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factorizations of Finite Groups and Related Topics

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Algebra Colloquium 2020年第1期27卷 149-180页

作者： Lev Kazarin Department of Mathematics Yaroslavl State UniversityRussia

This is a survey on the recent progress in the theory of finite groups with factorizations and around it,done by the author and his coauthors,and this has no pretensions to cover all topics in this wide area of *** particular,we only touch the great consequences of the fundamental paper of Liebeck,Praeger and Saxl on maximal factorizations of almost simple finite groups for the theory of groups with *** each case the reader can find additional references at the end of Section *** of the methods of investigation can be used to obtain information about finite groups in general,nilpotent algebras and related nearrings.

关键词： finite groups products of groups conjugacy classes Sylow subgroups Lie type groups factorizations

On Rank factorizations and Factor Prime factorizations for Multivariate Polynomial Matrices

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Journal of Systems Science & Complexity 2018年第6期31卷 1647-1658页

作者： GUAN Jiancheng LI Weiqing OUYANG Baiyu College of Mathematics and Computer Science Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China)Hunan Normal UniversityChangsha 410081China Department of Mathematics Xiangnan UniversityChenzhou 423000China College of Mathematics and Computer Science Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China)Hunan Normal UniversityChangsha 410081China

In this paper,rank factorizations and factor left prime factorizations are *** authors prove that any polynomial matrix with full row rank has factor left prime *** for a class of polynomial matrices,the authors give an algorithm to decide whether they have rank factorizations or factor left prime factorizations and compute these factorizations if they exist.

关键词： Factor prime factorizations multivariate polynomial matrices rank factorizations

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factorizations Orthogonal to a Subgraph in Graphs

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数学进展 1997年第5期26卷 81-83页

作者：李国君刘桂真山东大学数学与系统科学系山东济南

ＦａｃｔｏｒｉｚａｔｉｏｎｓＯｒｔｈｏｇｏｎａｌｔｏａＳｕｂｇｒａｐｈｉｎＧｒａｐｈｓＬｉＧｕｏｊｕｎ（李国君）ＬｉｕＧｕｉｚｈｅｎ（刘桂真）（Ｄｅｐｔ．，ｏｆＭａｔｈ．ａｎｄＳｙｓｔｅｍｓＳｃｉｅｎｃｅ，Ｓｈａｎｄ... 详细信息

ＦａｃｔｏｒｉｚａｔｉｏｎｓＯｒｔｈｏｇｏｎａｌｔｏａＳｕｂｇｒａｐｈｉｎＧｒａｐｈｓＬｉＧｕｏｊｕｎ（李国君）ＬｉｕＧｕｉｚｈｅｎ（刘桂真）（Ｄｅｐｔ．，ｏｆＭａｔｈ．ａｎｄＳｙｓｔｅｍｓＳｃｉｅｎｃｅ，ＳｈａｎｄｏｎｇＵｎｉｖｅｒｓｉｔｙ，Ｊｉｎ...

关键词：李国君刘桂 Graphs Orthogonal Subgraph factorizations

Aurifeuillian factorizations of q~q±1(q=p~n)

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数学进展 1997年第1期 75-76页

作者：孙琦洪绍方

ＡｕｒｉｆｅｕｉｌｉａｎＦａｃｔｏｒｉｚａｔｉｏｎｓｏｆｑｑ±１（ｑ＝ｐｎ）ＳｕｎＱｉ（孙琦）ＨｏｎｇＳｈａｏｆａｎｇ（洪绍方）（Ｄｅｐｔ．ｏｆＭａｔｈ．，ＳｉｃｈｕａｎＵｎｉｖｅｒｓｉｔｙ，Ｃｈｅｎｇｄｕ，Ｓｉｃｈ... 详细信息

ＡｕｒｉｆｅｕｉｌｉａｎＦａｃｔｏｒｉｚａｔｉｏｎｓｏｆｑｑ±１（ｑ＝ｐｎ）ＳｕｎＱｉ（孙琦）ＨｏｎｇＳｈａｏｆａｎｇ（洪绍方）（Ｄｅｐｔ．ｏｆＭａｔｈ．，ＳｉｃｈｕａｎＵｎｉｖｅｒｓｉｔｙ，Ｃｈｅｎｇｄｕ，Ｓｉｃｈｕａｎ，６１００６４）Ｃｏｍｍｕｎ...

