Topic
Matrix group
About: Matrix group is a research topic. Over the lifetime, 1067 publications have been published within this topic receiving 20436 citations.
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Papers
Book•
1 Jan 1955
TL;DR: In this article, the authors define the notion of permutation groups as a group of linear substitutions, and show that a group can be represented as a permutation-group.
Abstract: Preface to the second edition Preface to the first edition 1. On permutations 2. The definition of a group 3. On the simpler properties of a group which are independent of its mode of representation 4. Further properties of a group which are independent of its mode of representation 5. On the composition-series of a group 6. On the isomorphism of a group within itself 7. On Abelian groups 8. On groups whose orders are the powers of primes 9. On Sylow's theorem 10. On permutation-groups: transitive and intransitive groups 11. On permutation-groups: transitivity and primitivity 12. On the representation of a group of finite order as a permutation-group 13. On groups of linear substitutions 14. On the representation of a group of finite order as a group of linear substitutions 15. On group-characteristics 16. Some applications of the theory of groups of linear substitutions and of group-characteristics 17. On the invariants of groups of linear substitutions 18. On the graphical representation of a group 19. On the graphical representation of groups 20. On congruence groups Notes Index of technical terms Index of authors quoted General index.
1,226 citations
1 Dec 1985
TL;DR: The aim of this paper is to replace most of the (proven and unproven) group theory of [BS] by elementary combinatorial arguments and defines a new hierarchy of complexity classes “just above NP
Abstract: In a previous paper [BS] we proved, using the elements of the theory of nilpotent groups, that some of the fundamental computational problems in matriz groups belong to NP. These problems were also shown to belong to coNP, assuming an unproven hypothesis concerning finite simple groups.The aim of this paper is to replace most of the (proven and unproven) group theory of [BS] by elementary combinatorial arguments. The result we prove is that relative to a random oracle B, the mentioned matrix group problems belong to (NP∩coNP)B.The problems we consider are membership in and order of a matrix group given by a list of generators. These problems can be viewed as multidimensional versions of a close relative of the discrete logarithm problem. Hence NP∩coNP might be the lowest natural complexity class they may fit in.We remark that the results remain valid for black box groups where group operations are performed by an oracle.The tools we introduce seem interesting in their own right. We define a new hierarchy of complexity classes AM(k) “just above NP”, introducing Arthur vs. Merlin games, the bounded-away version of Papdimitriou's Games against Nature. We prove that in spite of their analogy with the polynomial time hierarchy, the finite levels of this hierarchy collapse to AM=AM(2). Using a combinatorial lemma on finite groups [BE], we construct a game by which the nondeterministic player (Merlin) is able to convince the random player (Arthur) about the relation [G]=N provided Arthur trusts conclusions based on statistical evidence (such as a Slowly-Strassen type “proof” of primality).One can prove that AM consists precisely of those languages which belong to NPB for almost every oracle B.Our hierarchy has an interesting, still unclarified relation to another hierarchy, obtained by removing the central ingredient from the User vs. Expert games of Goldwasser, Micali and Rackoff.
1,003 citations
Book•
13 Nov 1979
TL;DR: In this paper, the basic subject matter of algebraic matrix groups is discussed, including the following: 1.1 What We are Talking About.- 1.2 Representable Functors, 2.3 Natural Maps and Yoneda's Lemma, 3.4 Realization as Matrix Groups, 4.5 Translating from Group to Algebraic Matrix Group, 5.5 Construction of All Representations, and 6.
Abstract: I The Basic Subject Matter.- 1 Affine Group Schemes.- 1.1 What We Are Talking About.- 1.2 Representable Functors.- 1.3 Natural Maps and Yoneda's Lemma.- 1.4 Hopf Algebras.- 1.5 Translating from Groups to Algebras.- 1.6 Base Change.- 2 Affine Group Schemes: Examples.- 2.1 Closed Subgroups and Homomorphisms.- 2.2 Diagonalizable Group Schemes.- 2.3 Finite Constant Groups.- 2.4 Cartier Duals.- 3 Representations.- 3.1 Actions and Linear Representations.- 3.2 Comodules.- 3.3 Finiteness Theorems.- 3.4 Realization as Matrix Groups.- 3.5 Construction of All Representations.- 4 Algebraic Matrix Groups.- 4.1 Closed Sets in kn.- 4.2 Algebraic Matrix Groups.- 4.3 Matrix Groups and Their Closures.- 4.4 From Closed Sets to Functors.- 4.5 Rings of Functions.- 4.6 Diagonalizability.- II Decomposition Theorems.- 5 Irreducible and Connected Components.- 5.1 Irreducible Components in kn.- 5.2 Connected Components of Algcbraic Matrix Groups.- 5.3 Components That Coalesce.- 5.4 Spec A.- 5.5 The Algebraic Meaning of Connectedness.- 5 6 Vista: Schemes.