Book contents
- Frontmatter
- Contents
- Preface
- 1 Groups and permutations
- 2 The real numbers
- 3 The complex plane
- 4 Vectors in three-dimensional space
- 5 Spherical geometry
- 6 Quaternions and isometries
- 7 Vector spaces
- 8 Linear equations
- 9 Matrices
- 10 Eigenvectors
- 11 Linear maps of Euclidean space
- 12 Groups
- 13 Möbius transformations
- 14 Group actions
- 15 Hyperbolic geometry
- Index
12 - Groups
Published online by Cambridge University Press: 05 September 2012
- Frontmatter
- Contents
- Preface
- 1 Groups and permutations
- 2 The real numbers
- 3 The complex plane
- 4 Vectors in three-dimensional space
- 5 Spherical geometry
- 6 Quaternions and isometries
- 7 Vector spaces
- 8 Linear equations
- 9 Matrices
- 10 Eigenvectors
- 11 Linear maps of Euclidean space
- 12 Groups
- 13 Möbius transformations
- 14 Group actions
- 15 Hyperbolic geometry
- Index
Summary
Groups
In Chapter 1 we defined what it means to say that a set G is a group with respect to an operation *, and we studied groups of permutations of a set. In this chapter we shall study ‘abstract’ groups. Although this may seem like a more difficult task, every ‘abstract’ group is the permutation group of some set so that, in some sense, this apparent change of direction is only an illusion.
We have already seen many examples of groups: the sets ℤ, ℝ and ℂ of integers, real numbers, and complex numbers, respectively, the spaces ℝn and ℂn, the set of matrices of a fixed size, and indeed any vector space, all form a group with respect to addition. Likewise, the set ℝ+ of positive real numbers, the set ℂ* of non-zero complex numbers, the set of complex numbers of modulus one, the set of non-singular n × n matrices, and the set of n-th roots of unity, all form a group with respect to multiplication. Finally, the set of bijections of a given set onto itself, the set of isometries of ℂ, and of ℝn, the permutations of {1, …, n}, and the set of non-singular (invertible) linear transformations of a vector space onto itself, all form a group with respect to the usual composition of functions (that is, if f * g is the function defined by (f * g)(x) = f(g(x))).
- Type
- Chapter
- Information
- Algebra and Geometry , pp. 215 - 253Publisher: Cambridge University PressPrint publication year: 2005