Automorphism Groups of Maps, Surfaces and Smarandache Geometries (second edition), graduate text book in mathematics

Automorphisms of a system survey its symmetry and appear nearly in all mathematical branches, such as those of algebra, combinatorics, geometry, ... and theoretical physics or chemistry. The main motivation of this book is to present a systemically introduction to automorphism groups on algebra, graphs, maps, i.e., graphs on surfaces and geometrical structures with applications. Topics covered in this book include elementary groups, symmetric graphs, graphs on surfaces, regular maps, lifted automorphisms of graphs or maps, automorphisms of maps underlying a graph with applications to map enumeration, isometries on Smarandache geometry and CC conjecture, etc., which is suitable as a textbook for graduate students, and also a valuable reference for researchers in group action, graphs with groups, combinatorics with enumeration, Smarandache multispaces, particularly, Smarandache geometry with applications. The most importance of Smarandache geometries was the introduction of the degree of negation of an axiom (and more general the degree of negation of a theorem, lemma, scientific or humanistic proposition) which works somehow like the negation in fuzzy logic (with a degree of truth, and a degree of falsehood) or more general like the negation in neutrosophic logic (with a degree of truth, a degree of falsehood, and a degree of neutrality (neither true nor false, but unknown, ambiguous, indeterminate) [not only Enclid s geometrical axioms, but any scientific or humanistic proposition in any field] or partial negation of an axiom (and, in general, partial negation of a scientific or humanistic proposition in any field). These geometries connect many geometrical spaces with different structures into a heterogeneous multispace with multistructure. In general, a rule in a system is said to be Smarandachely denied if it behaves in at least two different ways within the same set, i.e. validated and invalided, or only invalided but in multiple distinct ways. A Smarandache system is a system which has at least one Smarandachely denied rule . In particular, a Smarandache geometry is such a geometry in which there is at least one Smarandachely denied rule, and a Smarandache manifold is an n-dimensional manifold that supports a Smarandache geometry. In a Smarandache geometry, the points, lines, planes, spaces, triangles, ... are respectively called s-points, s-lines, s-planes, s-spaces, s-triangles, ... in order to distinguish them from those in classical geometry. Howard Iseri constructed the Smarandache 2-manifolds by using equilateral triangular disks on Euclidean plane R^2. Such manifold came true though paper models in R^3 for elliptic, Euclidean and hyperbolic cases. It should also be noted that a more general Smarandache n-manifold, i.e. combinatorial manifold and a differential theory on such manifold were constructed by Linfan Mao. Contents Preface