As mentioned before in the video series, mathematicians would like to study the group structure. But
sometimes, the group is too big, like that of the Rubik’s cube has more than 43 quintillion
symmetries.
Let’s represent the entire group we want to study as a big rectangle. What we can do
is to divide and conquer, so that we can handle these smaller rectangles. This setup is very
similar to what you see in the Lagrange’s theorem. In fact, we are going to find some
substantially smaller subgroup H of the whole group G, and study its group structure intensely.
Then we will study the structure of the cosets intensely.
A metaphor I like to use is to treat the entire group as a passage, and the individual symmetries
can then be regarded as words of a passage. Then treat the cosets, including the subgroup
H, as paragraphs. So what we do is to analyse a central paragraph intensely, then see the
megastructure of the passage formed by the paragraphs to analyse the entire passage.
Then we will gain more insight into the passage, or the entire group.
To talk about the structure of the cosets, mathematicians require that they form a group.
It turns out that these cosets form a group when H is a normal subgroup, a concept that
we have derived together in the last video. For our purposes, these cosets form a group
when we can define some consistent operations on the cosets.
A natural definition of operations on cosets is that if a symmetry sigma is in this yellow
coset, another symmetry tau is in this blue coset, and the symmetry sigma tau lies in
the red coset, then we want to define yellow coset dot blue coset to equal red coset. Again,
notice that this is not normal multiplication in the sense that we do not necessarily have
blue coset dot yellow coset to be the red coset. In other words, order matters, like
normal symmetries.
For a general subgroup H, if we pick sigma prime from the same yellow coset, and tau
prime from the same blue coset instead, sigma prime tau prime might end up elsewhere. Of
course, we want it to lie in the red coset for consistency, so we need to impose some
conditions on the subgroup H. These conditions turn out to be that H needs to be normal.
The resulting group of cosets is called the quotient group, denoted G forward slash H,
and read as G mod H. This is a more abstract concept, because it is not a group of symmetries
anymore. It is a group of sets, which themselves contain a lot of symmetries.
But wouldn’t it be nice if we can still kind of treat these sets like symmetries?
This yellow coset acts very much like sigma itself in terms of operations of these cosets
within the quotient group, and this blue coset again acts very much like tau itself. So maybe
we can just call this yellow coset sigma, and this blue coset tau. Actually mathematicians
do that, just that we stick an H at the end to remind ourselves that it is not just a
symmetry, but a coset. This notation actually comes from our definition of a coset. For
example, this yellow coset is just a collection of symmetries of the form sigma h, where little
h is in the subgroup capital H. This definition was introduced in the context of stabilisers
in Chapter 3 of this video series. But there is something special for quotient groups.
You can choose anything inside the yellow coset in place of sigma, and it does not affect
the operation of the cosets within the quotient group. Same for tau.
The quotient group is the most popular name for this concept. However, there is another
name, called the factor group. However, I myself like to think of it as a group of remainders.
You probably just need to learn one name and be stuck with it, but knowing why there are
different names helps you build better intuitions on what a quotient group really is.
It is not very difficult to make sense of why quotient groups are named this way. We
basically divide the group into a collection of cosets, so the act of division is definitely
related to quotients here. A more direct way to see this is by Lagrange’s theorem. Here,
the number of cosets is just the size of this quotient group by definition. By simple rearrangement,
we can see that the size of the quotient group is really the quotient when the size of the
whole group G is divided by the size of the subgroup H.
Another perspective starts off with this yellow coset containing the symmetry sigma. As said
before, the coset contains all elements of the form sigma h, where h is in the subgroup
H. Here, sigma looks like a factor of the whole coset because the notation here looks
like normal multiplication. So the set of cosets can be seen as just a group of these
factors.
What about thinking of this as a remainder group? This intuition is particularly useful
when we are dealing with rings, a specific type of groups. The rings we consider will
only be the set of integers, Z, and the set of real numbers, R, as they can be visualised
as symmetries. More specifically, each integer or real number can be thought of as translational
symmetries of the number line. For example, the integer -2 can be thought of as the translational
symmetry to the left by 2 units. And the real number pi can be thought of as the translational
symmetry to the right by pi units. We now focus only on integers, where we can talk
about remainders.
The notations for group operations for this group are a bit tricky. In the previous chapter,
we say that if there is a symmetry corresponding to g then h, where g and h are both symmetries,
we call this symmetry hg, simply concatenating the symmetries. However, if we are considering
the set of integers as translational symmetries, simply concatenating does not make sense.
For example, if we do -3 then 2, then simply concatenating the symmetries looks like we
get the result to be -6. This is obviously not true. Translating to the left by 3 units
then translating to the right by 2 units is equal to a translation to the left by 1 unit,
i.e. -1, and not -6. An obvious patch to this notation would be to attach an addition sign
between the two symmetries. And this will then make sense in our example. But this is
the only time we don’t simply concatenate symmetries.
Another notation that we need to talk about is undoing symmetries. We denote this by g
to the power -1. Again, if we apply this to this situation, we get some nonsense like
2 to the power -1, which isn’t even an integer. The patch here would be to just stick the
negative sign in front of g. So if we consider the conjugation of h by g, it should change
to g plus h plus negative g, because instead of concatenation, we add plus sign between
these symmetries, and the inverse symbol is replaced by the minus sign. By our usual arithmetic,
we have this to be simply h itself.
You can also see this more intuitively by our conjugation intuition introduced in Chapter
4 of the video series. Conjugation means the symmetry h, viewed in the perspective stated
by the symmetry g. In whatever translated perspective, it is going to be the same translation
by h units anyway. All this is to say conjugating h by anything is still h itself. A natural
consequence of this is that for any subgroup H, conjugating by any symmetry g, would be
equal to the subgroup H itself, which means that all subgroups of Z are normal.
For Z, we can actually write down the form of the subgroups. We usually denote these
subgroups as nZ, which means the set of all multiples of n. Since nZ is a subgroup of
Z, it is also normal, which means we can consider the quotient group Z mod nZ.
So consider this diagrammatic representation of Z. Let’s consider a subgroup of Z, which
is 12Z. We just pick n in the previous discussion to be 12. Now 1 is outside this subgroup because
it is not a multiple of 12. So this is why we can then construct a coset with 1 in it.
We call this coset 1+12Z, because one of the elements is 1, and the subgroup in consideration
is 12Z. And again, instead of simply concatenating the symmetry 1, and the subgroup 12Z, we add
an addition sign here to avoid confusion.
Similarly, 2 is not included in 12Z or 1+12Z, so we can then form another coset, which is
similarly named as 2+12Z. Very similarly, we can construct all the other cosets of 12Z.
And we can cover the whole set of integers. For example, we want to work out where -15
lie. Since it is equal to 9 plus negative 24, and negative 24 is a multiple of 12, it
must be in this coset 9+12Z. In fact, this 9 here is just the remainder when -15 is divided
by 12. So this Z mod 12Z can really be viewed as just a group of remainders when divided
by 12. As a side note, this is one of the formal definitions of operations in modular
arithmetic, at least for addition.
So now you have seen how the quotient group can be thought of in different ways, which
hopefully builds up better intuition for this crucial concept. The next video will be about
group homomorphisms, which gives paramount importance to quotient groups.
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