In group theory, we often want to study the group structure, and in this context, it means
how the symmetries relate to each other. So for example, if we consider the real number
line, then translation to the right by 2 units is a symmetry, because it preserves both distances
and the number line. And of course, translation to the right by 1 unit is also a symmetry.
But we of course know that doing the translation by 2 units then by 1 unit, is the same as
doing the translation by 3 units to the right, which is itself a symmetry. These kinds of
relationships for all the symmetries are what we are trying to study. For this real number
line, there is an infinite number of symmetries even just by translation, but for the group
of symmetries of, say, the tetrahedron, there is only a finite number of symmetries. And
we often count the number of symmetries first before analysing the group structure. That
is because for a given number of elements in a group, there is usually only a handful
of group structures possible, so the number of elements is a useful starting point.
Let’s start with a regular polygon, and here we have an octagon, but the argument
would make sense for all regular n-gons. Let’s label the vertices as 1 to 8. Let’s say
sigma is some random symmetry of the regular polygon. And suppose the vertex 1 is sent
to vertex 4 via this sigma symmetry. Then now, we do any symmetry you can think of the
send vertex 4 back to vertex 1. Here we pick rotation, which is denoted as r_4, which just
means sending 4 back to 1. You can also pick a reflection or some combinations of rotations
and reflections. So that means r_4 acts on 4 to give 1. Here, I replaced 4 by sigma star
1, because we have established earlier, that sigma star 1 equals 4. So this means that
doing sigma then r_4 will fix the vertex 1. Now consider where vertex 2 can go. It was
next to 1, and should be next to 1 after all these symmetries because distances are preserved.
This means that after all these symmetries, 2 can go to 2, or 8, because both vertices
are adjacent to 1. So depending on whether it goes to 2 or 8, we will or will not do
another reflection, which we denote as s. And then this sequence of symmetries will
fix 2 as well. And with 2 points fixed in this regular polygon, every point is fixed.
So this sequence of symmetries is the same as doing nothing, that is, the identity.
So to summarise and generalise to an n-gon, for every symmetry sigma that we can think
of, we try to revert back to identity first by fixing 1. We do this by doing one of these
symmetries. If vertex 1 ends up being vertex k, we can revert by using r_k. Then we want
to fix 2 as well so that everything is fixed. This is done by either not doing anything,
if 2 is already fixed, or doing a reflection, if 2 is not fixed yet. After this process,
this has to be the identity symmetry, that is, not doing anything. And we can identify
sigma uniquely by the combination of these two processes. And so now, in the first process
of fixing the vertex 1, there are n different symmetries possible for choice. And the second
process offers 2 options. That means, when we count the total number of symmetries, we
just multiply the two numbers together, and we get 2n. And this group structure is known
as the dihedral group of order 2n, denoted as D_2n, but this remark is not important
here. What’s more important is that this is already an illustration of the Orbit-stabiliser
theorem.
But what exactly is the Orbit-stabiliser theorem? Recall the group action intuition we had in
the previous video. We can choose anything from a given set, which can be vertices, edges
or faces, and let a symmetry act on it. Now let’s say x is something from the set we
are interested in. Then we just ask the question, where x can go under all the symmetries. The
set of things that x can go is called the orbit of x. So in the example of regular n-gon,
the orbit of the vertex 1 is the set of all vertices because we can always rotate the
polygon from vertex 1 to any vertex we want. So now we have explained what an orbit is.
This name does take time getting used to, but the definition of orbit is somehow related
to its name. After all, I do not name them, but mathematicians.
So now, we explain the stabiliser bit. Stabiliser of a particular element x is the set of symmetries
fixing x. And in the example of a regular n-gon, of course the identity symmetry fixes
1. In fact, it fixes everything by definition. But reflectional symmetry along the green
axis here also fixes 1. So that explains the stabiliser bit. Hopefully, the name stabiliser
makes sense here. Now, knowing what the orbits and stabilisers are, what exactly is the theorem
about, and why does the previous discussion illustrate the theorem already? The theorem
relates the two concept by saying that the number of symmetries is the product of the
sizes of the orbit and the stabiliser. In this previous counting method, how did we
illustrate this theorem?
The n choices here really is the size of the orbit of 1 under the symmetries. Why? For
all k inside the orbit of 1, there must be some symmetry sigma that acts on 1 to give
k. That’s the definition of an orbit. Then for any k that we have, we just need to find
a way to make it go back to 1. We denote the symmetry of choice as r_k. In this process,
for any k in the orbit of 1, we just need 1 such r_k so that we can get the symmetry
to fix the vertex 1. So this is why this n actually corresponds to the size of the orbit.
And the 2 here really is the size of the stabiliser, because any symmetries you do in this second
box should fix the vertex 1. Otherwise, the work we have done in the first process would
be useless, because ultimately we want to fix everything, and we have already fixed
1 in this process. If we move 1 elsewhere in this process, we are not going to fix everything
in the end, because at least the vertex 1 is not fixed. So these symmetries are actually
the stabiliser of 1, and hence the 2 here refers to precisely the size of the stabiliser.
Now what we did was multiplying the size of the orbit and the size of the stabiliser,
hence illustrating the Orbit-stabiliser theorem. In fact this even offers a way to prove the
theorem in a more general case when you think about it.
In the general case, let’s still take sigma to be any symmetry of an object. Again, we
want to revert this symmetry. This is done by first finding a way to fix x. Remember
that x can be a vertex, an edge, or a face. Then we apply any of these symmetries. These
symmetries are constructed such that g of x_k sends x_k back to x. There can be more
than one that does this, but we just pick one. So that means, if the original symmetry
sigma sends x to x_k, we just do one of these so that x is fixed. Then we try to do fix
every other thing as well, while keeping x fixed. This is basically the definition of
the stabiliser of x, because inside the stabiliser, all the symmetries can do whatever they want,
as long as x is fixed. So in the first process, we have the number of choices to be the size
of the orbit, because for anything inside the orbit, we find exactly one corresponding
to it. And of course, the number of choices for the second process is exactly the size
of the stabiliser, because these symmetries exactly form the stabiliser. And so here,
the total number of symmetries is simply multiplying these two things, hence the orbit-stabiliser
theorem proved.
Well, there is just one caveat, which is a pretty big one actually. This is only true
for the group of symmetries being finite. That is because talking about sizes of infinite
sets doesn’t make any sense. To make this work, we need to introduce the concept of
cosets, which we will do next time; and bijection, which is actually not a group theory concept
at all, but it makes what we mean more precise. But this finite version is definitely way
more intuitive to understand.
The next chapter will be about Lagrange’s theorem, which is another important theorem
in group theory. This is a bit strange because the normal order that you see in a book or
lectures might look something like this, placing Lagrange’s theorem before introducing group
actions and the orbit-stabiliser theorem, probably because the logic work that way better,
but I find orbit-stabiliser theorem more intuitive to understand than Lagrange’s theorem, and
you can compare these two after we talk about it next video.
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