Book contents
- Frontmatter
- Contents
- Preface
- Chapter 1 Introduction to Groups
- Chapter 2 Group Axioms
- Chapter 3 Examples of Groups
- Chapter 4 Multiplication Table of a Group
- Chapter 5 Generators of a Group
- Chapter 6 Graph of a Group
- Chapter 7 Definition of a Group by Generators and Relations
- Chapter 8 Subgroups
- Chapter 9 Mappings
- Chapter 10 Permutation Groups
- Chapter 11 Normal Subgroups
- Chapter 12 The Quaternion Group
- Chapter 13 Symmetric and Alternating Groups
- Chapter 14 Path Groups
- Chapter 15 Groups and Wallpaper Designs
- Appendix: Group of the Dodecahedron and the Icosahedron
- Solutions
- Bibliography
- Index
Chapter 3 - Examples of Groups
- Frontmatter
- Contents
- Preface
- Chapter 1 Introduction to Groups
- Chapter 2 Group Axioms
- Chapter 3 Examples of Groups
- Chapter 4 Multiplication Table of a Group
- Chapter 5 Generators of a Group
- Chapter 6 Graph of a Group
- Chapter 7 Definition of a Group by Generators and Relations
- Chapter 8 Subgroups
- Chapter 9 Mappings
- Chapter 10 Permutation Groups
- Chapter 11 Normal Subgroups
- Chapter 12 The Quaternion Group
- Chapter 13 Symmetric and Alternating Groups
- Chapter 14 Path Groups
- Chapter 15 Groups and Wallpaper Designs
- Appendix: Group of the Dodecahedron and the Icosahedron
- Solutions
- Bibliography
- Index
Summary
If we want to decide whether a given set of elements with a specific binary operation constitutes a group, we must test to see whether the axioms are satisfied. Let us examine the following sets for eligibility as groups. We begin with Group A (p. 4).
Example 1
Set of elements: All integers (positive, negative, and zero).
Binary operation: Addition.
Associativity: Addition of numbers is associative.
Identity: The set contains zero as an element and, for every integer u, u + 0 = 0 + u = u. Zero is the identity element.
Inverses: If u is an integer, its negative –u is an integer and u + (–u) = (–u) + u = 0; –u is the inverse of u, or, in group notation, u-1 = –u.
Thus, the set under test is a group. Since this group has infinitely many elements, we say the group is infinite. This group will sometimes be referred to as an infinite additive group or the additive group of integers.
Example 2
Let the set be the same as in Example 1, but now consider multiplication. The reader can check for himself that multiplication is a binary operation on the set of all integers and that the axioms on associativity and the existence of an identity element are satisfied. To see if the set satisfies Axiom 3, we try to determine the inverse of the element 2.
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- Groups and Their Graphs , pp. 15 - 25Publisher: Mathematical Association of AmericaPrint publication year: 1992