1. Sporadic and Related groups Lecture 15 Introduction to Fusion.

2. Atlas notations for groups Given any finite group, a minimal normal subgroup must be characteristically simple – (fixed by all automorphisms) – which is the direct product of isomorphic simple groups. We denote the direct product of n copies of a group G by G n.

3. Abelian characteristically simple groups. The direct product of cyclic groups of order p. We denote this by p n, denoting the direct product of n cyclic groups of order p. Examples 3 5, 7 3, 107 44

4. Non-abelian characteristically simple groups These are denoted by their name, and in this work it is quite rare that there is more than one of them. Examples M 11, L 3 (7), U 5 (2), HS, Co 2, M

5. Then we use a normal series If A and B are groups, then A.B is any group with normal subgroup A and quotient group B. So we can have groups like 2 8.3 3 or 2 11.M 24 If A, B and C are groups, then we can use the notation A.B.C. This is “associative” since such a group has both a normal subgroup A with quotient B.C and a normal subgroup A.B with quotient C.

6. Sometimes we want a bit more detail So we say p 1+n is an extraspecial group, with centre of order (precisely) p, quotient group p n. In the case of odd p, it is always the case that all elements have order p. This is more specific than p.p n since not all groups of this form are extraspecial.

7. Other groups S 3 is really 3.2 but S 3 gives more structure information. Similarly S 4 is 2 2.3.2 but I prefer S 4. S n we might call A n.2 but surely S n is a better name. If a group is the direct product, I prefer to say so. Hence 2 2 2 11 2 22 (S 3 x M 24 ) is a perfectly good group.

8. Definition. Given any finite group G, H is a local subgroup of G for the prime p if it is the normalizer in G of a non- trivial p-group. Some local subgroups are contained in others, of course, and as often we are only interested in the maximal local subgroups, we might as well take only the normalizers of elementary abelian subgroups.

9. One idea in the classification Since a simple group does not have any non-trivial normal subgroups, the smallest unknown simple group (if there is one) has all its proper subgroups known, and in particular the composition factors of the local subgroups must be known. Usually it is the prime 2 that we are interested in.

10. What happens in L n (2) A good example of the sort of thing that happens is obtained by looking at the 2-local subgroups of L n (2) – the set of all n x n matrices over the field of 2 elements. A Sylow-2 subgroup can be taken as the upper- triangular matrices.

11. Sylow-2 subgroup of L 8 (2) 1 * * * * * * * 0 1 * * * * * * 0 0 1 * * * * * 0 0 0 1 * * * * 0 0 0 0 1 * * * 0 0 0 0 0 1 * * 0 0 0 0 0 0 1 * 0 0 0 0 0 0 0 1

12. A 2 7 normalizer 1 * * * * * * * 0 X X X X X X X

13. A 2 12 normalizer Z Z * * * * * * 0 0 Y Y Y Y Y Y

14. Their joint normalizer 1 * * * * * * * 0 1 * * * * * * 0 0 Y Y Y Y Y Y

15. If you didn't know about L 8 (2) You could construct the groups 2 7 L 7 (2) and 2 12 (S 3 xL 6 (2)) and then be amazed that the subgroup 2 13 (2xL 6 (2)) in the two groups were isomorphic. They need to be for L 8 (2) to exist! The idea of local analysis is to find all the cases where this sort of thing happens

16. Of course you need to find the group Given the fusion system for L 8 (2), we would need to know that the only group that contained it was indeed L 8 (2), and a fair chunk of the size of the classification is proving a whole series of identifications of this sort.

17. Just for the record... The Sylow-2 subgroup of L 5 (2) – a group of order 2 10 – is also the Sylow-2 subgroup of M 24 and the Held sporadic group. Nevertheless the fusion systems in these three cases are totally different. Moral... there are not many p-groups where lots of subgroups have lots of automorphisms. If you find one, you've got to make good use of it!

18. Glauberman's Z* theorem Another “just for the record” statement. In any finite group G, if an involution is not conjugate to any involution that commutes with it, then it is in the centre of G modulo the greatest normal odd-order (solvable) group. In particular, in a simple group, every involution must commute with a conjugate.

19. Application – 2.A 8 Suppose we ask – is there a simple group G with an involution centralizer which is the double cover of A 8 ? 2.A 8 has only two classes of involution – the centre and the double cover of (12)(34)(56)(78), since (12)(34) squares, in the double cover, to the central element so has order 4. Hence, by Z*, the central involution must be conjugate in G to all the others.

20. 2A 8 continued. In 2A 8 the other involutions are all conjugate, and centralize 2 4 L 2 (7). Hence the normalizer of the 2 4 in G must be transitive on the 15 involutions and the stabilizer of one is 2 4 L 2 (7). Hence it must have order 15.168=2520 and it is not too hard to see that this must be A 7.