- 6 Connected Components and Separable Algebras.- 6.1 Components That Decompose.- 6.2 Separable Algebras.- 6.3 Classification of Separable Algebras.- 6.4 Etale Group Schemes 49 6 5 Separable Subalgcbras.- 6.5 Separable Subalgcbras.- 6.6 Connected Group Schemes.- 6.7 Connected Components of Group Schemes.- 6.8 Finite Groups over Perfect Fields.- 7 Groups of Multiplicative Type.- 7.1 Separable Matrices.- 7.2 Groups of Multiplicative Type.- 7.3 Character Groups.- 7.4 Anisotropic and Split Tori.- 7.5 Examples of Tori.- 7.6 Some Automorphism Group Schcmes.- 7.7 A Rigidity Theorem.- 8 Unipotent Groups.- 8.1 Unipotent Matrices.- 8 2 The Kolchin Fixed Point Theorem.- 8.3 Unipotent Group Schemes.- 8.4 Endomorphisms of Ga..- 8.5 Finite Unipotent Groups.- 9 Jordan Decomposition.- 9.1 Jordan Decomposition of a Matrix.- 9.2 Decomposition in Algebraic Matrix Groups.- 9.3 Decomposition of Abelian Algebraic Matrix Groups.- 9.4 Irreducible Representations of Abelian Group Schemes.- 9.5 Decomposition of Abelian Group Schemes.- 10 Nilpotent and Solvable Groups.- 10.1 Derived Subgroups.- 10.2 The Lie-Kolchin Triangularization Theorem.- 10.3 The Unipotent Subgroup.- 10.4 Decomposition of Nilpotent Groups.- 10.5 Vista: Borel Subgroups.- 10.6 Vista: Differential Algebra.- III The Infinitesimal Theory.- 11 Differentials.- 11.1 Derivations and Differentials.- 11.2 Simple Properties of Differentials.- 11.3 Differentials of Hopf Algebras.- 11.4 No Nilpotents in Characteristic Zero.- 11.5 Differentials of Field Extensions.- 11.6 Smooth Group Schemes.- 11.7 Vista: The Algebro-Geomctric Meaning of Smoothness.- 11.8 Vista: Formal Groups.- 12 Lie Algebras.- 12.1 Invariant Operators and Lie Algebras.- 12.2 Computation or Lie Algebras.- 12.3 Examples.- 12.4 Subgroups and Invariant Subspaces.- 12.5 Vista: Reductive and Semisimple Groups.- IV Faithful Flatness and Quotients.- 13 Faithful Flatness.- 13.1 Definition of Faithful Flatness.- 13.2 Localization Properties.- 13.3 Transition Properties.- 13.4 Generic Faithful Flatness.- 13.5 Proof of the Smoothness Theorem.- 14 Faithful Flatness of Hopf Algebras.- 14.1 Proof in the Smooth Case.- 14.2 Proof with Nilpotents Present.- 14.3 Simple Applications.- 14.4 Structure of Finite Connected Groups.- 15 Quotient Maps.- 15.1 Quotient Maps.- 15.2 Matrix Groups over$$ bar k $$/k.- 15.3 Injections and Closed Kmbeddings.- 15.4 Universal Property of Quotients.- 15.5 Sheaf Property of Quotients.- 15.6 Coverings and Sheaves.- 15.7 Vista: The Etale Topology.- 16 Construction of Quotients.- 16.1 Subgroups as Stabilizers.- 16.2 Difficulties with Coset Spaces.- 16.3 Construction of Quotients.- 16.4 Vista: Invariant Theory.- V Descent Theory.- 17 Descent Theory Formalism.- 17.1 Descent Data.- 17.2 The Descent Theorem.- 17.3 Descent of Algebraic Structure.- 17.4 Example: Zariski Coverings.- 17.5 Construction of Twisted Forms.- 17.6 Twisted Forms and Cohomology.- 17.7 Finite Galois Extensions.- 17.8 Infinite Galois Extensions.- 18 Descent Theory Computations.- 18.1 A Cohomology Exact Sequence.- 18.2 Sample Computations.- 18.3 Principal Homogeneous Spaces.- 18.4 Principal Homogeneous Spaces and Cohomology.- 18.5 Existence of Separable Splitting Fields.- 18.6 Example: Central Simple Algebras.- 18.7 Example: Quadratic Forms and the Arf Invariant.- 18.8 Vanishing Cohomology over Finite Fields.- Appendix: Subsidiary Information.- A.1 Directed Sets and Limits.- A.2 Exterior Powers.- A.3 Localization. Primes, and Nilpotents.- A.4 Noetherian Rings.- A.5 The Hilbert Basis Theorem.- A.6 The Krull Intersection Theorem.- A.7 The Nocthcr Normalization Lemma.- A.8 The Hilbert Nullstellensatz.- A.9 Separably Generated Fields.- A.10 Rudimentary Topological Terminology.- Further Reading.- Index of Symbols.
957 citations
[...]
TL;DR: In this paper, the geometry of groups of Lie type was studied and the generalized Fitting subgroup was proposed. But it was not shown how to represent groups on groups and how to express finite groups.
Abstract: 1. Preliminary results 2. Permutation representations 3. Representation of groups on groups 4. Linear representations 5. Permutation groups 6. Extensions of groups and modules 7. Spaces with forms 8. p-groups 9. Change of field of a linear representation 10. Presentation of groups 11. The generalized Fitting subgroup 12. Linear representation of finite groups 13. Transfer and fusion 14. The geometry of groups of Lie type 15. Signalizer functors 16. Finite simple groups References List of symbols Index.
881 citations
Book•
21 Mar 2006
TL;DR: The Frobenius algebra of the symmetric group is studied in this article, where the characters of the group characters and the structure of continuous matrix groups and invariant matrices of unitary matrices are discussed.
Abstract: Matrices Algebras Groups The Frobenius algebra The symmetric group Immanants and $S$-functions $S$-functions of special series The calculation of the characters of the symmetric group Group characters and the structure of groups Continuous matrix groups and invariant matrices Groups of unitary matrices Appendix Bibliography Supplementary bibliography Index.
758 citations
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