The case of regular polygons, such as the hexagon, is a very useful place to talk about symmetry. I want to know which linear transformations preserve the regular polygons. By ‘preserve’, in this section I mean that the new shape is precisely the same as the old: same size and same positions. Some of the vertices and edges may have been moved, but the result ends up tracing over exaclty the same original shape.
Definition9.1.1.
A Dihedral Group is the group of linear symmetries of a regular polygon. If the polygon has \(n\) edges, then I write \(D_n\) for the associated dihedral group. Thus, \(D_3\) is the group of symmetries of the triangle, \(D_4\) is the group of symmetries of the square, \(D_5\) is the group of symmetries of the pentagon, and so on. (Some texts use \(D_{2n}\) instead of \(D_n\text{.}\))
You may have noticed that I used the word ‘group’ instead of ‘set’ to describe this collection. ‘Group’ is a technical word in mathematics, referring to a set with some extra properties. Let me give the definition in somewhat formal language. (In this definition, a binary operation is a process that takes two elements and combines them into one. It is a general term that includes addition of numbers, multiplication of numbers, or composition of transformations: in each operation, two things are combined together).
Definition9.1.2.
Let \(G\) be a set with an (associative) binary operation. I represent the operation by writing two elements next to each other (as is common with multiplication of variables). \(G\) is a group if it satisfies these properties.
There is an element \(e\text{,}\) called the identity element. If \(g\) is any other element of the group, then \(ge =
eg = g\text{.}\) (Multiplication by the identity doesn’t do anything to any group element).
For each element of the group \(g\text{,}\) there is a unique element \(g^{-1}\) such that \(gg^{-1} = g^{-1}g =
e\text{.}\) (Everything in a group has an inverse).
You already know a number of groups.
\(\ZZ\) is a group under the operation of addition. \(0\) is the identity element, and the inverse of any \(a\) is \(-a\text{.}\)
\(\QQ \setminus \{0\}\) is a group under multiplication. \(1\) is the identity, and the inverse of any fraction \(\frac{a}{b}\) is \(\frac{b}{a}\text{.}\) Obviously, I needed to exclude \(0\) since I cannot divide by \(0\text{,}\) and therefore it wouldn’t have an inverse in the group.
\(\RR \setminus \{0\}\) is also a group under multiplication.
In Definition 7.1.4, I defined the general linear group \(GL_n(\RR)\text{.}\) This was all \(n
\times n\) invertible matrices. The identity matrix (of size \(n\)) is the identity element; the composition is the ‘multiplication,’ and the matrix inverse is the inverse operation. By restriction to only invertible matrices of a fixed size, all elements can be composed together, and all elements have inverses.
In Definition 9.2.1, I defined another matrix group: the special linear group \(SL_n(\RR)\text{.}\) This was like the general linear group, but with the added restriction of determinant 1.
In Definition 9.2.2, I defined two more matrix groups: the orthogonal group \(O_n(\RR)\) and the special orthogonal group \(SO_n(\RR)\text{.}\) These are smaller sets of invertible matrices with particular properties. Each of them contains the identity matrix and has inverses, which are matrices with the same restricted properties.
In this course, I will primarily discuss dihedral groups to understand what groups are all about. In other institutions with larger mathematics programs, courses are devoted exclusively to group theory, which is an active and important area of mathematical research.
Subsection9.1.2The Group \(D_4\)
Figure9.1.3.Symmetries of the Square
Consider the symmetries of the square. I assume the origin is at the centre of the square, and I can put the vertices of the square at \((1,1)\text{,}\)\((1,-1)\text{,}\)\((-1,-1)\) and \((-1,1)\text{.}\) Conveniently, the vertices are all I need to think about: since linear transformations preserve lines, if I know where the vertices go, then I know that the lines connecting them are preserved, and hence the shape is preserved. Which transformations preserve the vertices?
There are two relatively obvious classes that might occur. First, there is rotation about the origin. If I rotate by \(\pi/2\) radians (a quarter turn), the vertices are preserved. Likewise for rotations by \(\pi\) and \(3\pi/2\) radians. If I am really paying attention, I might also think of rotation by \(2\pi\) radians, a full turn. This might seem silly, but the identity transformation (the transformation that leaves everything fixed) is a very important transformation and should be included in the list. So, for now, the symmetries of the square include the identity and three rotations. I’ll call these three rotations \(R_1\text{,}\)\(R_2\) and \(R_3\text{,}\) in the order I defined them.
