《Generators and Relations For Discrete Groups》

1．Cyclic,Dicyclic and Metacyclic Groups1

1．1Generators and relations1

1．2 Factor groups2

1．3 Direct products3

1．4 Automorphisms5

1．5 Some well-known finite groups6

1．6 Dicyclic groups7

1．7 The quaternion groups8

1．8 Cyclic extensions of cyclic groups8

1．9 Groups of order less than 3211

2．Systematic Enumeration of Cosets12

2．1Finding an abstract definition for a finite group12

2．2 Examples14

2．3 The corresponding permutations17

2．4 Finding whether a given subgroup is normal17

2．5 How an abstract definition determines a group17

3．Graphs,Maps and Cayley Diagrams18

3．1Graphs19

3．2 Maps19

3．3 Cayley diagrams21

3．4 Planar diagrams23

3．5 Unbounded surfaces24

3．6 Non-planar diagrams28

3．7 SCHREIER＇S coset diagrams31

4．Abstract Crystallography33

4．1The cyclic and dihedral groups33

4．2 The crystgallographic and non-crystallographic point groups33

4．3 Groups generated by reflections35

4．4 Subgroups of the reflection groups38

4．5 The seventeen two-dimensional space groups40

4．6 Subgroup relationships among the seventeen groups51

5．Hyperbolic Tessellations and Fundamental Groups52

5．1Regular tessellations52

5．2 The Petrie polygon54

5．3 DYCK＇s groups54

5．4 The fundamental group for a non-orientable surface,obtained as a group generated by glide-reflections56

5．5 The fundamental region for an orientable surface,obtained as a group of translations58

6．The Symmetric,Alternating,and other Special Groups61

6．1ARTIN＇s braid group62

6．2 The symmetric group63

6．3 The alternating group66

6．4 The polyhedral groups67

6．5 The binary polyhedral groups68

6．6 MILLER＇s generalization of the polyhedral groups71

6．7 A new generalization76

6．8 BURNSIDE＇s problem80

7．Modular and Linear Fractional Groups83

7．1Lattices and modular groups83

7．2 Defining relations when n = 285

7．3 Defining relations when n ？ 388

7．4 Linear fractiona groups92

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 group98

8．Regular Maps100

8．1Automorphisms100

8．2 Universal covering102

8．3 Maps of type {4,4} on a torus102

8．4 Maps of type {3,6} or {6,3} on a torus106

8．5 Maps having specified holes108

8．6 Maps having specified Petrie polygons110

8．7 Maps having two faces113

8．8 Maps on a two-sheeted Riemann surface115

8．9 Symmetrical graphs116

9．Groups Generated by Reflections117

9．1Reducible and irreducible grops117

9．2 The graphical notation117

9．3 Finite groups118

9．4 A brief description of the individual groups122

9．5 Commutator subgroups124

9．6 Central quotient groups126

9．7 Exponents and invariants129

9．8 Infinite Euclidean groups131

9．9 Infinite non-Euclidean groups132

Tables 1-12134

Bibliography144

Index152