Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- 1 Functions
- 2 Permutations of a finite set
- 3 Groups of permutations of ℝ and ℂ
- 4 The Möbius group
- 5 The regular solids
- 6 Abstract groups
- 7 Inversions of the Möbius plane and stereographic projection
- 8 Equivalence relations
- 9 Cosets
- 10 Direct product
- 11 Fields and vector spaces
- 12 Linear transformations
- 13 The general linear group GL(2, F)
- 14 The vector space V3(F)
- 15 Eigenvectors and eigenvalues
- 16 Homomorphisms
- 17 Conjugacy
- 18 Linear fractional groups
- 19 Quaternions and rotations
- 20 Affine groups
- 21 Orthogonal groups
- 22 Discrete groups fixing a line
- 23 Wallpaper groups
- Bibliography
- Index
Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- 1 Functions
- 2 Permutations of a finite set
- 3 Groups of permutations of ℝ and ℂ
- 4 The Möbius group
- 5 The regular solids
- 6 Abstract groups
- 7 Inversions of the Möbius plane and stereographic projection
- 8 Equivalence relations
- 9 Cosets
- 10 Direct product
- 11 Fields and vector spaces
- 12 Linear transformations
- 13 The general linear group GL(2, F)
- 14 The vector space V3(F)
- 15 Eigenvectors and eigenvalues
- 16 Homomorphisms
- 17 Conjugacy
- 18 Linear fractional groups
- 19 Quaternions and rotations
- 20 Affine groups
- 21 Orthogonal groups
- 22 Discrete groups fixing a line
- 23 Wallpaper groups
- Bibliography
- Index
Summary
Some groups exhibit, in miniature, the structure of other groups. A group H contains a ‘miniature version’ of a group G when there is a function of G to H, usually a many-to-one function, which preserves the group structure of G. Such a function is called a homomorphism. What we mean when we say that a function ‘preserves the group structure’ is that if g1 ↦ h1 and g2 ↦ h2, then g1g2 ↦ h1h2. In ordinary language we describe this by saying that the image of a product is the product of the images.
Concurrent reading: Fraleigh, sections 11 and 12; Green, chapter 7.
1 If A and B are matrices in GL(2, F) and A and B belong to the same right coset of SL(2, F), what can you say about det A and det B?
2 If A and B are matrices in GL(2, F) and det A = det B, does it follow that A and B belong to the same right coset of SL(2, F)?
3 What is the index of SL(2, ℤ3) in GL(2, ℤ3) and what is the order of SL(2, ℤ3)? (See qn 13.32)
4 What is the index of SL(2, ℤ5) in GL(2, ℤ5)? What is the order of SL(2, ℤ5)?
5 If A and B are matrices in GL(2, F) and det A = det B, does it follow that A and B belong to the same left coset of SL(2, F)?
6 […]
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- GroupsA Path to Geometry, pp. 148 - 154Publisher: Cambridge University PressPrint publication year: 1985
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