1. Sporadic and Related groups Lecture 16 The Monster.

2. It's BIG Order is roughly 808. 10 51 Smallest representation over the complex numbers is 196883 (over any field 196882). Smallest permutation representation on nearly 10 20 points. Divisible by 15 primes – largest 71.

3. It is a 6+ transposition group It has a class of involution (2A) where every pair from this class has product order 6 or less, and if the order is 6, the cube is also in the class. Many sporadic groups have this property, usually because they are involved in the monster.

4. It's subgroup structure is extraordinary. Every subgroup normalizes a “small” subgroup – order less than ten thousand. All its big subgroups are local – mainly element centralizers..

5. All it acts on is itself! A very arrogant group. There is no big simple subgroup available to fix anything. Co 1, by contrast, contains a big simple subgroup Co 2 which is the stabilizer in the Leech Lattice of a shortest vector. But the monster... all it can permute is its own elements. So we have no choice but to look at the elements.

6. Two classes of involution. 2A is centralized by the double cover of the “baby monster”. This is the 6+ transposition class, and the centralizer is a 4-transposition group (all products of order at most 4). 2B has centralizer 2 1+24 Co 1

7. Elements of order 3 Three classes, all with sporadic centralizers. 3A has normalizer 3.Fi 24.2 3B has normalizer 3 1+12.2Suz.2 3C has normalizer S 3 x Th

8. Elements of order 5 Two classes, both with sporadic normalizers One has normalizer (D 10 x HN).2 The other has normalizer 5 1+6.4.J2.2

9. Other normalizers (7.3 x He).2 (11.5 x M 12 ).2

10. Lots of sporadic groups in the monster Indeed 20 of the 26 sporadic groups are sub-quotients of the monster. The other six are called the “pariah” groups. So if we want to understand sporadic groups, we must clearly start by understanding the monster. Unfortunately, I don't.

11. Patterns of subgroups across primes For every prime p where p+1 divides 24, there is a subgroup corresponding to the appropriate centralizer in the Conway Group 2.Co 1 2.Co 1 Monster 2.Co 1 2 1+24.Co 1 3.(2.Suz.2)3 1+12.2.Suz.2 5.(4.J2.2)5 1+6.4.J2.2 7.(3x2.S 7 )7 1+4.3x2.S 7 13.(3x4.S 4 )13 1+2.3x4.S 4

12. These are the fixed-point-free elements. The elements in the previous list are (in the Conway group) precisely the elements that fix no vector at all. And in the monster there are alternating groups up to A 12 whose single-cycle elements of prime order are exactly those in the list. This is the A 12 of the Y555 diagram.

13. A bit like matrix groups mod p These centralizers with large extraspecial parts are much like you'd expect in matrix groups with that characteristic. Hence the monster has characteristic 2,3,5,7 and 13 all at the same time!

14. Other prime centralizers All more-or-less simple groups 2.B 3.Fi 24 5 x HN 7 x He 11 x M 12 13 x L 3 (3) 17 x L 3 (2) 19 x A 5

15. And the other one 3 x Th – the Thompson sporadic group – that does not fit into the pattern. Moonshine also picks this element out as unusual.

16. Moonshine - I If one considers differentiable complex functions f(z) on the upper half-plane (x+yi where y > 0) and such that f( [pz+q] / [rz+s] ) = f(z) whenever ps-qr = 1, it turns out that they are all rational functions of a particular function j(z). Although any function a.j+b / c.j+d would do, there is a nice one to pick...

17. Moonshine - II Taking q = e 2πiz, clearly q(z) = q(z+a) for any integer a, so we have some of the group of 2 x 2 matrices already. In fact we usually take j(z) = 1/q + 744 + 196884.q + 21493760.q 2... McKay has a t-shirt with his most famous theorem which states that 196884 = 196883 + 1.

18. Moonshine III Thompson then noticed that 21493760 = 1 + 196883 + 21296876 where the last number is the next smallest degree for a complex representation of the monster. These are huge numbers. This could not be a coincidence.

19. Moonshine IV Since then, these numerological coincidences have been extended and extended to work with all the elements of the monster, and other modular forms besides j. They have been proved to hold indefinitely Borcherds has constructed a vertex algebra with the monster as its automorphism group.

20. I'm sorry This is clearly a very important branch of mathematics (Borcherds got the Fields medal for his part) but I just do not know what a vertex algebra is, nor really what a modular form is.

21. Construction Conway's proof that the group exists was by no means the first, but is perhaps the simplest found so far... It starts with a loop of order 2 13. It is the double cover of the Golay Code (octads, dodecads, 16-sets etc.) but it is not a group. It is actually a Moufang Loop.

