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| Paul-Hermann Zieschang received a Doctor of Natural Sciences and the Habilitation in Mathematics from the Christian-Albrechts-Universität zu Kiel. He is also Extraordinary Professor of the Christian-Albrechts-Universität zu Kiel. Presently, he holds the position of an Associate Professor at the University of Texas at Brownsville. He held visiting positions at Kansas State University and at Kyushu University in Fukuoka. |
| 1 Basic Facts 1.1 Structure Constants 1.2 Symmetric Elements 1.3 The Complex Product 1.4 Complex Products and Valencies 1.5 Complex Products of Subsets of Cardinality 1 2 Closed Subsets 2.1 Basic Facts 2.2 Dedekind Identities 2.3 Structure Constants 2.4 Maximal Closed Subsets 2.5 Normalizer and Strong Normalizer 2.6 Conjugates of Closed Subsets 3 Generating Subsets 3.1 Basic Facts 3.2 The Thin Residue 3.3 Elements of Valency 2 3.4 Closed Subsets Generated by Involutions 3.5 Basic Results on Constrained Sets of Involutions. 3.6 Basic Results on Coxeter Sets 4 Quotient Schemes 4.1 Basic Definitions 4.2 General Facts 4.3 Valencies 4.4 Hall Subsets 4.5 Sylow Subsets 5 Morphisms 5.1 Basic Facts 5.2 Isomorphisms 5.3 The Isomorphism Theorems 5.4 Composition Series 5.5 The Group Correspondence 5.6 Residually Thin Schemes 6 Faithful Maps 6.1 Basic Facts 6.2 Faithfully Embedded Closed Subsets 6.3 The Schur Group of a Closed Subset 6.4 Elements of Valency 2 6.5 More About Elements of Valency 2 6.6 Constrained Sets of Involutions 6.7 Thin Thin Residues 7 Products 7.1 Direct Products of Closed Subsets 7.2 Quasidirect Products of Schemes 7.3 Semidirect Products 7.4 A Characterization of Semidirect Products 8 From Thin Schemes to Modules 8.1 Rings and Modules 8.2 Integrality in Associative Rings with 1 8.3 Completely Reducibility 8.4 Irreducible Modules over Associative Rings with 1 8.5 Semisimple Associative Rings with 1 8.6 Characters of Associative Rings with 1 8.7 Roots of Unity in Integral Domains 9 Scheme Rings 9.1 Basic Facts 9.2 Algebraically Closed Base Fields 9.3 Scheme Rings over the Field of Complex Numbers 9.4 Closed Subsets 9.5 Schemes with at most Five Elements 9.6 Constrained Sets of Involutions 10 Dihedral Closed Subsets 10.1 General Remarks 10.2 The Spherical Case 10.3 Arithmetic of the Length Function 10.4 Two Characteristic Subsets 10.5 The Constrained Spherical Case 10.6 Dihedral Closed Subsets of Finite Valency 11 Coxeter Sets 11.1 Parabolic Subsets 11.2 Direct Products 11.3 Faithful Maps 11.4 The Extension Theorem 12 Spherical Coxeter Sets 12.1 Elements of Maximal Length 12.2 Faithful Maps 12.3 The Main Theorem 12.4 Coxeter Schemes of Finite Valency and Rank 2 . 12.5 Valencies and Multiplicities 12.6 Polarities References Index |
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