Automorphism Groups of Maps, Surfaces and Smarandache Geometries (second edition), graduate textbook in mathematics 9781599731544, 1599731541

Table of contents :
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