Book contents
- Frontmatter
- INTRODUCTION
- Contents
- CHAPTER 1 COMBINATORIAL GROUP THEORY
- CHAPTER 2 SPACES AND THEIR PATHS
- CHAPTER 3 GROUPOIDS
- CHAPTER 4 THE FUNDAMENTAL GROUPOID AND THE FUNDAMENTAL GROUP
- CHAPTER 5 COMPLEXES
- CHAPTER 6 COVERINGS OF SPACES AND COMPLEXES
- CHAPTER 7 COVERINGS AND GROUP THEORY
- CHAPTER 8 BASS-SERRE THEORY
- CHAPTER 9 DECISION PROBLEMS
- CHAPTER 10 FURTHER TOPICS
- NOTES AND REFERENCES
- BIBLIOGRAPHY
- INDEX
CHAPTER 1 - COMBINATORIAL GROUP THEORY
Published online by Cambridge University Press: 08 January 2010
- Frontmatter
- INTRODUCTION
- Contents
- CHAPTER 1 COMBINATORIAL GROUP THEORY
- CHAPTER 2 SPACES AND THEIR PATHS
- CHAPTER 3 GROUPOIDS
- CHAPTER 4 THE FUNDAMENTAL GROUPOID AND THE FUNDAMENTAL GROUP
- CHAPTER 5 COMPLEXES
- CHAPTER 6 COVERINGS OF SPACES AND COMPLEXES
- CHAPTER 7 COVERINGS AND GROUP THEORY
- CHAPTER 8 BASS-SERRE THEORY
- CHAPTER 9 DECISION PROBLEMS
- CHAPTER 10 FURTHER TOPICS
- NOTES AND REFERENCES
- BIBLIOGRAPHY
- INDEX
Summary
FREE GROUPS
Let x be a generating subset of a group G. Certain products of members of X and their inverses will be 1 whatever X and G are; for instance, xyyz−1zy−1y−1x−1. Other products, such as xyz or xx, will be 1 for some choices of X and G but not for other choices. Those pairs G and X for which a product of elements in X ∪ X−1 is 1 only when the properties holding in all groups require it to be 1 are obviously of interest.
They are called free groups; a more formal definition will be given later. If G is such a group, any function f from x to a group H can be extended uniquely to a homomorphism from G to H. For any g ∈ G can be written as xi1ε1 … xin εn where εt = ±1 and xir ε X for r = 1, …, n. Now suppose that g can also be written as xj1δ1 … xjm, δm where δs = ±1 and xjs ε X for s − 1, …, m. Then
and our assumption on G and X then tells us we must have
Hence the element of H given by (xi1 ƒ)ε1 … (xin ƒ)εn depends only on g and not on how g is written as a product of elements of X ∪ X−1. I t follows that we can define a function φ:G → H by requiring gφ to be this element. It is easy to check that φ is a homomorphism and that xφ − xƒ for all x ε X.
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- Combinatorial Group TheoryA Topological Approach, pp. 1 - 48Publisher: Cambridge University PressPrint publication year: 1989