Book contents
- Frontmatter
- Contents
- Preface
- Notation and special symbols
- 0 Historical introduction
- 1 Plane Euclidean geometry
- 2 Affine transformations in the Euclidean plane
- 3 Finite groups of isometries of E2
- 4 Geometry on the sphere
- 5 The projective plane P2
- 6 Distance geometry on P2
- 7 The hyperbolic plane
- Appendix A The axiomatic approach
- Appendix B Sets and functions
- Appendix C Groups
- Appendix D Linear algebra
- Appendix E Proof of Theorem 2.2
- Appendix F Trigonometric and hyperbolic functions
- References
- Index
- Frontmatter
- Contents
- Preface
- Notation and special symbols
- 0 Historical introduction
- 1 Plane Euclidean geometry
- 2 Affine transformations in the Euclidean plane
- 3 Finite groups of isometries of E2
- 4 Geometry on the sphere
- 5 The projective plane P2
- 6 Distance geometry on P2
- 7 The hyperbolic plane
- Appendix A The axiomatic approach
- Appendix B Sets and functions
- Appendix C Groups
- Appendix D Linear algebra
- Appendix E Proof of Theorem 2.2
- Appendix F Trigonometric and hyperbolic functions
- References
- Index
Summary
A group is a set G together with a function from G × G to G satisfying certain conditions. In order to express these conditions in a compact way, we adopt the following notation. If a and b are elements of G, we denote by a ✻ b the result of applying the function to the ordered pair (a, b). The conditions for G to be a group are
(a ✻ b) ✻ c = a ✻ (b ✻ c) for all a, b, and c in G. This is called the associative law.
There is an element I of G such that a ✻ I = I ✻ a = a for all a in G. Such an element is called an identity.
For each element a in G there is an element b in G such that a ✻ b = b ✻ a = I. The elements b and a are said to be inverses of each other.
Remark: It is not difficult to show that there is only one identity element in a group and that each element has exactly one inverse.
When discussing group operations, we often use informal terminology that exploits the analogy with mutiplication of numbers. For instance, a ✻ b is sometimes called the product of a and b.
- Type
- Chapter
- Information
- Euclidean and Non-Euclidean GeometryAn Analytic Approach, pp. 189 - 192Publisher: Cambridge University PressPrint publication year: 1986