In the last video, we talked about Orbit-stabiliser theorem, which states that the total number
of symmetries is the product of the sizes of the orbit and stabiliser of x, where x
is any element in the set X, which the symmetries act on. One deduction that we can make is
that the size of the stabiliser always divide the total number of symmetries. This is a
weaker statement in the sense that it does not tell you what the other factor is. Of
course, the orbit-stabiliser theorem is one way to prove this, but there is another way
that directly explains this fact.
We have this schematic diagram for the symmetries, and the whole rectangle represents all the
symmetries for a particular object. Let’s say this bit here represents those that fixes
x, and we have counted them. If all the symmetries fix x, then the statement that we want to
prove is obvious, because the symmetry is the stabiliser. So let’s say there are other
symmetries out there that don’t fix x, and we call this new symmetry tau. Then we generate
a bunch of new symmetries, as many as the size of the stabiliser. These new symmetries
are all compositions of a symmetry sigma and tau, where sigma is any symmetry in the stabiliser
of x. Then pick another symmetry we haven’t counted, say tau prime. Then again we generate
a bunch of symmetries, in the same way as before. Repeating this process, we eventually
tile up this large rectangle. Since these small rectangles represent that these sets
are of the same size, namely the size of the stabiliser, it is now apparent that the size
of the stabiliser has to divide the total number of symmetries.
That’s very intuitive, but there are some details we glossed over. The first is why
we generate the same number of symmetries in each process. The second is why we generate
new symmetries. This can be further branched out to why these symmetries generated not
already in the stabiliser, and why these generated symmetries are mutually distinct from each
other.
Okay, so for the first detail, we want to know why these generated symmetries have the
same size. This just means that these rectangles are of the same size. This process could have
generated rectangles like this mess, and we don’t know at this stage. In each step,
every set of new symmetries has this particular form. So let’s say we pick two elements
sigma and sigma prime from the stabiliser, and compose it with the new symmetry tau that
we found. Suppose these two symmetries are actually the same, then by undoing tau, we
are left with sigma and sigma prime being equal in the first place. So if we pick two
different elements from the stabiliser, there is no way these two compose with tau to give
the same symmetry. So different elements in the stabiliser compose differently with tau,
and so we have generated as many symmetries as the stabiliser.
The second detail we need to take care of is why these sets of symmetries generated
are different from the stabiliser. Again, although we have this mental picture, after
seeing why these are different from the stabiliser, it can still be a mess, because they can intersect
with one another. Let’s say sigma is a symmetry in the stabiliser of x, then a symmetry that
undoes sigma should also be in the stabiliser. Why? By being in the stabiliser, sigma fixes
x. If this undoing does not fix x, doing sigma then undoing it will move x elsewhere, which
is impossible, so this undoing symmetry stays within the stabiliser.
This means that somewhere in this blue box, there is a symmetry that is equivalent to
undoing sigma first, then do the symmetry tau, because this undoing symmetry is inside
the stabiliser as discussed earlier. Suppose this symmetry in the blue box is also inside
the green rectangle, then we add sigma in the beginning of this symmetry sequence. This
sigma surely is inside the stabiliser. Doing something that fixes x then do something that
also fixes x definitely fixes x. So this symmetry, whatever this is, must be in the stabiliser.
But look more carefully on what this symmetry really is. If we do sigma then undo it, this
is the same as doing nothing, and so this symmetry is tau all along. But this shows
that tau is inside the stabiliser of x. But we deliberately choose tau so that it is not
inside the stabiliser, so we have achieved a contradiction. This means the initial assumption
that undoing sigma then do tau being inside the stabiliser is rubbish. So these rectangles
do not overlap with the original green one.
The last detail we need to look into is why these symmetries generated are all new, different
from one another. That would mean this mental picture is a correct depiction, with all rectangles
separated from one another. Again, we need the fact that if sigma fixes x, then undoing
sigma also fixes x. Then somewhere in this yellow box, there would be a symmetry equivalent
to first undoing sigma then do tau prime. This is because, again, the undoing symmetry
is inside the stabiliser. We want to prove that it is not inside the blue rectangle.
