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Section3.1Integer Equivalence Classes and Symmetries

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

Subsection3.1.1The Integers mod \(n\)

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 \}. \end{align*}

When no confusion can arise, we will use \(0, 1, \ldots, 11\) to indicate the equivalence classes \({[0]}, {[1]}, \ldots, {[11]}\) respectively. We can do arithmetic on \({\mathbb Z}_n\text{.}\) For two integers \(a\) and \(b\text{,}\) define addition modulo \(n\) to be \((a + b) \pmod{n}\text{;}\) that is, the remainder when \(a + b\) is divided by \(n\text{.}\) Similarly, multiplication modulo \(n\) is defined as \((a b) \pmod{ n}\text{,}\) the remainder when \(a b\) is divided by \(n\text{.}\)

\begin{equation*} \begin{array}{c|cccccccc} \cdot & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 2 & 0 & 2 & 4 & 6 & 0 & 2 & 4 & 6 \\ 3 & 0 & 3 & 6 & 1 & 4 & 7 & 2 & 5 \\ 4 & 0 & 4 & 0 & 4 & 0 & 4 & 0 & 4 \\ 5 & 0 & 5 & 2 & 7 & 4 & 1 & 6 & 3 \\ 6 & 0 & 6 & 4 & 2 & 0 & 6 & 4 & 2 \\ 7 & 0 & 7 & 6 & 5 & 4 & 3 & 2 & 1 \end{array} \end{equation*}
Figure3.1.3Multiplication table for \({\mathbb Z_8}\)

Subsection3.1.2Symmetries

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Figure3.1.5Rigid motions of a rectangle

A symmetry of a geometric figure is a rearrangement of the figure preserving the arrangement of its sides and vertices as well as its distances and angles. A map from the plane to itself preserving the symmetry of an object is called a rigid motion. For example, if we look at the rectangle in Figure Figure 3.1.5, it is easy to see that a rotation of \(180^{\circ}\) or \(360^{\circ}\) returns a rectangle in the plane with the same orientation as the original rectangle and the same relationship among the vertices. A reflection of the rectangle across either the vertical axis or the horizontal axis can also be seen to be a symmetry. However, a \(90^{\circ}\) rotation in either direction cannot be a symmetry unless the rectangle is a square.

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Figure3.1.6Symmetries of a triangle

Let us find the symmetries of the equilateral triangle \(\bigtriangleup ABC\text{.}\) To find a symmetry of \(\bigtriangleup ABC\text{,}\) we must first examine the permutations of the vertices \(A\text{,}\) \(B\text{,}\) and \(C\) and then ask if a permutation extends to a symmetry of the triangle. Recall that a permutation of a set \(S\) is a one-to-one and onto map \(\pi :S \rightarrow S\text{.}\) The three vertices have \(3! = 6\) permutations, so the triangle has at most six symmetries. To see that there are six permutations, observe there are three different possibilities for the first vertex, and two for the second, and the remaining vertex is determined by the placement of the first two. So we have \(3 \cdot 2 \cdot 1 = 3! = 6\) different arrangements. To denote the permutation of the vertices of an equilateral triangle that sends \(A\) to \(B\text{,}\) \(B\) to \(C\text{,}\) and \(C\) to \(A\text{,}\) we write the array

\begin{equation*} \begin{pmatrix} A & B & C \\ B & C & A \end{pmatrix}. \end{equation*}

Notice that this particular permutation corresponds to the rigid motion of rotating the triangle by \(120^{\circ}\) in a clockwise direction. In fact, every permutation gives rise to a symmetry of the triangle. All of these symmetries are shown in Figure Figure 3.1.6.

A natural question to ask is what happens if one motion of the triangle \(\bigtriangleup ABC\) is followed by another. Which symmetry is \(\mu_1 \rho_1\text{;}\) that is, what happens when we do the permutation \(\rho_1\) and then the permutation \(\mu_1\text{?}\) Remember that we are composing functions here. Although we usually multiply left to right, we compose functions right to left. We have

\begin{align*} (\mu_1 \rho_1)(A) & = \mu_1( \rho_1( A ) ) = \mu_1( B ) = C\\ (\mu_1 \rho_1)(B) & = \mu_1( \rho_1( B ) ) = \mu_1( C ) = B\\ (\mu_1 \rho_1)(C) & = \mu_1( \rho_1( C ) ) = \mu_1( A ) = A. \end{align*}

This is the same symmetry as \(\mu_2\text{.}\) Suppose we do these motions in the opposite order, \(\rho_1\) then \(\mu_1\text{.}\) It is easy to determine that this is the same as the symmetry \(\mu_3\text{;}\) hence, \(\rho_1 \mu_1 \neq \mu_1 \rho_1\text{.}\) A multiplication table for the symmetries of an equilateral triangle \(\bigtriangleup ABC\) is given in Figure Figure 3.1.7.

Notice that in the multiplication table for the symmetries of an equilateral triangle, for every motion of the triangle \(\alpha\) there is another motion \(\beta\) such that \(\alpha \beta = id\text{;}\) that is, for every motion there is another motion that takes the triangle back to its original orientation.

\begin{equation*} \begin{array}{c|cccccc} \circ & id & \rho_1 & \rho_2 & \mu_1 & \mu_2 & \mu_3 \\ \hline id & id & \rho_1 & \rho_2 & \mu_1 & \mu_2 & \mu_3 \\ \rho_1 & \rho_1 & \rho_2 & id & \mu_3 & \mu_1 & \mu_2 \\ \rho_2 & \rho_2 & id & \rho_1 & \mu_2 & \mu_3 & \mu_1 \\ \mu_1 & \mu_1 & \mu_2 & \mu_3 & id & \rho_1& \rho_2\\ \mu_2 & \mu_2 & \mu_3 & \mu_1 & \rho_2 & id & \rho_1\\ \mu_3 & \mu_3 & \mu_1 & \mu_2 & \rho_1 & \rho_2& id \end{array} \end{equation*}
Figure3.1.7Symmetries of an equilateral triangle