Grade 9/10 Math Circles
An Introduction to Group Theory Part 2
Review from last
time
Today we are going to learn about symmetric groups, and the
relationship between symmetry groups and symmetric groups! But before we
jump in, let’s do a quick recap of last weeks lesson. Recall that a
group is a set ,
together with a binary operation , such that the following group
axioms hold:
(Associativity) For every , .
(Identity element) There exists such that for all , .
(Inverse element) For every , there exists such that .
We write to
represent the group with binary
operation .
As an example, we saw that the integers
with addition form a group. In other words, is a group.
We also saw that the set of symmetries of a shape , such as the equilateral triangle
below, with composition form a group. We will return to symmetry groups
later in this lesson.
Symmetric Groups
Today we are going to focus on a special class of groups called
symmetric groups. Like most classes of groups, symmetric groups
appear all over mathematics. They are in particular important to the
area of combinatorics! At the end of the lesson, we will see how the
theory of symmetric groups can be used to solve a problem that has to do
with "switching minds"! Recall again that a group is a set together with
a binary operation. To get started, we first learn about this underlying
set.
The
underlying set of a symmetric group
The underlying set of a symmetric group is a certain collection of
objects called permutations. You likely come across
permutations in your every day life! For example, consider a full deck
of cards:
Figure 1: Standard deck of 52 cards
Source: Wikimedia Commons. English pattern playing cards deck [Online]. Available from: https://commons.wikimedia.org/wiki [Accessed March 24 2023].
A standard deck of cards contains 52 distinct cards. We consider
cards with the same number to be distinct because they have different
suits. Similarly, cards with the same suit are considered distinct
because they have different numbers. Any shuffling of this deck of cards
is an example of a permutation. So, in this example, we are
looking at permutations of cards. In card games, the ordering
of the cards in the deck matters and can affect the outcome of the
game!
There are two key things to note in this example:
Cards are being shuffled in various arrangements, and
The order of cards matters.
Together, these two key points describe what a permutation is!
Definition 1
If is a set with a finite
number of elements, meaning that
has elements for some , then we call a finite set.
Note: The definition of set requires that all
elements of a set are distinct. In other words, we don’t allow duplicate
objects to appear in a set. So, if has elements, this means that there are
distinct
elements in .
If is a finite set, then
roughly put, a permutation of is an arrangement of its elements into
some order.
Definition 2
Let be a finite set. A
permutation of
is an arrangement of the elements of into some order such that every element
of appears in the arrangement
exactly once.
The verb we use to describe the action of arranging the elements is
permute. In present tense, we say that "we are
permuting the elements of ".
Example 1
Let . This is a
finite set since there are 3 elements in it. Below are all possible
permutations of :
Note that the definition of permutation requires that every element
of appears exactly
once. We see that the arrangements in Example 1 satisfy this.
Now, for instance, is not a
permutation of since
does not appear. Also, is not a permutation of since appears more than once.
Example 2
For another example, consider . Below are all possible
permutations of :
Exercise 1
Let . List all of
the permutations of .
Stop and Think
Do you see a connection between the permutations in Example 1 and
Example 2?
You may have noticed that the permutations in Example 1 and Example 2
are basically the same. More precisely, if we label with the number ,
with the number , and with the number , then the permutations of are exactly the same as the
permutations of . In fact,
this phenomenon is true in general! The permutations of a given set do
not depend on what the elements look like, it only matters
how many elements are in the set.
Let be a finite set with elements. Going forward,
we may assume that because
if , then is the set containing no elements
(i.e., the empty set) and so there are no elements to permute. Recall
that the elements of a set are always distinct, meaning that there are
no repeating elements. Since has
distinct elements, we can write
where are considered to be
all distinct from each other. In other words, we label each element of
with the symbol for some . For example, we can label
the elements of as , and .
Notice that the permutations of are the same as the permutations of
. Why? Well, if we
permute the elements of around, it’s the same as permuting the
indices of the around. So, really, we are just
permuting elements of the set . Because of this, we can
assume that our sets take the form for some . This phenomenon tells us
that finite sets with the same number of elements have the same
permutations. So, we can make the following definition.
Definition 3
Let be a finite set with elements. The set of
permutations of is
denoted by .
Given a finite set with elements, we can form the
set . The set together with a certain binary
operation forms a group. This group is called the symmetric
group on , or the
symmetric group on . Let’s learn about this
binary operation on .
Binary
operation for symmetric groups
For each ,
consider the set . We want to
find a binary operation on . In
other words, we want to find a way to combine two permutations of into another permutation of
. To do this, we
introduce some new notation called two-line notation.
It’s a bit tricky to define this new notation without introducing more
symbols, so we are going to explain the new notation using examples.
Example 3
Let and consider the
permutation . The
permutation shuffles the
natural ordering into . We can think of this as being replaced by ,
being replaced by , and being replaced by . We can illustrate this information
using the following diagram:
In fact, we can think of
as an operation that takes in and
outputs , takes in and outputs , and takes in 3 and outputs 1. In
words, it is common to say "1 goes to 2, 2 goes to 3, and 3 goes to 1".
