This video will be a summary based on all the videos in this series, so please watch
all these videos first. But here is a quick recap on the concepts in all these videos.
In the first chapter, instead of using the standard definition using axioms, we introduce
groups as symmetries of objects, which lays the foundation for the entire video series,
as well as making the concept of group actions easier to understand, because symmetries naturally
act on the vertices, edges or faces. Then we derived the orbit-stabiliser theorem which
states that the size of the orbit multiplying the size of the stabiliser of any element
in the set X is the size of the group G.
Next we discover the fact that the size of the stabiliser divides the size of the group
can be argued in a different way, and then we extract the properties of the stabiliser
we used to define what a subgroup is. Then we discover Lagrange’s theorem, which states
that the size of a subgroup divides the size of the whole group. Then we introduce the
concept of conjugation and figure out its intuition to be just viewing the symmetry
being conjugated in another perspective. This allows us to introduce the concept of normal
subgroups and simple groups, which in turn allows us to define what a quotient group
is, which is a group of the cosets where we can define a consistent operation. The video
also consisted of seeing the concept in different perspectives.
In the sixth chapter, we then introduce the concept of homomorphism and then understand
the three statements of the isomorphism theorem more intuitively. then finally in the last
chapter, we discover that group actions can be thought of as a homomorphism, and we also
use Cayley’s theorem as a direct application of this change of perspective. With all these
recaps done, now we are going to see an example that will use all the concepts that we have
seen before.
Suppose G is a simple group with 60 elements, and H is a subgroup of G. By considering an
action of G on the cosets of H, show that H cannot have 2,3, or 4 cosets. Let’s dissect
what this means. G being a simple group means that its only normal subgroups are the identity
and the group G itself. So whatever we need to do, it would somehow be related to normal
subgroups. Next, we have the number 60, but the only concepts that we discussed that will
involve some sort of numbers are the Orbit-stabiliser theorem and the Lagrange theorem, because
they refer to products of something.
For the next line, we just note that subgroup is a concept that we have seen in the third
chapter. Nothing too special here. The next line is quite special because it specifically
tells us what to do, so this should be a useful starting point. Just as a reminder of what
a coset is, we have this mental picture of the subgroup and its cosets, so the cosets
are just the rectangles tiling up the entire group G. The last line is what we need to
prove. And this result is not immediately obvious either. Apparently, it must have something
to do with simplicity, and the number 60 somehow. And we are going to figure out how these things
are connected. The third line provides us with the starting point, and the line of reasoning
would be what if G is simple with 60 elements but H has 2,3, or 4 cosets. It should have
some contradiction with the theorems that we know, then we can show that H cannot have
2,3, or 4 cosets. Pause the video if you want to have a go yourself.
The first step is to consider an action of G on cosets of H, which sounds abstract, but
the action we will be considering is very straightforward. Again, we consider this mental
picture we have for this more abstract action. Each dot represents a coset, including the
subgroup H. Considering the action of g on the cosets, this simple action will transform
any coset, to some other coset. This action sends H to the coset gH, g inverse H to H
itself, and so on.
So we now have a group action of G on the cosets of H, given by this relation where
you just stick g_1 in front, we would have the corresponding homomorphism, say phi, from
the group G to the symmetric group on the cosets of H. This opens up the possibility
of analysing the group action with the isomorphism theorem. The isomorphism theorem states three
things once we have a homomorphism between two groups. Let’s look at the first statement
here. It is given that G is simple, which means that the kernel can only be the identity
set or the group G itself, because these are the only normal subgroups that G has. Now
consider the case where this kernel is the entire group G.
Because the kernel is everything that gets mapped to the identity, and in this case,
the identity of the symmetric group of the cosets of H, would be the identity permutation,
which is the permutation where everything stays fixed. If the kernel is the entire group
G, then for all g in the group, the corresponding permutation would look like this, not moving
anything. To show this is not possible, consider g acting on H. H would be sent to gH. Now
this isn’t done yet, because sometimes gH would be the subgroup H itself, like when
g is actually a member of the subgroup H. But this cannot happen for ALL g inside the
entire group, because the only case all elements of the group G will send H to itself is that
there is only one coset to begin with. Since our line is reasoning to assume that H has
2,3, or 4 cosets and see what happens, this case when Kernel of phi is the entire group
G is not possible.
Back to where we were a minute ago. All the discussion just now shows that the kernel
can only be the identity set. Next up, we can use the second statement here. Since it
uses the concept “subgroup”, we are going to use Lagrange’s theorem, which states,
in this context, that the size of the image divides the size of the symmetric group. Lastly,
from the last statement, all we need to use is the fact that isomorphic groups must have
the same size, because the isomorphism has to be one-to-one. Then recall from the quotient
group video where I explained why it is called a quotient group, the size of the quotient
group is just a quotient of the sizes. This also just comes directly from Lagrange’s
theorem. So that means in the last statement, we have the size the quotient group to be
size of G divided by size of kernel. Since the group has size 60, and the kernel is the
identity set, the size of the image can then be just calculated as 60. So in this second
implication that we draw, we can replace the size of the image as 60. So far everything’s
good. But this implication here will be the main punchline.
Let’s rewrite this bit here at the top. The only mystery that we need to tackle is
the size of the symmetric group. Let’s do this in general. If we want to find out the
size of the symmetric group of n things, then the first thing under consideration can be
put in n different places, including the option to not move. then the second thing can be
put in (n-1) different places and so on. This means that the size of the symmetric group,
that is, all the permutations of n things would be n times n-1 all the way to 1 because
the first thing has n places for choice during permutation, and the second thing would have
1 fewer choice, and so on.
So we apply the result we just discovered. If we have only 2 cosets of H, the size of
the symmetric group would be 2 times 1 equals 2. This is not possible because 60 does not
divide 2. What if there are 3 cosets of H? The symmetric group would then have size 6,
not possible. In a similar fashion, if there are 4 cosets, 60 still does not divide the
size of the symmetric group, which completely proves the result that we need to show at
hand. I know that this proof is rather involved because it covers most of the concepts that
we have covered in this video series, so feel free to replay the entire proof and pause
where necessary.
If you are ready for some more, it can be shown that there is only 1 group up to isomorphism
that is simple and has 60 elements. The proof of this is way, way beyond the scope of the
video series, but we can show what this group k structure looks like. It is precisely the
group of rotational symmetries of an icosahedron. To link to really all the concepts we have
covered, try using orbit-stabiliser theorem to see why this group has 60 elements. And
use the intuition of conjugation to see why this group is simple.
Great thanks to all your support to this group theory video series, and I really hope that
you can grasp the intuition to aid your learning whether you are studying or about to study
this interesting field of mathematics.
If you enjoyed this video, give it a like and subscribe to the channel with notifications
on. See you next time!