of conjugacy classes of *** object of the paper is to present criteria for finite groups (more precisely for solublegroups G and H which are both semi-direct products of a similar type) when1. G and H have isomorphic spectral tables (i.e., they form a Brauer pair),2. G and H have isomorphic table of marks (in particular the Burnside rings are isomorphic),3. G and H have the same *** this the authors construct two *** soluble groups for which all these threerepresent at iont heor et ical invar taut s coincide.

关键词： finite group Spectral table Burnside matriX Isomorphism.

On finite Non-Solvable groups Whose Gruenberg-Kegel Graphs are Isomorphic to the Paw

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Communications in Mathematics and Statistics 2022年第4期10卷 653-667页

作者： A.S.Kondrat’ev N.A.Minigulov N.N.Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences 16 S.Kovalevskaya StrYekaterinburg 620108Russia

The Gruenberg-Kegel graph(or the prime graph)Γ(G)of a finite group G is a graph,in which the vertex set is the set of all prime divisors of the order of G and two different vertices p and q are adjacent if and only if there exists an element of order pq in *** paw is a graph on four vertices whose degrees are 1,2,2,*** consider the problem of describing finite groups whose Gruenberg-Kegel graphs are isomorphic as abstract graphs to the *** example,the Gruenberg-Kegel graph of the alternating group A_(10)of degree 10 is isomorphic as abstract graph to the *** this paper,we describe finite non-solvable groups G whose Gruenberg-Kegel graphs are isomorphic as abstract graphs to the paw in the case when G has no elements of order 6 or the vertex of degree 1 ofΓ(G)divides the order of the solvable radical of G.

关键词： finite group Non-solvable group Gruenberg-Kegel graph The paw