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.