I might also think about reflections. Looking at the square, I can find four lines of reflection: the \(x\)-axis, the \(y\)-axis, the line \(x=y\) and the line \(x=-y\text{.}\) I’d like to label the reflection by taking the lines of reflection in counter-clockwise order starting from the \(x\)-axis: the reflection over the \(x\)-axis is \(F_1\text{;}\) the reflection over \(y=x\) is \(F_2\text{;}\) the reflection over the \(y\)-axis is \(F_3\text{;}\) and the reflection over \(y=-x\) is \(F_4\text{.}\) The lines in Figure 9.1.3 are labelled with these reflection names.
That brings the total number of transformations to eight: the identity, three rotations, and four reflections. These eight are, in fact, the complete group.
So I’ve constructed a group of symmetries of the square. I have eight elements, one of which is the identity and the operation of composition. The abstract question that mathematicians ask now is: what is the structure of this group? By ‘structure,’ I mean the elements but, more importantly, how the elements relate to each other when composed. To show this structure, I can build a multiplication table for the group. Such a table would explicitly show all the compositions. It is structured very much like the multiplication tables for numbers that you learned in elementary school — you are now just studying a new kind of multiplication.
To make the multiplication table, I need to know how to calculate the composition of elements. I could do this by writing their matrices and doing the matrix multiplication. That would work, but I want to show you an easier method — one much more connected to the geometry of the square. I’m going to represent each element of the group simply by how it permutes the vertices. In Figure 9.1.3, I labelled the vertices. By showing where each vertex goes, I can describe each transformation. For example, the rotation \(R_1\) sends each vertex to the next vertex counterclockwise. I can represent this by the following vertex movements.
The reflection \(F_2\) is over the line \(y=x\text{.}\) It fixes the two vertices which lie on that line and switches the other two. I can represent this by the following vertex movements.
Now I can use these vertex movements to understand composition. Consider \(R_1 \circ F_2\text{.}\) Since composition works from right to left, this is doing the reflection first, then the rotation. I can put the two sets of vertex movements together.
Then I can ask: what is this transformation? This switches vertices \(1\) and \(2\text{,}\) and also switches the other two vertices. This must be reflected over the \(y\)-axis, which is \(F_3\text{.}\) Therefore, \(R_1 \circ F_2 = F_3\text{.}\) I’ve calculated a multiplication. What happens if I reverse the order: \(F_2 \circ R_1\text{?}\) Now the rotation happens first, but I can still calculate the vertex movements by doing the rotation then the reflection.
This transformation switches vertices \(1\) and \(4\text{,}\) and also switches vertices \(2\) and \(3\text{.}\) It must be the reflection over the \(x\)-axis, which is called \(F_1\text{.}\) Therefore, \(F_2 \circ R_1 = F_1\text{.}\)
In this way, I can calculate all the compositions and put them into a multiplication table. In this table, each entry is the composition of the row and then the column; therefore, the column entry happens first (again, right to left) and the row entry second.
Table9.1.4.
\(\circ\)
\(\Id\)
\(R_1\)
\(R_2\)
\(R_3\)
\(F_1\)
\(F_2\)
\(F_3\)
\(F_4\)
\(\Id\)
\(\Id\)
\(R_1\)
\(R_2\)
\(R_3\)
\(F_1\)
\(F_2\)
\(F_3\)
\(F_4\)
\(R_1\)
\(R_1\)
\(R_2\)
\(R_3\)
\(\Id\)
\(F_2\)
\(F_3\)
\(F_4\)
\(F_1\)
\(R_2\)
\(R_2\)
\(R_3\)
\(\Id\)
\(R_1\)
\(F_3\)
\(F_4\)
\(F_1\)
\(F_2\)
\(R_3\)
\(R_3\)
\(\Id\)
\(R_1\)
\(R_2\)
\(F_4\)
\(F_1\)
\(F_2\)
\(F_3\)
\(F_1\)
\(F_1\)
\(F_4\)
\(F_3\)
\(F_2\)
\(\Id\)
\(R_3\)
\(R_2\)
\(R_1\)
\(F_2\)
\(F_2\)
\(F_1\)
\(F_4\)
\(F_3\)
\(R_1\)
\(\Id\)
\(R_3\)
\(R_2\)
\(F_3\)
\(F_3\)
\(F_2\)
\(F_1\)
\(F_4\)
\(R_2\)
\(R_1\)
\(\Id\)
\(R_3\)
\(F_4\)
\(F_4\)
\(F_3\)
\(F_2\)
\(F_1\)
\(R_3\)
\(R_2\)
\(R_1\)
\(\Id\)
From the table, I can see a number of patterns, all of which are part of the structure of this group. Even with eight elements, there is already a fair bit to talk about. The composition of any two rotations is another rotation. Moreover, rotations commute with each other (they can be composed in either order without changing the result). The composition of any two reflections is also a rotation. The composition of any rotation and any reflection is some other reflection. The rotations and reflections do not commute, nor do reflections commute with each other. In each row and each column, each group element shows up exactly once.
