Book contents
- Frontmatter
- Contents
- Preface
- Preliminaries
- 1 Basic properties of the integers
- 2 Congruences
- 3 Computing with large integers
- 4 Euclid's algorithm
- 5 The distribution of primes
- 6 Finite and discrete probability distributions
- 7 Probabilistic algorithms
- 8 Abelian groups
- 9 Rings
- 10 Probabilistic primality testing
- 11 Finding generators and discrete logarithms in
- 12 Quadratic residues and quadratic reciprocity
- 13 Computational problems related to quadratic residues
- 14 Modules and vector spaces
- 15 Matrices
- 16 Subexponential-time discrete logarithms and factoring
- 17 More rings
- 18 Polynomial arithmetic and applications
- 19 Linearly generated sequences and applications
- 20 Finite fields
- 21 Algorithms for finite fields
- 22 Deterministic primality testing
- Appendix: Some useful facts
- Bibliography
- Index of notation
- Index
8 - Abelian groups
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Preliminaries
- 1 Basic properties of the integers
- 2 Congruences
- 3 Computing with large integers
- 4 Euclid's algorithm
- 5 The distribution of primes
- 6 Finite and discrete probability distributions
- 7 Probabilistic algorithms
- 8 Abelian groups
- 9 Rings
- 10 Probabilistic primality testing
- 11 Finding generators and discrete logarithms in
- 12 Quadratic residues and quadratic reciprocity
- 13 Computational problems related to quadratic residues
- 14 Modules and vector spaces
- 15 Matrices
- 16 Subexponential-time discrete logarithms and factoring
- 17 More rings
- 18 Polynomial arithmetic and applications
- 19 Linearly generated sequences and applications
- 20 Finite fields
- 21 Algorithms for finite fields
- 22 Deterministic primality testing
- Appendix: Some useful facts
- Bibliography
- Index of notation
- Index
Summary
This chapter introduces the notion of an abelian group. This is an abstraction that models many different algebraic structures, and yet despite the level of generality, a number of very useful results can be easily obtained.
Definitions, basic properties, and examples
Definition 8.1.An abelian group is a set G together with a binary operation ⋆ on G such that
(i) for all a, b, c ∈ G, a ⋆ (b ⋆ c) = (a ⋆ b) ⋆ c (i.e., ⋆ is associative),
(ii) there exists e ∈ G (called the identity element) such that for all a ∈ G, a ⋆ e = a = e ⋆ a,
(iii) for all a ∈ G there exists a′ ∈ G (called the inverse of a) such that a ⋆ a′ = e = a′ ⋆ a,
(iv) for all a, b ∈ G, a ⋆ b = b ⋆ a (i.e., ⋆ is commutative).
While there is a more general notion of a group, which may be defined simply by dropping property (iv) in Definition 8.1, we shall not need this notion in this text. The restriction to abelian groups helps to simplify the discussion significantly. Because we will only be dealing with abelian groups, we may occasionally simply say “group” instead of “abelian group.”
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- A Computational Introduction to Number Theory and Algebra , pp. 180 - 210Publisher: Cambridge University PressPrint publication year: 2005