to the Second Edition CHAPTER 1.Groups §1.1 SETS 1.1.1 Set 1.1.2 Cardinality 1.1.3 Subset Enumeration §1.2 GROUPS 1.2.1 Algebra System 1.2.2 Group 1.2.3 Group Property 1.2.4 Subgroup 1.2.5 Symmetric Group 1.2.6 Regular Representation §1.3 HOMOMORPHISMTHEOREMS 1.3.1 Homomorphism 1.3.2 Quotient Group 1.3.3 Isomorphism Theorem §1.4 ABELIAN GROUPS 1.4.1 Direct Product 1.4.2 Basis 1.4.3 Finite Abelian Group Structure §1.5 MULTIGROUPS 1.5.1 MultiGroup 1.5.2 Submultigroup 1.5.3 Normal Submultigroup 1.5.4 Abelian Multigroup 1.5.5 Bigroup 1.5.6 ConstructingMultigroup §1.6 REMARKS CHAPTER 2.Action Groups §2.1 PERMUTATION GROUPS 2.1.1 Group Action 2.1.2 Stabilizer 2.1.3 Burnside Lemma §2.2 TRANSITIVE GROUPS 2.2.1 Transitive Group 2.2.2 Multiply Transitive Group 2.2.3 Sharply k-Transitive Group §2.3 AUTOMORPHISMS OF GROUPS 2.3.1 Automorphism Group 2.3.2 Characteristic Subgroup 2.3.3 Commutator Subgroup §2.4 P-GROUPS 2.4.1 Sylow Theorem 2.4.2 Application of Sylow Theorem 2.4.3 Listing p-Group §2.5 PRIMITIVE GROUPS 2.5.1 Imprimitive Block 2.5.2 Primitive Group 2.5.3 Regular Normal Subgroup 2.5.4 O’Nan-Scott Theorem §2.6 LOCAL ACTION AND EXTENDED GROUPS 2.6.1 Local Action Group 2.6.2 Action Extended Group 2.6.3 Action MultiGroup §2.7 REMARKS CHAPTER 3.Graph Groups §3.1 GRAPHS 3.1.1 Graph 3.1.2 Graph Operation 3.1.3 Graph Property 3.1.4 Smarandachely Graph Property §3.2 GRAPH GROUPS 3.2.1 Graph Automorphism 3.2.2 Graph Group 3.2.3 􀀀-Action §3.3 SYMMETRIC GRAPHS 3.3.1 Vertex-Transitive Graph 3.3.2 Edge-Transitive Graph 3.3.3 Arc-Transitive Graph §3.4 GRAPH SEMI-ARC GROUPS 3.4.1 Semi-Arc Set 3.4.2 Graph Semi-Arc Group 3.4.3 Semi-Arc Transitive Graph §3.5 GRAPH MULTIGROUPS 3.5.1 GraphMultigroup 3.5.2 Multigroup Action Graph 3.5.3 Globally Transitivity §3.6 REMARKS CHAPTER 4.Surface Groups §4.1 SURFACES 4.1.1 Topological Space 4.1.2 Continuous Mapping 4.1.3 Homeomorphic Space 4.1.4 Surface 4.1.5 Quotient Space $4.2 CLASSIFICATION THEOREM 4.2.1 Connected Sum 4.2.2 Polygonal Presentation 4.2.3 Elementary Equivalence 4.2.4 Classification Theorem 4.2.5 Euler Characteristic $4.3 FUNDAMENTAL GROUPS 4.3.1 HomotopicMapping 4.3.2 Fundamental Group 4.3.3 Seifert-Van Kampen Theorem 4.3.4 Fundamental Group of Surface $4.4 NEC GROUPS 4.4.1 Dianalytic Function 4.4.2 Klein Surface 4.4.3 Morphism of Klein Surface 4.4.4 Planar Klein Surface 4.4.5 NEC Group $4.5 AUTOMORPHISMS OF KLEIN SURFACES 4.5.1 MorphismProperty 4.5.2 Double Covering of Klein Surface 4.5.3 Discontinuous Action 4.5.4 AutomorphismofKlein Surface $4.6 REMARKS CHAPTER 5.Map Groups §5.1 GRAPHS ON SURFACES 5.1.1 Cell Embedding 5.1.2 Rotation System 5.1.3 Equivalent Embedding 5.1.4 Euler-Poincar ́e Characteristic §5.2 COMBINATORIAL MAPS 5.2.1 Combinatorial Map 5.2.2 Dual Map 5.2.3 Orientability 5.2.4 Standard Map §5.3 MAP GROUPS 5.3.1 Isomorphism of Maps 5.3.2 Automorphism of Map 5.3.3 Combinatorial Model of Klein Surface §5.4 REGULAR MAPS 5.4.1 RegularMap 5.4.2 Map NEC-Group 5.4.3 Cayley Map 5.4.4 Complete Map §5.5 CONSTRUCTING REGULAR