Apart from the first two chapters of the video series, there is not a mention of group action
in the series, for a good reason, as we will see. Let’s revisit what we have said about
group action in the first video.
A group action is where a group of symmetries act on a set, usually of vertices, edges or
faces, to give the same set back, and we often used the example of a particular rotational
symmetry of the regular tetrahedron to explain the concept.
Let’s use another example of the group of symmetries of a regular octagon acting on
the set of vertices. If we denote r as the anticlockwise rotation by 45 degrees, then
r acting on 1 will give 2, acting on 2 will give 3 and so on. Note that the output, which
is the rightmost column is just a rearrangement of the input, which is the middle column.
This is not a coincidence and it comes straight from the definition of a symmetry, because
in this case, a symmetry preserves the octagon, and so the vertices as well, so acting any
symmetry on the vertices would just rearrange them.
This means, if we discard the octagon and leave only the vertices behind, then every
symmetry will correspond to some rearrangement of the vertices. However, not all rearrangement
might correspond to a symmetry. For example, this rearrangement of swapping the two vertices
will result in this octagon, and I don’t think we want to call it a symmetry of the
octagon. Still, it is more convenient to consider all these rearrangements, not just some of
them. This is more officially named as permutation. So now we have established that all symmetries
can correspond to a permutation of the vertices, but not necessarily the other way around.
The reason why we consider permutations is that in some sense, it is still a symmetry.
After all, if we discard the notion of distances, then a permutation like this one would preserve
the object just that the object in consideration is just the collection of the 8 vertices.
And in fact, all the core properties of symmetries as discussed in the first video would also
hold if you replace the word symmetry by permutation. Hence, the permutations would form a group.
And for some reason, it is confusingly named “symmetric group”, even though it is not
about symmetries.
So to summarise what we have so far, we have a group of symmetries of an octagon, acting
on the set of vertices. For every symmetry, say the horizontal reflection, we can always
discard the figure and just look at how the symmetry acts on the vertices itself, which
means all symmetries correspond to a permutation. This correspondence is more formally constructed
as a function, say phi. This function sends an element from the group of symmetries, to
something from the symmetric group of the set of vertices. It looks complicated, but
what this function does is simply sending any symmetry g, say a horizontal reflection
in this illustration, to the corresponding permutation, which we just abbreviate as g*.
However, this function is not just a normal function, it is specifically a homomorphism.
Why? Consider phi of h phi of g, which means that we are doing the permutation corresponding
to g first then do the permutation corresponding to h. This is a permutation corresponding
to the symmetry hg. So it is just phi of hg. So it fulfils this definition of homomorphism
that we discussed in the previous video in this series. In short, whenever you have an
action of a group of symmetries G on a set X, you will have a homomorphism from G to
the symmetric group, which is the collection of all permutations of elements in the set
X. Here, we just use the example where G is the group of symmetries of an octagon, and
the set X is the set of the 8 vertices. And this correspondence between an action and
a homomorphism will hold in general.
This correspondence between two concepts is very useful, because previously, to analyse
a group action, we only have the orbit-stabiliser theorem as a tool. But because we link it
to a homomorphism, we also have the tools described in the last video, i.e. the isomorphism
theorem. A very direct application of this change in perspective is the proof of Cayley’s
theorem. Before we move on, I just want to point out that everything from now on like
the set that G acts on can be more abstract than just vertices, but it is still very useful
to have this mental picture where the elements in the abstract set X are just these little
dots, and a group action just somehow rearranges these dots. With that said, let’s look at
what Cayley’s theorem says.
Cayley’s theorem states that every group G is isomorphic to a subgroup of the symmetric
group of G. Looking at the correspondence once again, it looks like the set X is the
group G itself, which is quite different from what we did, but the action we will be considering
is very simple. If we act g_1 on the symmetry g_2, then we get the symmetry corresponding
to doing g_2 first then g_1. So going back to the visualisation of group actions using
these colorful dots, each dot now represents a symmetry. So one dot would represent the
identity symmetry, and one dot would represent symmetry g, and there would also be its inverse
somewhere, and we also might have another symmetry h, and then we would have gh, and
so on. Because of the action we defined has this form, when we consider the action of
g on these symmetries, the identity would become g itself, because doing identity symmetry
first then g is the same as doing g altogether. Then g inverse would be sent to the identity
because undoing g then do g is the same as doing nothing, that is the identity. Then
the action of g on other symmetries would look something like this. So now, every single
symmetry, say h, corresponds to a permutation of these dots, which are symmetries themselves.
This looks like we have already proved Cayley's theorem, because we showed that every symmetry
will correspond to a permutation of these symmetries, but there is still one very subtle
thing to consider. Even though every symmetry will correspond to a permutation, it can happen
that two symmetries can correspond to the same permutation. In which case, the theorem
does not necessarily hold. Luckily, in this particular case, we can show that this cannot
happen. The trick is to see where the identity goes. If we are considering the permutation
corresponding to symmetry g, then the identity will be sent to symmetry g. If however we
are considering the action of h, then the identity would be sent to h instead. In other
words, the worries that we had were unnecessary, because different symmetries would correspond
to different permutations.
A more rigorous way to prove Cayley’s theorem rather than moving the dots around is to use
the isomorphism theorem on the corresponding homomorphism. You can show that this is a
set only including the identity. And as part of the isomorphism theorem, the image has
to be a subgroup of the symmetric group. Next, you can show further that this quotient group
is isomorphic to G, by constructing an isomorphism directly. This theorem provides a motivation
for studying symmetric groups in particular. If we can study all the subgroups of a symmetric
group, then we will have studied all the possible groups as well. This is why most courses will
continue to include symmetric groups as well.
However, even though symmetric groups would be a big topic in group theory, I would put
an end to this video series because most concepts in symmetric groups as listed here are very
algebraic. Even for something seemingly simple like the sign of permutation, which is literally
just whether it is positive or negative, the justification for such assignment is not intuitive.
So for this topic of symmetric groups at least, I don’t think I could bring some very unique
insights and intuition to YouTube, especially when a quick search reveals that a lot of
YouTube channels talking about symmetric groups are available.
But on the up side, the next videos would be first a summary of the entire video series,
including a very cool example to illustrate ALL the concepts covered in the video series,
which should be out in a few days, and then a second video on coronavirus, but this time
not about the SIR model, but a stochastic process known as branching process. Then I
will release yet another group theory video, but this time not “essence” anymore, because
it is an application of the orbit-stabiliser theorem. After that, I don’t have a very
solid plan now, so leave a comment for some topics for me to cover. Even if you don’t
have a request, leave a comment anyway.
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on. See you next time!