Again, we suppose what we want to prove is false, and arrive at a contradiction somehow.
Suppose there is an overlap between the yellow and blue rectangles, which means that this
symmetry over here is somehow the same as another symmetry zeta then tau, where zeta
is just some other symmetry in the stabiliser of x. We then apply sigma to both symmetries.
Since both symmetries are the same in the first place, doing the same thing to same
symmetries yield same symmetries. For the left side, we use the same trick and see that
this is tau prime. For the right side, we notice that sigma is also in the stabiliser
of x. And again, composing any two symmetries inside the stabiliser should also stay inside
the stabiliser. So this symmetry is a composition of a symmetry in the stabiliser then tau,
which means this symmetry here is still in the blue box. But notice that this is also
equivalent to tau prime. But tau prime is not in the blue box in the first place! Tau
prime is something we haven’t counted before! So again, this is rubbish. And all of this
is because we believe there is an overlap somewhere between the blue and the yellow
rectangles. So that means no overlap between all rectangles.
This is an insane amount of effort and rigour to justify this rectangular tiling proof.
But the stabiliser of x is not the only subset of the group of symmetries that has this property.
Why? Let’s look back again at the three details we have gone through.
For the first part of the argument, we haven’t actually used any property of the stabiliser.
In fact, these rectangles generated must be of the same size whatever set of symmetries
we take from G.
For the second part of the argument, we actually used some properties of the stabiliser. We
first used the fact that undoing a symmetry in the stabiliser stays inside the stabiliser.
So let’s just point that out because it is important. And over here on the right side,
we use another property of the stabiliser. Doing any two symmetries inside the stabiliser
is still inside the stabiliser. In fact, combining these two, we get that the identity is in
the stabiliser, because the identity is the same as first doing sigma, then undoing it,
and by this first property we discovered, undoing of sigma is itself inside the stabiliser,
and by this second property, this sequence of symmetries, which equals the identity,
is also inside the stabiliser.
Now let’s see if the last part of the argument gives us more properties of the stabiliser.
This bit is really the same, so no new properties found here. And here, we also used the property
that composition of symmetries stay within the stabiliser, which we have also discovered
before.
So this means that these three properties of the stabiliser is crucial to the entire
proof. But are these properties only enjoyed by the stabilisers? In other words, if I say
a subset of symmetries H enjoy these properties, must it be the stabiliser of x? Not necessarily.
But what we know is that proving this only requires these properties, if any H fulfils
these properties, then the size of H also divides the size of the whole group of symmetries.
Such an H as a subset of the whole group of symmetries G fulfilling these properties is
called a subgroup of G. For example, when we have the whole group, then all these properties
really hold. So G is always a subgroup of itself. Going back to the rectangular tiling,
but now replacing the stabiliser by any subgroup H, all of these rectangles have a special
name called cosets of H in G. So now, we have this relation. If we want to count the number
of symmetries in group G, we need to count how many things there are in a subgroup H,
then multiply it with the number of cosets you have. This is known as Lagrange’s theorem,
the topic of this video.
So now, we have derived Lagrange’s theorem and the orbit-stabiliser theorem. Now notice
that the stabiliser is a subgroup of G, so what Orbit-stabiliser theorem is saying is
that the number of cosets of the stabiliser is equal to the size of the orbit. So in a
normal lecture or book, they will prove this statement, then we can prove Orbit-stabiliser
from Lagrange. In the last video, we are left with the question of what happens when the
group G is actually infinite, like the group of symmetries of the real number line. Instead
of saying this statement, we would say that the orbit of x bijects with the set of cosets
of the stabiliser of x. This concept will generalise well for infinite sets and is actually
the statement of the full Orbit-stabiliser theorem. But don’t worry if you don’t
know about bijection or infinite sets, because it is not very important in developing intuitions
for group theory. This is just some extra stuff to finish off something that I have
hinted on in previous videos.
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