关键词：孙琦 q~q 绍方 factorizations Aurifeuillian

Analytic Theory of Finite Asymptotic Expansions in the Real Domain. Part II-B: Solutions of Differential Inequalities and Asymptotic Admissibility of Standard Derivatives

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Advances in Pure Mathematics 2015年第8期5卷 481-502页

作者： Antonio Granata Department of Mathematics and Computer Science University of Calabria Cosenza Italy

Part II-B of our work continues the factorizational theory of asymptotic expansions of type (*) , , where the asymptotic scale , , is assumed to be an extended complete Chebyshev system on a one-sided neighborhood of x0. The main result states that to each scale of this type it remains as-sociated an important class of functions (namely that of generalized convex functions) enjoying the property that the expansion (*), if valid, is automatically formally differentiable n ? 1 times in the two special senses characterized in Part II-A. A second result shows that formal applications of ordinary derivatives to an asymptotic expansion are rarely admissible and that they may also yield skew results even for scales of powers.

关键词： Asymptotic Expansions Formal Differentiation of Asymptotic Expansions factorizations of Ordinary Differential Operators Chebyshev Asymptotic Scales

On Certain Distributive Lattices of Subgroups of Finite Soluble Groups

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Acta Mathematica Sinica,English Series 2007年第11期23卷 2069-2078页

作者： L.M.EZQUERRO X.SOLER-ESCRIV InformáticaUniversidad Pública de Navarra Campus de Arrosadía.31006 PamplonaSpain Departamentde Matemàtica Aplicada Universitat d'AlacantCampus de Sant Vicent.Ap.Correus 99-03080 AlacantSpain

In this paper, we prove the following result. Let ξ be a saturated formation and ∑ a Hall system of a soluble group G. Let X be a w-solid set of maximal subgroups of G such that ∑ reduces into each element of X. Consider in G the following three subgroups： the ξ-normalizer D of G associated with ∑; the X-prefrattini subgroup W = W（G, X） of G; and a hypercentrally embedded subgroup T of G. Then the lattice ζ（T, W, D） generated by T, D and W is a distributive lattice of pairwise permutable subgroups of G with the cover and avoidance property. This result remains true for the lattice ,ζ（V, W, D）, where V is a subgroup of G whose Sylow subgroups are also Sylow subgroups of hypercentrally embedded subgroups of G such that ∑ reduces into V.

关键词： lattice properties permutability factorizations cover and avoidance properties

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The Factorizational Theory of Finite Asymptotic Expansions in the Real Domain: A Survey of the Main Results

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Advances in Pure Mathematics 2015年第1期5卷 1-20页

作者： Antonio Granata Dipartimento di Matematica e Informatica Università della Calabria Cosenza Italy

After studying finite asymptotic expansions in real powers, we have developed a general theory for expansions of type (*) ,x → x0 where the ordered n-tuple forms an asymptotic scale at x0 , i.e. as x → x0, 1 ≤ i ≤ n – 1, and is practically assumed to be an extended complete Chebyshev system on a one-sided neighborhood of x o. As in previous papers by the author concerning polynomial, real-power and two-term theory, the locution “factorizational theory” refers to the special approach based on various types of factorizations of a differential operator associated to . Moreover, the guiding thread of our theory is the property of formal differentiation and we aim at characterizing some n-tuples of asymptotic expansions formed by (*) and n -1 expansions obtained by formal applications of suitable linear differential operators of orders 1,2,…,n-1. Some considerations lead to restrict the attention to two sets of operators naturally associated to “canonical factorizations”. This gives rise to conjectures whose proofs build an analytic theory of finite asymptotic expansions in the real domain which, though not elementary, parallels the familiar results about Taylor’s formula. One of the results states that to each scale of the type under consideration it remains associated an important class of functions (namely that of generalized convex functions) enjoying the property that the expansion(*), if valid, is automatically formally differentiable n-1 times in two special senses.

关键词： Asymptotic Expansions Formal Differentiation of Asymptotic Expansions factorizations of Ordinary Differential Operators Chebyshev Asymptotic Scales