21. Typical local problem Involution centralizer 2.A 8 2 4 normalizer 2 4 A 7 The two 2 4 L 3 (2) subgroups are isomorphic. Yes, there is a group like that. The McLaughlin group

22. Another example The extra-special group 2 1+8 has automorphism group 2 8.O 8 (2).2. By the way, there is no subgroup O 8 (2) in there. It is not a split extension. Since both 2 8 and O 8 (2) have double covers, there are actually several different groups 2 1+8 O 8 (2), but I am talking about a particular one of them.

23. In this 2 1+8 O 8 (2) We can take the central 2, and an elementary abelian 2 4 from the 2 1+8, and a 2 6 A 8 from the O 8 (2) and we get (perhaps rather surprisingly) an elementary abelian 2 11 with 2 4 A 8 on top [where the 2 4 is the rest of the 2 1+8 ]. Of course :) this is also a subgroup of the split extension 2 11.M 24 so we can ask whether they fit together.

24. And they do. In Co 1 Involution centralizer 2 1+8 O 8 (2) 2 11 normalizer 2 11.M 24 2 1+4.[2 4 ].2 6.A 8 = 2 11.[2 4 ].A 8 These two groups actually are isomorphic! And this occurs only in the Conway group Co 1

25. What about the monster. 2 1+24 Co 1 is the involution centralizer If we take an M 24 -involution in the 2 24 we get a group 2 2.2 23.2 11.M 24 and the fact that half of this (2 2.2 22.2 11.M 24 ) has automorphisms permuting the three central involutions is the amazing fact. 2 2.2 11.2 22.(S 3 x M 24 ) is a 2 2 normaliser in the monster, so it is not so amazing, I suppose.

26. Fusion The idea of fusion is to take a p-subgroup P of some group G, and ask, for every pair of subgroups of P, whether they are conjugate (or “fuse”) in G. Usually we take a Sylow-p subgroup of G, as this is the maximal case and there are extra results (basically Sylow's theorems) to help us.

27. Well... not quite Actually the idea of fusion is to do this the other way round. Take a p-group P and ask, for every pair of isomorphisms between pairs of subgroups of P, whether the is an element of G which effects precisely that isomorphism (as a map) by conjugation.

28. Alperin's Theorem A fundamental result says that fusion is locally generated. That means that if you take all the isomorphisms that you get looking at the normalizers of elementary abelian groups, and then compose them (A->B and B- >C gives a map A->C) you get all the things that conjugation can do to p-groups.

29. Alperin's theorem is fairly elementary Given a Sylow subgroup P of a group G, and a conjugacy class of p-subgroups Q in G, Sylow's theorems tell us that a Sylow subgroup of the normalizer N G (Q) is in a Sylow p subgroup of G, and as all are conjugate, that a conjugate is actually in P. Hence there is a conjugate of Q such P contains a Sylow subgroup of the normalizer of Q.

30. The real game, then Given a p-group P and a normal subgroup Q (we might as well shrink P if necessary so that Q is normal), what can the normalizer of Q be with Sylow subgroup P? Of course if there is a normal p' subgroup (often called the “core”), this will make no difference so we might as well quotient that out. With that proviso, there can only be finitely many answers.

31. Another statement of Alperin's theorem If you know the answer to all these local questions, you know precisely which isomorphisms between subgroups of P are effected by conjugation in G.

32. What are the axioms of fusion? There are quite a few conditions that need to be satisfied by a fusion system on a p-group P before it has any chance of being the fusion system of some group G. But the fact remains that all these necessary conditions are not sufficient – the are some interesting fusion systems that look fine but do not come from a group. Nevertheless it is one way of classifying.

33. Chris Parker's (no relation) fusion patterns on p 1+2 If you take P as the group p 1+2, for large p the only interesting fusion pattern is the one that occurs in L 3 (p), but for small p there are others. The idea is that you want a group p 1+2.X which fixes a subgroup of order p 2 that can be extended to larger groups in two ways, one suitable for normalizing the p 2 and the other for normalizing the p 1+2.

34. His 13-local characterization of the Monster If you take P as the group 13 1+2, there is an exceptional fusion pattern where the normalizer is 13 1+2.(3x4S 4 ) and the 13 2 normalizer is 4.L 2 (13).2. Chris proves (using classification) that this fusion pattern only occurs in the monster, and dreams that perhaps there might be a direct proof.

35. All this was done by hand. I would very much like to see a computer-automated way of doing this sort of fusion analysis. It is very error-prone. Actually Chris Parker is pretty keen on that too.

36. Of course we'd like to handle the monster So we would need to be able to handle the Sylow-2 subgroup of the monster – a group of order 2 46. And then we'd need to be able to hold automorphisms of subgroups of this. I just do not know how to do that, although I also feel that it should be possible.