There's a thing called the Monster Group
which is a beautiful, very large,
symmetrical thing. - We're gonna have a
look at what's called the monster group.
(Brady: That's a cool name isn't it?) - It's a really really cool name, yeah.
There's never been any kind of
explanation of why it's there and it's
obviously not there just by coincidence.
Okay, so groups are algebraic structures
that arise naturally in the study of
symmetry. Now that can be symmetry in a
sort of a vast range of different
contexts. Let's take a geometric- familiar
sort of geometric shapes, so let's take an
equilateral triangle, okay and let's
label the vertices 1, 2 & 3. I want to
consider the symmetries of an
equilateral triangle, so what what I'm
really thinking about here are rotations
and reflections. Ok, so these are natural
symmetries of this triangle. For example,
we can consider certain rotations about
that centre point.
Okay now let's let's write down this
these collections of symmetries, let's
call this collection G. If I sort of
think of putting a pin in the paper at
that centre point and then moving the
triangle around; well I can rotate
through certain angles and the triangle
will move back onto itself. Ok I have to
be careful which angle I choose here but-
sort of, it's not too difficult to see
that I could say rotate through 120
degrees - let's write that R120 - that
would have the effect of permuting those
vertices, the corners of the triangle. And
I could also go through say 240 degrees,
and they're really the only two
non-trivial rotations that I can do
which will move that triangle back onto
itself. - (Because 360 doesn't count?)
Well 360 does count actually and that's
really an important one so let's
actually include that one, we're gonna
need there to get this idea of a group,
so let's call that R0. That means do
nothing, okay, so that's the do nothing
symmetry that's really really important
in the concept of a group. Ok so we've
got our rotations now what else have we
got? Let's consider sort of mirror
symmetry. Okay so we could think about
putting a mirror down that line and that
would also give us a symmetry of the of
the triangle. Ok, it would fix the top
vertex and it would swap these two bottom corners.
Ok so let's call that a, so let's include
that over here, and then similarly we
have b and c. Ok so we've got three lines
of reflectional symmetry of this
triangle as well; ok so a, b and c. So what
I've written down here, these are- it's not
too difficult to check that this is this
is a complete set of symmetries of the
equilateral triangle, we've got six of them
in total.
This is what's called the symmetry group
of the equilateral triangle. The key
concept with a group is it's more than
just a collection of symmetries. There's
a rule or there's an operation which
allows us to combine two symmetries
together. So if we take two symmetries we
could- we could take the first one and
combine it with the second and that will
give us a third symmetry of the triangle. - (How does that work?)
Okay let me give you an example. So let's
say consider- let's draw this triangle
out again a little bit smaller so I've got
our 1, 2, 3.
Okay let's actually see what happens to
the vertices when we apply these symmetries;
so let's say apply- let's do a, okay. Okay
so a, we defined it to be this reflection
in this vertical line of symmetry here.
So what does that do to the do to the
vertices? Well it swaps the bottom two,
okay, so it looks like that. That's what
would that will look like after we've
applied that reflectional symmetry. And
let's do another one, let's now follow
that up with say our anti-clockwise
rotation through 120
degrees. Okay now what does that give us?
Okay so what does that mean? So this
becomes 2, this is 1, and this is 3. Ok so
that's what we've got. Now we could have
gone straight from the initial triangle
to this end triangle with just one
operation, what would that operation be?
Well what have we done here? We've fixed
the bottom right vertex 3, we've
swapped these two. Well that corresponds
to what we called c. Okay, that's that
reflection so we've got c. Okay so what
we could do we could write that down
sort of mathematically, R120 composed
with a equals c. What we're trying to do
here is we're trying to define a sort of
like a multiplication on this set. Okay
so it's like a- like some sort of
natural operation which allows us to
combine these two symmetries together
and get another one. That's really, that's
really what's going on and that's really
the key idea of a group. It's a
collection of objects with some
operation defined on it which satisfies
these sort of properties, this allows-
this allows us to define an
abstract notion of a group. So this
object here this is the set, this is the
collection of symmetries. Ok, now the
group, strictly speaking, is this
collection of symmetries together with
this rule for combining symmetries
together. We can
perhaps try and understand groups by
breaking them up into smaller pieces. The
analogy to keep in mind here is the idea
of factorising a positive number into
prime factors, okay, so this should be
familiar to everybody. Okay. now we can do
something like that for a group; it's a
little bit more subtle, it's a little bit
more complicated. - Let's say 60 is 2 times
2 times 3 times 5; and those numbers I
just said were all prime numbers. In
group theory there's a sort of
similar-sounding theorem that every
group is composed of a number of prime
groups, except the term is simple groups.
They're not actually terribly simple.
