Abstract Algebra: Theory and Applications

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:

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

Let us find the symmetries of the equilateral triangle $△ABC\text{.}$ To find a symmetry of $△ABC\text{,}$ 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 $\pi :S\to S\text{.}$ 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

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 3.6.

A natural question to ask is what happens if one motion of the triangle $△ABC$ 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

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, ${\rho }_1{\mu }_1\ne {\mu }_1{\rho }_1\text{.}$ A multiplication table for the symmetries of an equilateral triangle $△ABC$ is given in Figure 3.7.