MAPS BY GROUPS 5.5.1 Regular Tessellation 5.5.2 Regular Map on Finite Group 5.5.3 RegularMap on FiniteMultigroup §5.6 REMARKS CHAPTER 6.Lifting Map Groups §6.1 VOLTAGE MAPS 6.1.1 Covering Space 6.1.2 Covering Mapping 6.1.3 Voltage Map with Lifting §6.2 GROUP BEING THAT OF A MAP 6.2.1 LiftingMap Automorphism 6.2.2 Map Exponent Group §6.3 MEASURES ON MAPS 6.3.1 Angle on Map. 6.3.2 Non-Euclid Area on Map §6.4 A COMBINATORIAL REFINEMENT OF HURIWTZ THEOREM 6.4.1 Combinatorially Huriwtz Theorem 6.4.2 Application to Klein Surface §6.5 THE ORDER OF AUTOMORPHISM OF KLEIN SURFACE 6.5.1 The Minimum Genus of a Fixed-Free Automorphism 6.5.2 The Maximum Order of Automorphisms of a Map §6.6 REMARKS CHAPTER 7.Map Automorphisms Underlying a Graph §7.1 A CONDITION FOR GRAPH GROUP BEING THAT OF MAP 7.1.1 Orientation-Preserving or Reversing 7.1.2 Group of a Graph Being That ofMap §7.2 AUTOMORPHISMS OF A COMPLETE GRAPH ON SURFACES 7.2.1 Complete Map. 7.2.2 Automorphisms of CompleteMap §7.3 MAP-AUTOMORPHISM GRAPHS 7.3.1 Semi-Regular Graph 7.3.2 Map-Automorphism Graph §7.4 AUTOMORPHISMS OF ONE FACE MAPS 7.4.1 One-Face Map 7.4.2 Automorphisms of One-Face Map §7.5 REMARKS CHAPTER 8.EnumeratingMaps on Surfaces §8.1 ROOTS DISTRIBUTION ON EMBEDDINGS 8.1.1 Roots on Embedding 8.1.2 Root Distribution 8.1.3 Rooted Map §8.2 ROOTED MAP ON GENUS UNDERLYING A GRAPH 8.2.1 Rooted Map Polynomial 8.2.2 Rooted Map Sequence §8.3 A SCHEME FOR ENUMERATING MAPS UNDERLYING A GRAPH §8.4 THE ENUMERATION OF COMPLETE MAPS ON SURFACES §8.5 THE ENUMERATIONOFMAPS UNDERLYINGA SEMI-REGULAR GRAPH 8.5.1 Crosscap Map Group. 8.5.2 Enumerating Semi-RegularMap §8.6 THE ENUMERATION OF A BOUQUET ON SURFACES 8.6.1 Cycle Index of Group 8.6.2 Enumerating One-Vertex Map §8.7 REMARKS CHAPTER 9.Isometries on Smarandache Geometry §9.1 SMARANDACHE GEOMETRY 9.1.1 Geometrical Axiom 9.1.2 Smarandache Geometry 9.1.3 Mixed Geometry §9.2 CLASSIFYING ISERI’S MANIFOLDS 9.2.1 Iseri’s Manifold 9.2.2 A Model of Closed Iseri’s Manifold 9.2.3 Classifying Closed Iseri’s Manifolds §9.3 ISOMETRIES OF SMARANDACHE 2-MANIFOLDS 9.3.1 Smarandachely Automorphism 9.3.2 Isometry on R2 9.3.3 Finitely Smarandache 2-Manifold 9.3.4 Smarandachely Map 9.3.5 Infinitely Smarandache 2-Manifold §9.4 ISOMETRIES OF PSEUDO-EUCLIDEAN SPACES 9.4.1 Euclidean Space 9.4.2 Linear Isometry on Euclidean Space 9.4.3 Isometry on Euclidean Space 9.4.4 Pseudo-Euclidean Space 9.4.5 Isometry on Pseudo-Euclidean Space §9.5 REMARKS CHAPTER 10.CC Conjecture §10.1 CC CONJECTURE ON MATHEMATICS 10.1.1 Combinatorial Speculation 10.1.2 CC Conjecture 10.1.3 CC Problems in Mathematics §10.2 CC CONJECTURE TO MATHEMATICS 10.2.1 Contribution to Algebra 10.2.2 Contribution to Metric Space §10.3 CC CONJECTURE TO PHYSICS 10.3.1 M-Theory 10.3.2 Combinatorial Cosmos References Index Automorphism groups survey similarities on mathematical systems, which appear nearly in all mathematical branches, such as those of algebra, combinatorics, geometry, and theoretical physics, theoretical chemistry, etc.