Let's just consider, let me call R here,
let's just consider the rotations, ok, so
let's forget about the reflections for a
second. Ok, now as I said you can talk
about that you have this notion of an
abstract group. Now R itself is actually
a group, ok? It's not the symmetry group
of the equilateral triangle it's the
rotation- it's the group of rotational
symmetries of the equilateral triangle. - (Like a subset?)
It's like a subset but it's what we call
a subgroup. And you see it's not just
any old subset, the key property is that
it- first of all it has the do-nothing
element, that's really important, and the
other key property we're looking for here
is that it's- this set is closed under
this operation that we defined. In other
words if I take two rotations in here
and combine them then I get another
element of that particular set. I can't
combine two rotations together and then
suddenly get a reflection, that's not
going to happen, we can easily check that.
Ok so that's the key property that this
is actually a subgroup. Ok, so that's R
and let's define S. To take- ok, we're
gonna have R0 in there, we're going to
need R0 in any of our- in any of our
subgroups that's this key do-nothing element, and
let's just take a. Now this has got three
elements and this has got two elements,
and again this set S is a subgroup
because if you do- if you compose a with
itself that's a reflection done twice
but that gets you R0, that's the do
nothing- that's the do nothing operation. - (So to be a subgroup it always has)
(to hav- to leave you a method to get back home?)
Yeah that's right, so it must be- yeah that's
right. So I mean it must contain the do
nothing element and it must have the
property that if I take any two elements
in that smaller subset, if I combine them,
then I stay inside that set, that's the
key thing we're looking for here. It's
not too hard to see that we actually can
write G as R times S as a sort of a
factorisation. What this means is,
what I've written down here, is
that every element
of this of this symmetry group G is a
product of something in R with
something in S. Okay so that's why we
might say that this is a factorisation
of G into two smaller pieces. Okay so
these things are what are called simple
groups. So the idea would be then to try
and understand all groups, or in
other words to try to understand all
possible types of symmetry. Can we give a
complete description of what all these
basic building blocks, all these atoms of
symmetry, what they actually look like?
Okay so this has been sort of a
fundamental problem in group theory and in
algebra for more than a hundred years.
What's remarkable is that it is actually
possible to do this, okay, so there's this
there's a theorem called the
Classification of Finite Simple Groups
which was announced around 1980 which
provides us with this sort of periodic table
of symmetry. - (I can see it behind you!)
That's right it's up on the
wall there, that's right. Okay so what- what is
this theorem then? So it's it's um it's
this incredible amazing result
that describes what all the finite
simple groups look like. So this is a
really very unique theorem in all of
mathematics, I mean it was a huge
international collaborative effort to
prove this theorem over many many decades.
Its proof runs to between 10,000 and
100,000 pages and it was done by three
or four hundred mathematicians, I don't
think anybody's read through the whole
proof. - It's sort of an incredible
incredible achievement really,
certainly one of the highlights of 20th
century mathematics. - There were some
mistakes in it; in anything as long as
that they are bound to be some mistakes
so it's not terribly worrying that there are
some mistakes in it. - What we're looking
to do is we're just trying to sort of
identify various families of simple
groups. So for example one family will be
all of the groups which have size 'a
prime number'. Okay, and we don't know
what all the primes look like of course
there's infinitely many of them but but
we can give a descriptive definition of
this particular family. And it turns out
there's various other infinite families
of simple groups which are a little more
difficult to define so I won't do that
here, but there's various other infinite
families that arise, they don't all have
to have size 'a prime number'. Okay so it's
a little bit more complicated than sort
of the factorisation of integers. What
what's what's really interesting about
this theorem is that we have these
various infinite families that
come up but there were some exceptions.
Okay, so it turns out it's sort
of a remarkable fact that there are 20-
exactly 26 finite simple groups which do
not belong to any of these infinite
families. - (They're like black sheep?)
They're like black sheep, yeah, they they refuse to belong to any of
these- any of these sort of well-known
well behaved groups. - (And there are 26 of them?)
Exactly 26 of them yeah. So these are what are
called the sporadic simple groups. Okay,
so now they have a long long history so
the first five of them were discovered
in the 1860s - 1870s; and then the sixth
one wasn't discovered until about 100
years later in the 1960s. And then over
the subsequent 10-15 years a further 20
of these sporadic groups were discovered;
and this all came out of this intensive
effort going into the proof of the
classification theorem, this enormous
enormous project.
The result is probably true and I don't understand it.
It's absolutely amazing, incredible.
It's not incredible that I don't
understand it,
I mean it's the the fact that the theorem
is true,
apparently, and it's- we don't know why
it's true. - And this is where the monster
comes in, because the monster group is
the largest of these sporadic simple groups.
(Largest - what do you mean by largest? Has the most members or?)