By ignoring the arrows in the above diagram, we can represent using the following
two-line notation: The top row is the inputs, the elements of the
set . The bottom row is the
outputs, the values each number is mapped to.
Example 4
Let and consider
the permutation .
In our new notation, we write
Given permutations , we can combine them to make a new permutation in . The permutation is read as the
composition of with . In general, note that the
permutation is not
necessarily equal to the permutation .
Composition is a binary operation on .
Defining the binary operation composition on requires a lot of extra notation, so
we will learn how it works through examples.
Example 5
Let and consider the
following two permutations in The composition is given as follows:
Let’s explain how we got this. Looking at , we see that goes to . We then look at to see where goes. In see that goes to . So the result is that goes to . This is where the first column of
comes from. Next,
looking at we see that goes to . Then looking at we see that goes to . So the result is that goes to . This is where the second column of
comes from.
Lastly, looking at we see
that 3 goes to 2. Then looking at we see that 2 goes to 3. So the
result is that 3 goes to 3. This is where the third column of comes from. We can think
of as first
shuffling into and then shuffling according to .
Example 6
Let and consider
the following two permutations of Then This example illustrates that .
Here, the symbol means "not
equal".
Exercise 2
Let and consider
the following two permutations of Compute the compositions and .
Exercise 3
Convince yourself that is a group.
See solutions to Exercise 3 before reading on
The explanation in Exercise 3 can be generalized to show that is a group for any . So, in general, with composition forms a group! Let’s
sketch how this works. To do this, we need to convince ourselves that
the group axioms hold for with
composition . So, let .
Axiom 1: For associativity to hold, we need to be the
same permutation as . Similar to
symmetry groups, this axiom is a bit tough to argue given our current
mathematical tools. As mentioned in Example 5, we can view permutations
as operations, or more formally as functions. If you know a bit about
functions, then by viewing each permutations as a function, it isn’t too
hard to show that If
you were able to convince yourself that associativity holds in Exercise
3, that is plenty good enough!
Axiom 2: Not shuffling out of it’s natural ordering
is a valid permutation of . In other words, is a permutation in . This is the identity element in
because if you compose this
permutation with you get
back. We write
Axiom 3: Lastly, given a permutation in we can construct it’s inverse as follows. Write using the two-line notation. The
inverse of is obtained by switching the rows
of and then reordering the
columns so that the top row is in the order .
Definition 4
We call the group
the symmetric group on .
Relationship
between symmetry groups and symmetric groups
We are going to first compare the symmetry group of the equilateral
triangle with the symmetric group on . Last time we saw that the
symmetries of the equilateral triangle are
counter-clockwise rotations by multiples of 120 degrees,
and
reflections in the 3 axes that pass through the center and a tip
of the triangle.
If we label the tips of the triangle with the numbers , ,
then we can view the symmetries of the triangle as certain permutations
of . It doesn’t matter how
we label the triangle, but we will use the labelling below for this
lesson:
Just as a permutation in
shuffles the natural ordering , a symmetry of the triangle
shuffles the positions of the numbers on the triangle above. Let’s see
how symmetries of the triangle can be identified with permutations in
. First consider the symmetry
which is counter-clockwise rotation by degrees:
We see that this symmetry moves from the top tip of the triangle to the
left tip, from the left tip to
the right tip, and from the right
tip to the top tip. Alternatively, we can think of this as moving to where was, moving to where was, and moving to where was. In words, we can say " goes to ,
goes to , and goes to " (sound familiar? See Example 3). Given
this, this symmetry corresponds to the permutation Next, consider counter-clockwise rotation by
degrees:
Similar to above, this symmetry sends to ,
to , and to . In words, we can say goes to ,
goes to , and goes . Given this, this symmetry corresponds
to the the permutation The last rotation symmetry of the triangle is
counter-clockwise rotation by
degrees, or equivalently, the identity symmetry which "does
nothing":
This symmetry does nothing, and so the positions of the numbers on
the triangle do not change. Here, we say goes to ,
goes to , and goes to . To no surprise, the identity symmetry
of the triangle corresponds to the identity permutation Next, consider the symmetry that reflects in the
yellow axis drawn below:
This symmetry fixes in the top
position, but interchanges and
. So, we can say goes to ,
goes to , and goes to . This symmetry corresponds to the
permutation The permutations corresponding to the last two
symmetries shown below are
respectively.
The above shows us that every
permutation in corresponds to a
unique symmetry of the triangle. Note that if you compose two symmetries
of the triangle, it’s the same as composing the two permutations that
correspond to the symmetries. For example, the composition of
"counter-clockwise rotation by
degrees" with "counter-clockwise rotation by degrees" rotates the triangle a full
degrees, and so the composition
is the identity symmetry. So, we should expect that the composition of
the corresponding permutations is the identity permutation. Indeed, we
have that
Given these correspondences, or identifications, there is no harm in
saying that the symmetry group of the triangle is the same as
the symmetric group on .