22. For each Golay code word, you take two elements of the loop, x and -x (and for the zero word, we take 1 and -1) You then define... x.x = -1 or 1 depending on whether x is a dodecad or not. xy = -yx or yx depending on whether there are an odd number of dodecads in the set x,y,x+y. x(yz)=-(xy)z or (xy)z depending on whether the set x,y,z,x+y,x+z,y+z,x+y+z has an odd number of dodecads.

23. This defines a Moufang loop! The Golay code has the property that every four-space contains an even number of dodecads, and this is precisely the condition that the above definition actually works. This is a start.

24. Latin Squares Take three sets A, B and C of the same size n. Take n 2 triples (a,b,c) one from each set, and such that if you pick an element from two distinct sets there is precisely one triple containing them both. We usually make an n x n array of elements of C (so that A and B are the set of rows and columns respectively). It then must have every element of C once in each row and column.

25. When are two Latin squares the same? When there are three bijections A-A', B-B', C-C' such that the triples go to their dashed triples.

26. So what is a Latin Square “automorphism” It is a triple of maps α, β and γ on the sets A, B and C such that if (a,b,c) is a triple, then so is (aα,bβ,cγ). This is called an isotopy. And actually, while we are about it, we could permute the sets A, B and C bodily. For example swapping rows and columns is a natural thing to do. I call them tri-topies, but I don't think anyone else does.

27. Moufang Loop A Moufang loop is a Latin square where the isotopies are transitive on each of the sets A, B and C. The Golay construction does construct a Moufang Loop.

28. What is the tri-topy group It is a group of the form 2 2 2 11 2 22 (M 24 xS 3 ) Actually this is the group of “standard tri-topies” where the size (octad or 16-set) must also be preserved. Heard that structure before? Surely there can't be two important groups like that? Actually there are!

29. Again I cannot get the details right. From this group, and its connection with the Leech Lattice, one can construct the monster in 196883 dimensions, where the group 2 1+24 Co 1 acts on 98280 + 98304 + 299.

30. The Conway Representation 196883 = 98280 + 98304 + 299 where 98280 is the monomial action on the elements of 2 1+24 corresponding to the short (norm 4) vectors 98304 = 24 x 4096 – the tensor product of the natural 24-dimensional representation of Co 1 with the 4096 dimensional representation of 2 1+24.Co 1 299 is symmetric square of the 24.

31. Subtleties abound For example take that 24 x 4096. The group 2 1+24 has just one faithful representation. It's degree is 4096, and it can be written over the integers. Take the set of integral matrices of determinant 1 that normalize this set of matrices. It is of the form 2 1+24.O 24 (2)

32. Co 1 is in O 24 (2) So one can take the subset of these matrices to get a group of the form 2 1+24.Co 1. (Note – it contains a matrix -1 representing the central 1). Is this isomorphic to a subgroup of the monster. No. It is the wrong one.

33. 2.Co 1 The double cover of the Conway group is the automorphism group of the Leech Lattice. This also contains a matrix -1. If we tensor these matrices together, but only tensor matrices corresponding to the same element of Co 1 we get a 24 x 4096 = 98304 dimensional representation of the correct 2 1+24.Co 1.

34. Footballs If you take triples (a,b,c) of elements of the 2A class, and consider the action of the braid group [so, for example, (a,b,c) → (b,a b,c)] There is a natural way of associating a plane figure with this, and because the the monster is a 6 transposition group, this plane figure has all its faces 6- ogons or smaller, and so is a bit like a football (20 hexagons and 12 pentagons) although when we actually make them they are less pretty – more like a potato!

35. Representations over finite fields If the characteristic is odd, the smallest faithful representation of the subgroup 2 1+24. Co 1 is 98304, and a little consideration of character values forces the dimension to be at least 196883 over any field. In characteristic 2, the subgroup 3 1+12.2.Suz.2 similarly forces the dimension to be at least 196882, and indeed the representation 196883 is 1+196882 in characteristic 2.

36. Reminder of Y555 And I must mention again that the Monster has a very concise presentation as the Y444 diagram... And the Y555 diagram is the even more astonishing group (M x M).2

37. What is the purpose of the monster? Wheras J 1 may not have a purpose, I feel that the monster is such a pretty structure that it just must have a purpose. Is it vertex algebras, as Borcherds suggests? Maybe, but somehow it seems too big. I suppose I live in hopes that one day we will find a new area of mathematics which is elementary and where the monster arises in an elementary way.

38. But I am not yet satisfied. The hardest thing in life to find is a purpose. In My Humble Opinion, the Monster has yet to find its purpose. One reason I gave this course is to encourage some young people to strive to find it.