Let’s consider the group of symmetries of a regular dodecagon, that is a polygon with
12 sides.
Specifically, we are considering the subgroup of all rotational symmetries.
This means we are only considering rotations in either direction by a multiple of 30 degrees.
You can use this checklist introduced in Lagrange’s theorem video to check that
it is indeed a subgroup.
We temporarily call this group R_12, R for rotation.
However, it seems similar to the quotient group Z mod 12Z.
We will see why they are similar in a minute.
Just a reminder that Z mod 12Z is a quotient group, that is, a group of cosets
which themselves contain symmetries.
So on one hand, we have R_12, with symmetries corresponding to rotations by a multiple of
30 degrees, and on the other hand, we have Z mod 12Z, which is a group of cosets of 12Z
written this way.
It’s tempting to say that these two groups are the same, but R_12 is fundamentally a
group consisting of rotational symmetries of the dodecagon, while Z mod 12Z is fundamentally
a group consisting of cosets which themselves contain translational symmetries of the number
line.
Mathematicians wanted to make the sense of sameness more rigorous, and came up with the
concept of isomorphism.
This video deals with the concept of homomorphisms and isomorphisms, which is strange in terms
of naming because both “homo” and “iso” mean “same”, but isomorphism is a stronger
kind of sameness than homomorphism.
Let’s start with homomorphism.
As said before, “homo” means “same”, and “morphism” here could mean the group
structure.
Roughly speaking, it is a function from a group to another that preserves the group
structure.
Let’s say we have a group G, which can be illustrated more colorfully as such, where
each blob represents an element in G. Suppose that in the group G, when we concatenate yellow
and blue blobs in this particular order, we get a green blob.
Let’s say there is a function phi from G to another group H. And it maps these circles
to squares in the group H. Now if we concatenate yellow square and blue square in the exact
same order as before, we might get the green square.
If for every possible relation that you have for the group G, you have the correct corresponding
relation for group H, then this function phi is called a homomorphism from G to H.
To put it more succinctly, the yellow square is actually the image of the yellow circle
under the map phi.
Similarly for the blue square and the green one.
However, because of how yellow, blue and green circles are related to each other, we can
replace the green circle with this.
Basically, if this relation holds for all pairs of elements in the group G, then phi
is a homomorphism.
This is what you will see in a textbook, of course with symbols not these circles.
Now this function phi preserves group structure in the sense that all the compositions of
symmetries remain intact through this map.
Now we are going to see an example of a kind of unexpected homomorphism, first by denoting
R_n, as the group of rotational symmetries of a regular n-gon, generalising the case
when n equals 12 that we discussed.
It turns out that we can actually construct a homomorphism from R_8 to R_4, which is pretty
unexpected considering that they have different numbers of symmetries.
Let’s list= all the symmetries in each group.
The function maps the symmetries like what’s shown here.
Just to convince you that it is indeed a homomorphism, we consider this particular pair of symmetries.
The result of these two symmetries, should be a rotation by 225 degrees.
Now apply the function phi to all the symmetries involved.
We can verify that the result of the two yellow symmetries on the right does give us the blue
symmetry, because a combination of 180 and 270 degree rotations does give a 90 degree
rotation.
But we definitely don’t want to say these two groups are the same.
The main obstacle is that the function phi is not invertible.
If you don’t know what that means, this homomorphism phi is not a one-to-one function.
In fact, any invertible homomorphism is called an isomorphism.
To see an example of isomorphism, let’s go back to our example at the very beginning
of this video.
An obvious way to construct an isomorphism phi is to map the rotational symmetry by 30k
degrees to the coset k + 12Z.
To check that this is really an isomorphism, we need to first check that it is a homomorphism.
We take an arbitrary pair of symmetries on the left, and then map it to something in
the quotient group according to the rule described.
We then compose symmetries on the left, and compose the cosets on the right separately.
Then we really have to check that the symmetry on the left does map to the coset on the right
under the function phi.
If that is true, then this function phi is a homomorphism.
However, we also need to check that it is one-to-one in order to say it is an isomorphism.
This is obviously true in this case.
So this function phi is actually an isomorphism.
However, this is not the only isomorphism you can have.
For example, this similar-looking function is also an isomorphism.
This notation here is not really a standard one, but it means the remainder of 5k when
divided by 12.
You should check that this function is a homomorphism as well as a one-to-one function in order
to show it is an isomorphism.
The difficult part is to show that this function psi is one-to-one, but we will skip this for
this video.
In general, we say the group G is isomorphic to another group H, if there is just one isomorphism
between the groups.
We have already shown that the isomorphism might not be unique, but as long as there
is an isomorphism, we will call these two groups isomorphic.
So since there is at least one isomorphism between R_12 and Z mod 12Z, we say these two
groups are isomorphic to each other.
Let’s revisit this homomorphism again.
As with all functions, we can define the image of this function phi, which is something that
is hit by the function.
This is usually denoted as Im of phi.
Another thing we are interested in is what symmetries on the left get mapped to the identity
on the right.
In this case, there are two symmetries.
This set of symmetries is called the kernel, denoted as Ker of phi.
We can also define the kernel and image in a general homomorphism.
The isomorphism theorem states three things.
Firstly, the kernel is a normal subgroup of G, secondly, we have the image to be a subgroup
of H. Lastly, since kernel is a normal subgroup of G, we can define the quotient group, and
this quotient group is isomorphic to the image.
Mathematicians like to use symbols to denote all these concepts, listed on the right here.
The third statement is what most people think about when it comes to isomorphism theorem,
but the first statement is often glossed over, and actually pretty useful sometimes.
Proving these three statements is not super difficult, but the focus of the rest of the
video will be about how we can intuitively understand them.
The first statement can be understood as follows.
By definition, the kernel is the set of all symmetries that gets mapped to the identity
of the other group.
As a result, the kernel actually acts like the identity, which looks the same in all
possible perspectives, because it is just not doing anything.
So intuitively, we can see why the kernel is a normal subgroup.
If we really need to prove the second statement here, we will need the subgroup checklist.
However, we can use the type of thinking as before.
The whole group G is mapped to the image by definition of the image, so the image acts
like the entire group G itself.
Of course, the image is a subset of the group H, and because it is a group, it is a subgroup
of the entire group H.
Now what about this third statement?
Let’s say this is the entire group, and this subset is the kernel of the homomorphism.
And these are the cosets of the kernel within the group G.
And consider the image of the homomorphism.
Let’s say this yellow square represents the identity in the image, then by definition
of the kernel, it gets mapped to the yellow square.
It is not too wild to assume that all the symmetries in the same coset gets mapped to
the same thing in the image.
So one coset can get mapped to one symmetry in the image, and hence these two groups are
isomorphic.
Now of course all these are just intuitive understanding of the theorem, not a rigorous
proof.
As a side note, the reason why the first statement is important is that it offers a way to prove
something is a normal subgroup easily.
This is sometimes also called the first isomorphism theorem, because there are second and third
isomorphism theorems.
However, all these theorems can be derived from the first one, using clever choices of
the homomorphism phi.
Now in a normal course on group theory, you might spend a great deal of time on symmetric
and matrix groups, but I will not cover those topics because most concepts will be specific
to those types of groups and not applicable in general.
And this is also because this video series has already been up for so long, that I am
not sure whether viewers like you are bored with this.
That does not mean group theory will not be on this channel anymore, because I might talk
about some interesting applications, but it highly depends on your enthusiasm towards
this idea.
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See you next time!