We can play the same game as above by identifying symmetries of a
regular polygon that has sides
with permutations of .
However, in general it is not true that the symmetry group of a regular
polygon with sides is the same as
the symmetric group on .
Exercise 4
Show that the symmetry group of the square (i.e., regular polygon
with 4 sides) is not the same as the symmetric group on
. To get started,
consider the following labelling of the square:
Application to
symmetric groups: The Futurama Theorem
We are going to end this lesson with an application to symmetric
groups! The problem we are going to discuss today was developed and
proved in the episode "The Prisoner of Benda" for the TV show Futurama.
In fact, the mathematician Ken Keeler, who wrote this episode, came up
with this theorem and proved it specifically for the show. The statement
of the problem is known as The Futurama Theorem. Let’s discuss
the problem introduced in this episode.
In the episode, two characters of the show, Amy and Professor
Farnsworth, build a machine that allows them to switch minds! They
decide to test the machine out, and switch minds. They are initially
very excited, but then quickly realize that they aren’t able to switch
their minds back. After reviewing their calculations, they realize that
they failed to take into account the "cerebral immune response" of the
brain, and as a result, once two bodies switch minds they can never
directly switch back using the machine. In other words, the mind swap
machine can’t be used on the same pair of beings more than once.
Figure 2: Amy (left) and Professor Farnsworth (right) using
their mind switching machine.
Source: The Futurama Theorem [Online]. Available from: https://www.youtube.com/watch?v=J65GNFfL94c [Accessed March 24 2023].
To summarize so far:
We have a mind switching machine!
If two beings sit in the chairs, they can switch minds.
Rule: Any two beings can use the machine at most
once to switch minds.
Amy and Professor Farnsworth are panicking a bit now. They want to
switch their minds back, but the machine won’t allow them! They come up
with the idea of swapping minds back by using a third body for temporary
storage space. For a third body, the robot Bender comes to help. Before
we switch more minds, let’s introduce pictures that will help us keep
track of which mind is in which body.
Figure 3: The 3 characters. Bender is at the bottom.
Source: All character photos in this lesson are from: Futurama Fandom. Futurama wiki characters [Online]. Available from: https://futurama.fandom.com/wiki/Characters [Accessed March 24 2023].
We use 2 photos for each character. The photo with the brain is used
to represent the mind of that character. The photo without the brain is
used to represent the body of that character. We can represent the first
mind switch using the following photos:
Figure 4: Amy and Professor Farnsworth switch minds.
The 2 photos on the left indicate that Professor Farnsworth’s mind is
in Amy’s body. The 2 photos on the right indicate that Amy’s mind is in
Professor Farnsworth’s body. Now, they introduce Bender as a way to try
and get their minds back. In the show, the robot Bender sits down in the
mind machine with Amy’ body (Farnsworth’s mind). They switch minds so
that Bender’s mind is now in Amy’s body and Farnsworth’s mind is now in
Bender’s body. So far, we have:
Amy’s body and Farnsworth’s body already switched minds, so they
can’t switch again. Similarly, Amy’s body and Bender’s body already
switched minds, so they can’t switch again. Given this, the only swap
that can be made is between Bender’s body and Farnsworth’s body. This
results in:
This was somewhat successful in the sense that Farnsworth’s mind is
back in their body. However, since Amy’s body and Bender’s body already
swapped minds they can’t switch back! We are essentially back to the
same problem as before. In the episode, they try to fix the problem by
bringing even more bodies into play. After doing many mind switches,
they end up with a big mess!
Exercise 5
Consider the original situation where only Amy and Professor
Farnsworth have switched minds (see Figure 4).
Can we get Amy’s mind back into Amy’s body and Farnsworth’s mind
back into Farnworth’s body?
Further, can (1) be done in such a way that everyone who switches
minds ends up with their own mind in the end?
See solutions to Exercise 5 before reading on
As you have seen, the answer to both questions in Exercise 5 is yes.
And this is true in general. That is, if we switch as many minds as we
want, then it’s true that we can always reverse the minds swaps so that
everyone gets their own mind back. This leads us to the theorem
mentioned at the beginning of this section.
Futurama Theorem
Given a mind swap machine that can’t be used on the same pair of
beings more than once, it is possible to reverse any number of mind
swaps by introducing two more beings that did not have their minds
swapped with anyone before.
Figure 5: Ken Keeler’s proof of the Futurama Theorem
appearing in the episode
Source: The Futurama Theorem [Online]. Available from: https://www.youtube.com/watch?v=J65GNFfL94c [Accessed March 24 2023].
To understand Ken’s proof of the Futurama Theorem, as displayed in
Figure 4, it is enough to go through the proof for the case when only 2
beings switch minds. You technically have done this already! The
solution to Exercise 5 is the proof of this case. Ken’s proof is just in
the language of groups theory. We won’t go through the details of how to
translate the solution to Exercise 5 into group theory, but the basic
idea is as follows. By putting the characters bodies in a fixed order,
you can interpret mind swapping as permuting the brains of the
characters around. That is, mind swaps are permutations of brains!