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1．Cyclic,Dicyclic and Metacyclic Groups 1

1．1 Generators and relations 1

1．2 Factor groups 2

1．3 Direct products 3

1．4 Automorphisms 5

1．5 Some well-known finite groups 6

1．6 Dicyclic groups 7

1．7 The quaternion groups 8

1．8 Cyclic extensions of cyclic groups 8

1．9 Groups of order less than 32 11

2．Systematic Enumeration of Cosets 12

2．1 Finding an abstract definition for a finite group 12

2．2 Examples 14

2．3 The corresponding permutations 17

2．4 Finding whether a given subgroup is normal 17

2．5 How an abstract definition determines a group 17

3．Graphs,Maps and Cayley Diagrams 18

3．1 Graphs 19

3．2 Maps 19

3．3 Cayley diagrams 21

3．4 Planar diagrams 23

3．5 Unbounded surfaces 24

3．6 Non-planar diagrams 28

3．7 SCHREIER＇S coset diagrams 31

4．Abstract Crystallography 33

4．1 The cyclic and dihedral groups 33

4．2 The crystgallographic and non-crystallographic point groups 33

4．3 Groups generated by reflections 35

4．4 Subgroups of the reflection groups 38

4．5 The seventeen two-dimensional space groups 40

4．6 Subgroup relationships among the seventeen groups 51

5．Hyperbolic Tessellations and Fundamental Groups 52

5．1 Regular tessellations 52

5．2 The Petrie polygon 54

5．3 DYCK＇s groups 54

5．4 The fundamental group for a non-orientable surface,obtained as a group generated by glide-reflections 56

5．5 The fundamental region for an orientable surface,obtained as a group of translations 58

6．The Symmetric,Alternating,and other Special Groups 61

6．1 ARTIN＇s braid group 62

6．2 The symmetric group 63

6．3 The alternating group 66

6．4 The polyhedral groups 67

6．5 The binary polyhedral groups 68

6．6 MILLER＇s generalization of the polyhedral groups 71

6．7 A new generalization 76

6．8 BURNSIDE＇s problem 80

7．Modular and Linear Fractional Groups 83

7．1 Lattices and modular groups 83

7．2 Defining relations when n = 2 85

7．3 Defining relations when n ？ 3 88

7．4 Linear fractiona groups 92

7．5 The groups LF (2,p) 93

7．6 The groups LF (2,2m) 96

7．7 The Hessian group and LF (3,3) 97

7．8 The first Mathieu group 98

8．Regular Maps 100

8．1 Automorphisms 100

8．2 Universal covering 102

8．3 Maps of type {4,4} on a torus 102

8．4 Maps of type {3,6} or {6,3} on a torus 106

8．5 Maps having specified holes 108

8．6 Maps having specified Petrie polygons 110

8．7 Maps having two faces 113

8．8 Maps on a two-sheeted Riemann surface 115

8．9 Symmetrical graphs 116

9．Groups Generated by Reflections 117

9．1 Reducible and irreducible grops 117

9．2 The graphical notation 117

9．3 Finite groups 118

9．4 A brief description of the individual groups 122

9．5 Commutator subgroups 124

9．6 Central quotient groups 126

9．7 Exponents and invariants 129

9．8 Infinite Euclidean groups 131

9．9 Infinite non-Euclidean groups 132

Tables 1-12 134

Bibliography 144

Index 152