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Chapter 1 Groups

We begin our study of algebraic structures by investigating sets associated with single operations that satisfy certain reasonable axioms; that is, we want to define an operation on a set in a way that
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will generalize such familiar structures as the integers
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\({\mathbb Z}\) together with the single operation of addition, or invertible \(2 \times 2\) matrices together with the single operation of matrix multiplication. The integers and the \(2 \times 2\) matrices, together with their respective single operations, are examples of algebraic structures known as groups
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Section 1.1 Integer Equivalence Classes and Symmetries

Carl Friedrich Gauß, Leonhard Euler

Let us now investigate some mathematical structures that can be viewed as sets with single operations.

Subsection 1.1.1 The Integers mod \(n\)

Gottfried Wilhelm Leibniz

The integers mod \(n\) have become indispensable in the theory and applications of algebra. In mathematics they are used in cryptography, coding theory, and the detection of errors in identification codes.
We have already seen that two integers \(a\) and \(b\) are equivalent mod \(n\) if \(n\) divides \(a - b\text{.}\) The integers mod \(n\) also partition \({\mathbb Z}\) into \(n\) different equivalence classes; we will denote the set of these equivalence classes by \({\mathbb Z}_n\text{.}\) Consider the integers modulo 12 and the corresponding partition of the integers:
\begin{align*} {[0]} & = \{ \ldots, -12, 0, 12, 24, \ldots \}\\ {[1]} & = \{ \ldots, -11, 1, 13, 25, \ldots \}\\ & \vdots\\ {[11]} & = \{ \ldots, -1, 11, 23, 35, \ldots \}\text{.} \end{align*}
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