Book contents
- Frontmatter
- Contents
- Preface
- 1 Some mathematical language
- 2 Sets and functions
- 3 Equivalence relations and quotient sets
- 4 Number systems I
- 5 Groups I
- 6 Rings and fields
- 7 Homomorphisms and quotient algebras
- 8 Vector spaces and matrices
- 9 Linear equations and rank
- 10 Determinants and multilinear mappings
- 11 Polynomials
- 12 Groups II
- 13 Number systems II
- 14 Fields and polynomials
- 15 Lattices and Boolean algebra
- 16 Ordinal numbers
- 17 Eigenvectors and eigenvalues
- 18 Quadratic forms and inner products
- 19 Categories and functors
- 20 Metric spaces and continuity
- 21 Topological spaces and continuity
- 22 Metric spaces II
- 23 The real numbers
- 24 Real-valued functions of a real variable
- 25 Differentiable functions of one variable
- 26 Functions of several real variables
- 27 Integration
- 28 Infinite series and products
- 29 Improper integrals
- 30 Curves and arc length
- 31 Functions of a complex variable
- 32 Multiple integrals
- 33 Logarithmic, exponential and trigonometric functions
- 34 Vector algebra
- 35 Vector calculus
- 36 Line and surface integrals
- 37 Measure and Lebesgue integration
- 38 Fourier series
- Appendix 1 Some ‘named’ theorems and properties
- Appendix 2 Alphabets used in mathematics
- Index of symbols
- Subject index
Book contents
- Frontmatter
- Contents
- Preface
- 1 Some mathematical language
- 2 Sets and functions
- 3 Equivalence relations and quotient sets
- 4 Number systems I
- 5 Groups I
- 6 Rings and fields
- 7 Homomorphisms and quotient algebras
- 8 Vector spaces and matrices
- 9 Linear equations and rank
- 10 Determinants and multilinear mappings
- 11 Polynomials
- 12 Groups II
- 13 Number systems II
- 14 Fields and polynomials
- 15 Lattices and Boolean algebra
- 16 Ordinal numbers
- 17 Eigenvectors and eigenvalues
- 18 Quadratic forms and inner products
- 19 Categories and functors
- 20 Metric spaces and continuity
- 21 Topological spaces and continuity
- 22 Metric spaces II
- 23 The real numbers
- 24 Real-valued functions of a real variable
- 25 Differentiable functions of one variable
- 26 Functions of several real variables
- 27 Integration
- 28 Infinite series and products
- 29 Improper integrals
- 30 Curves and arc length
- 31 Functions of a complex variable
- 32 Multiple integrals
- 33 Logarithmic, exponential and trigonometric functions
- 34 Vector algebra
- 35 Vector calculus
- 36 Line and surface integrals
- 37 Measure and Lebesgue integration
- 38 Fourier series
- Appendix 1 Some ‘named’ theorems and properties
- Appendix 2 Alphabets used in mathematics
- Index of symbols
- Subject index
Summary
A group is an ordered pair, (G, ∗), consisting of a set G and a binary operation ∗: G × G → G((x, y) ↦ x ∗ y) satisfying
(a) (x ∗ y) ∗ z = x ∗ (y ∗ z) for all x, y, z ∈ G (associativity),
(b) there exists an element e ∈ G such that x ∗ e = x = e ∗ x for all x ∈ G,
(c) for each x ∈ G, there exists an element x′ ∈ G such that x ∗ x′ = e = x′ ∗ x.
The operation ∗ is frequently written as a product, i.e. (x, y) ↦ xy, or as a sum, (x, y) ↦ x + y. We then say that (G, ∗) is respectively a multiplicative or an additive group.
The element e, which can be shown to be unique, is called the identity or neutral element of G. When G is multiplicative it is denoted by 1 and when G is additive by o. Correspondingly, x′, the inverse element of x (again unique) is denoted by x–1 and –x respectively. (Cf. p. 15.)
Note that the definition of a group given above is wasteful in the sense that not all the conditions listed are independent (p. 6). Thus, for example, one can omit the condition x = e ∗ x from (b) and the condition e = x′ ∗ x from (c), since these relations can be deduced from the remaining conditions.
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- A Handbook of Terms used in Algebra and Analysis , pp. 25 - 29Publisher: Cambridge University PressPrint publication year: 1972
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