Yeah that's right, so as you say
a group is a collection of
symmetries or a collection of elements
with respect to some operation and
largest just means, you know, in terms of
the number of the number of members of
this of this particular group. - (So the)
(monster is- it's not unique, it's just the)
(biggest of this- of a little group of outliers?)
Well yeah I mean it's unique- it's unique
in the sense that there's only one- there's only
one monster okay? You can say it that way,
there's only one monster but its its
unique defining property might be to say
that it's a simple group which is one of
these 26 sporadics and has by
far the largest order. So it's sort of like
the Big Daddy of all these sporadic
groups. And in fact quite a few of the
other 26 sporadic- the other 25 sporadic
sort of live inside the monster in some
sense. Not all of them do but but many of
them can be found in some way inside the
monster so it's like a bit like a bit
a mother group for these sporadic simple
groups. - (How big are we talking here?)
We're talking absolutely enormous,
actually, so let me- let me see if I can
write this down, so what are we looking
at? So so we use M to denote the monster
okay, and this denotes the size, okay and
what is this thing? So, okay, let's write
it out in full- in full detail. Okay so
it's a big number so I hope we've got a
bit of time for this. Sadly I'm going to run
out of space, let me continue down here
still carrying on so- 8 and then-
nine zeroes on the end. So this is
approximately 8 times 10 to the 53. Okay,
that's the size of the monster.
(That's how many symmetries it contains?) - That's right so I
mean, we sort of talked about a group as
being sort of the symmetry group of some
object. So the monster can be defined in
this way as well, it is the symmetry
group of some structure. - (So at the start)
(of our little talk)
(you showed me an equilateral triangle)
(and we looked at its symmetries;)
(what does the object for which the)
(monster group represents symmetries look like?
Okay well this is pretty difficult,
this is pretty difficult, it's quite a
difficult thing to try and explain. - I think
of them as Christmas tree ornaments. You
can hang- you know, sometimes you see a
Christmas tree ornament which has a
number of spikes coming out of it and
it's covered all over with silvery paint
or paper or something and it turns
around and then from some points of view
you see it has fivefold symmetry and
then, oh look, it's three-fold symmetry right
now and so on. - It is the symmetry group
of some structure, it's not as simple as
an equilateral triangle, we can't draw on
a piece of paper unfortunately. We can't
even think about it in three-dimensional
space. So it turns out the initial
construct- the the construction of the
monster relies on the existence of some
mathematical structure in high
dimensional space and this- and the
monster's the symmetry group of this
object. And the number of dimensions we're talking about here is
196,883.
So this is a very very
difficult thing to try and try and
picture in our minds.
Why 196883? That's just the
way it is. - (Like I know that's a lot of)
(dimensions, but it seems arbitrarily)
(small at the same time. It seems strange)
(that there's this cut-off number and)
(there's nothing else above it, like it seems such)
(an exact number.) - Yeah, I mean this is the
business with sporadic groups, I mean
this is why they're so fascinating. I
mean why are there only 26 of them? Why not 27?
Why not 28? Do they belong-
can we sort of fit them into some other
infinite family of groups? But it's it's
a sort of a bit of a mystery, but this is-
these are the numbers that come out
of this of this classification theorem.
It was a famous thing yeah for sure, I
mean the monster group, I mean simple
groups being these basic building blocks
of of symmetry, I mean these are these
are sort of at the centre of study in in in
group theory. And so the monster plays a
central role in that because of its
starring role as the largest of these
sporadics. - Sometimes I think of these
things as not like Christmas tree
ornaments but like gems. Most people like
jewellery, you know, because the light
sparkles and somehow there's a slight
prism effect so that you may see- you
know, even if it's illuminated with white
light you may say you had some greens
and blues and so on. They look nice. Well,
so do these things in higher
dimensional space except that I haven't
got 196,883
dimensional eyes so I'll
never see them. - But the actual study of
the monster group is quite a niche
subject, I mean people there are- there are a
small group of people who've really
dedicated their careers to really
understanding and probing the structure
of this group and so a lot a lot is
known about the monster; we know a
lot about its internal structure, about
its subgroups like we saw here in the
subgroups of of this symmetry group
of the equilateral triangle. We do know quite
a lot about it but there still a lot of
things to be to be fully understood. It's
a difficu- due to its size it's
it's a very difficult group to work with.
I would just like to know what it's all
about, you know why it's there. And every
now and then - I've often said, I've said
for 25 or 30 years - that the one thing
I'd really
like to know before I die is why the
monster group exists.
You have the monster here then the second largest one's
called the baby monster. - (Who discovered the monster?)
(Not someone called monster!)
No, not someone called monster, actually no. So
the baby monster and the monster are the
only ones I think which don't have that
have that property. - You haven't got
hope but probably you don't care either.
I care. I'd like to understand what the
hell's going on, if you forgive me for
expressing it like that.