Group theory is an important area in mathematics that studies objects known as groups. Groups show up all over mathematics, and even appear in unexpected places. For example, we can solve a Rubik’s Cube using group theory! Group theory also has many important applications in physics and chemistry. As an example, consider the following Benzene molecule from chemistry:
In your head, imagine rotating the molecule by multiples of 60 degrees, or reflecting the molecule along axes that go through the center and 2 spheres. Notice that the Benzene molecule appears the same after performing these actions. In other words, the molecule is unchanged. We call such actions symmetries. We will see later on that these symmetries actually form a group! Chemists use symmetries of molecules, like the benzene molecule, to try and predict or explain their properties. In this lesson, we define what a group is and look at a special class of groups called symmetry groups.
Briefly put, a group is a set together with an operation that follows some specific rules, which are called group axioms. Before we define a group, we need to define what a set and a binary operation on a set are.
A set is a collection of objects (e.g., numbers, symbols, shapes). We refer to the objects in a set as elements of the set.
Let
We write
We write
The collection of all integers is a set which we denote by
The collection of letters in the english alphabet form a set. We
write this set as
For our purposes, an operation is a "machine" that takes in one or more inputs, and produces an output.
A binary operation on a set is an operation that takes in two elements of the set and outputs another element of the set.
If
Note: Depending on what
Binary often refers to 2 things, or a pair of things. And in mathematics, an operation roughly means combing a bunch of things in some way to produce something new. So, binary operation means combing 2 things to make another thing.
Addition is a binary operation on
Now that we familiarized ourselves with sets and binary operations, we are ready to define a group.
A group is a set
(Associativity) For every
(Identity element) There exists
(Inverse element) For every
We often write
Let’s go through each group axiom individually! For a concrete
example, we will see that the group axioms hold for
Axiom 1 says that the operation is
associative. Associative means that rearranging the parentheses
in an expression will not change the end result. In other words, the
order in which we evaluate operations doesn’t matter. Addition on the
integers is associative. For example
Axiom 2 says that the underlying set
Axiom 3 says that every element in the underlying
set
If a set
Solution:
We know from Example 3 that addition is a binary operation on
We know that it doesn’t matter what order we add numbers in. So the first axiom holds.
Recall that if we add any integer to zero, we just get the
integer back. That is, for any integer
If
Consider the set
In Example 4 we saw that
Recall that a rational number is of the form
At the beginning of the lesson, we said that certain rotations and reflections of the Benzene molecule form a group. In fact, they form a special kind of group called a symmetry group. For the reminder of the lesson we are going to explore symmetry groups and see how the rotations and reflections of the Benzene molecule form a group!
What does the word symmetry mean to you? Have you heard it before? If so, where?
If you think back, you may have learnt that the monarch butterfly exhibits what we call symmetry. If you look up the word "symmetry" there are all sorts of definitions, but they all amount to a sense of balance and harmony. The monarch butterfly exhibits balance in the sense that both of it’s wings are the same. Imagine drawing a vertical line between the two wings of the butterfly. You can see that the same image appears on both sides of the line, with the only difference being that each side looks like the opposite side after being flipped over the line. Here we say that this imaginary line is a line of symmetry.
Source: PNGWING. The Monarch Butterfly Insect [Online]. Available from: https://www.pngwing.com/en/free-png-dguvl [Accessed March 19 2023].
Source: Tes for Teachers. Symmetry Lessons [Online]. Available from: https://www.pinterest.ca/pin/329114685250253709/ [Accessed March 19 2023].
Look at the flower in Figure 3. Do you see the symmetry that the flower has? What are the lines of symmetry for this flower?
In mathematics, we like to make our definitions as precise as possible. A "sense of harmony and balance" isn’t a very precise definition for symmetry.
In mathematics, a symmetry of a shape is an action (e.g., rotation, reflection) such that when applied to the shape you get the shape back.
There are other types of actions in mathematics, but for our purposes, we will only be considering rotations and reflections.
Consider the butterfly in Figure 3. Reflection in the line of symmetry discussed earlier is a symmetry of the butterfly because when we reflect the butterfly in this line it is unchanged.
Consider the image below. We label the tips of the left triangle with numbers to help us keep track of actions performed on the triangle. In other words, the numbers help us see how actions move the triangle. For example, in this image we see that if we rotate the left triangle counter clockwise by 120 degrees, we end up with the triangle on the right. This action sends tip 1 to where tip 2 was, tip 2 to where tip 3 was, and tip 3 to where tip 1 was.
Aside from the labels, the resulting triangle on the right is identical to the original triangle on the left. This means that rotation counter clockwise by 120 degrees is a symmetry of this triangle.
Consider the image below. The action here is the same as in Example 6, but on a different triangle. The sides of the triangle in Example 6 are all the same length, whereas the sides of the triangle below are not all the same. If we rotate the left triangle counter clockwise by 120 degrees, we end up with the triangle on the right.
The resulting triangle on the right does not look like the original triangle on the left. Because of this, rotation counter clockwise by 120 degrees is not a symmetry of this triangle.
Consider the image below. The action in this example is reflection in the axis (or line) which is depicted as the dotted yellow line.
If we
reflect in this axis, we see that
An equilateral triangle is a triangle whose 3 sides all have the same length. The triangle in Example 6 is an equilateral triangle. Write down all of the symmetries of an equilateral triangle.
Recall that a group is a set, together with a binary operation, that satisfies the group axioms. Given any shape, we can form a group which we call the symmetry group of that shape. The underlying set for the symmetry group of a shape is the set of symmetries of that shape. Next, we define the binary operation for symmetry groups.
Let
Consider the following images. The first symmetry is from Example 6.
The second symmetry is from Example 8, but on an equilateral triangle.
The trio of triangles in this image depicts the composition of "reflect in yellow axis" with "rotate counter clockwise by 120 degrees". The composition is computed by first rotating the left most triangle by 120 degrees counter clockwise, and then reflecting this resulting triangle (middle triangle) in the yellow axis.
Let
Note that this composition is the same as the following symmetry:
In Example 9, we see that the composition of the two symmetries is
itself a symmetry of the triangle. This is true in general. We take it
as a fact that if
Let
Let
The proof for Exercise 5 can be generalized to show that
Axiom 1: For associativity to hold, we need
Axiom 2: The action "do nothing" on
Axiom 3: Lastly, if we apply a symmetry
Consider the benzene molecule below and denote it by
Write down the symmetry group of the benzene molecule