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PreTeXt Sample Book: Abstract Algebra (SAMPLE ONLY)

Thomas W. Judson, Isaac Newton (Editor)

Section 3.1 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.
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Subsection 3.1.1 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.
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We have already seen that two integers a and b are equivalent mod n if n divides a−b. The integers mod n also partition Z into n different equivalence classes; we will denote the set of these equivalence classes by Zn. Consider the integers modulo 12 and the corresponding partition of the integers:
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[0]={…,−12,0,12,24,…}[1]={…,−11,1,13,25,…}⋮[11]={…,−1,11,23,35,…}.
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When no confusion can arise, we will use 0,1,…,11 to indicate the equivalence classes [0],[1],…,[11] respectively. We can do arithmetic on Zn. For two integers a and b, define addition modulo n to be (a+b)(modn); that is, the remainder when a+b is divided by n. Similarly, multiplication modulo n is defined as (ab)(modn), the remainder when ab is divided by n.
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Example 3.1.1. Modular Addition.

The following examples illustrate integer arithmetic modulo n:
7+4≡1(mod5)7⋅3≡1(mod5)3+5≡0(mod8)3⋅5≡7(mod8)3+4≡7(mod12)3⋅4≡0(mod12).
In particular, notice that it is possible that the product of two nonzero numbers modulo n can be equivalent to 0 modulo n.

Example 3.1.2. Modular Arithmetic.

Most, but not all, of the usual laws of arithmetic hold for addition and multiplication in Zn. For instance, it is not necessarily true that there is a multiplicative inverse. Consider the multiplication table for Z8 in Figure 3.1.3. Notice that 2, 4, and 6 do not have multiplicative inverses; that is, for n=2, 4, or 6, there is no integer k such that kn≡1(mod8).
⋅01234567000000000101234567202460246303614725404040404505274163606420642707654321
Figure 3.1.3. Multiplication table for Z8

Proof.

We will prove (1) and (6) and leave the remaining properties to be proven in the exercises.
(1) Addition and multiplication are commutative modulo n since the remainder of a+b divided by n is the same as the remainder of b+a divided by n.
(6) Suppose that gcd(a,n)=1. Then there exist integers r and s such that ar+ns=1. Since ns=1−ar, it must be the case that ar≡1(modn). Letting b be the equivalence class of r, ab≡1(modn).
Conversely, suppose that there exists an integer b such that ab≡1(modn). Then n divides ab−1, so there is an integer k such that ab−nk=1. Let d=gcd(a,n). Since d divides ab−nk, d must also divide 1; hence, d=1.
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Subsection 3.1.2 Symmetries

Figure 3.1.5. Rigid motions of a rectangle
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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 3.1.5, it is easy to see that a rotation of 180∘ or 360∘ 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∘ rotation in either direction cannot be a symmetry unless the rectangle is a square.
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Figure 3.1.6. Symmetries of a triangle
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Let us find the symmetries of the equilateral triangle △ABC. To find a symmetry of △ABC, we must first examine the permutations of the vertices A, B, 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 π:S→S. 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⋅2⋅1=3!=6 different arrangements. To denote the permutation of the vertices of an equilateral triangle that sends A to B, B to C, and C to A, we write the array
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(ABCBCA).
Notice that this particular permutation corresponds to the rigid motion of rotating the triangle by 120∘ in a clockwise direction. In fact, every permutation gives rise to a symmetry of the triangle. All of these symmetries are shown in Figure 3.1.6.
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A natural question to ask is what happens if one motion of the triangle △ABC is followed by another. Which symmetry is μ1ρ1; that is, what happens when we do the permutation ρ1 and then the permutation μ1? Remember that we are composing functions here. Although we usually multiply left to right, we compose functions right to left. We have
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(μ1ρ1)(A)=μ1(ρ1(A))=μ1(B)=C(μ1ρ1)(B)=μ1(ρ1(B))=μ1(C)=B(μ1ρ1)(C)=μ1(ρ1(C))=μ1(A)=A.
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This is the same symmetry as μ2. Suppose we do these motions in the opposite order, ρ1 then μ1. It is easy to determine that this is the same as the symmetry μ3; hence, ρ1μ1≠μ1ρ1. A multiplication table for the symmetries of an equilateral triangle △ABC is given in Figure 3.1.7.
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Notice that in the multiplication table for the symmetries of an equilateral triangle, for every motion of the triangle α there is another motion β such that αβ=id; that is, for every motion there is another motion that takes the triangle back to its original orientation.
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∘idρ1ρ2μ1μ2μ3ididρ1ρ2μ1μ2μ3ρ1ρ1ρ2idμ3μ1μ2ρ2ρ2idρ1μ2μ3μ1μ1μ1μ2μ3idρ1ρ2μ2μ2μ3μ1ρ2idρ1μ3μ3μ1μ2ρ1ρ2id
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Figure 3.1.7. Symmetries of an equilateral triangle
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