GRA Workshop 1

Last week was the introductory/instructional workshop for the Isaac Newton Institute’s six-month programme on Groups, Representations and Applications.

We were thrown in the deep end right at the start. The first two talks were on what were claimed to be two of the major themes of the programme, representations of reductive groups, and fusion systems. Neither of these are things that are within my comfort zone, and I felt considerably out of my depth for the first couple of days. Later things got better, with very nice talks by Colva Roney-Dougal on Aschbacher’s Theorem, Eamonn O’Brien on algorithms for matrix groups, and Martin Liebeck on subgroup structure of almost simple groups, to name just a few.

Anyway, after the first day, I felt a bit sorry for some of the beginning PhD students, and offered them a one-hour crash course on ordinary representation theory, a subject which in my view underlies much of what went on. (At St Andrews, for example, we have no module on representation theory on the books, though students can take it by independent study.)

It may be that these notes will be of wider interest, so I have uploaded them here. They should appear among my lecture notes.

A bit about fusion systems, which are one of the coming topics in group theory. This is my understanding, which might be quite wrong. In the Classification of Finite Simple Groups, it was often necessary to consider a p-local subgroup H of a finite simple group G (the normaliser of a non-trivial p-subgroup P of G), where p is prime (especially in the case p = 2). Now there may be additional conjugacies of subgroups of P not induced by elements of H. A fusion system is an axiomatic way of describing this, as a category whose objects are the subgroups of P and whose morphisms are conjugations induced by elements of G. Of course, once something like this is axiomatised, it is possible (and indeed happens, though it seems to be rare) that there exotic fusion systems not arising in the manner described. Now if we replace fusion arguments in subgroups of G by abstract fusion system arguments, some advantages appear. In particular, information about centralisers causes quite some difficulty in the proof of CFSG, and it is hoped that this can be avoided using fusion systems, since centralisers are invisible.

In addition, fusion systems appear in other parts of mathematics. Markus Linckelmann showed us how to associate a fusion system to a block of a modular group algebra, and Radha Kessar hinted that they also arise in a topological context.

Perhaps I have completely misunderstood, so please don’t take the above too seriously. If I have missed the point, perhaps the practitioners didn’t explain it clearly enough?

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The Frattini argument

The Frattini subgroup of a finite group G can be defined in two equivalent ways:

  • it is the intersection of all the maximal proper subgroups of G;
  • it is the set of all non-generators of G, that is, elements which can be dropped from any generating set (so g is a non-generator if, whenever G is generated by {g}∪X for some set X, then G is generated by X).

To see the equivalence, suppose first that g lies outside some maximal subgroup M. Then G is generated by M and g but not by M, so g is not a non-generator. Conversely, suppose that g is contained in every maximal subgroup, and let X be such that {g}∪X generates G. If X does not generate G, then it is contained in a maximal subgroup M; but g is also in M, so {g}∪X cannot generate G.

A little job I am doing at the moment requires me to remember and explain various facts about elementary group theory. One of these is:

Theorem The Frattini subgroup of a finite group is nilpotent.

I couldn’t remember the argument, so, following my standard practice, I went to bed last night thinking about it, and when I awoke I had the proof. It is, not so surprisingly, the Frattini argument.

My first permanent teaching job was at Bedford College, University of London (which now no longer exists, having fallen on hard times financially and been rescued by a merger with Royal Holloway College). I was there for a bit over a year in the mid-1970s. Paul Cohn was the head of department; he ran a study group which was reading a paper by Procesi about invariants of n-tuples of matrices. The participants were stuck on a certain point; I managed to notice that it was a ring-theoretic version of the Frattini argument. Paul was delighted that Procesi had adapted an argument by his compatriot Frattini.

It is an argument, not a theorem, and can be applied in a number of different contexts. But it appears in a pure form in the following theorem. (I discussed Sylow’s Theorem very recently here.)

Theorem Let G be a finite group, N a normal subgroup of G, and P a Sylow p-subgroup of N. Then G = NG(P).N.

To prove this, we have to take an arbitrary element g of G, and factorise it as hn, where h belongs to the normaliser of P and n belongs to n.

Consider Pg, the conjugate of P by g. This is contained in N (since N is a normal subgroup of G), and has the same order as P, so it is a Sylow p-subgroup of N. According to the second part of Sylow’s Theorem, it is conjugate to P in N; that is to say, there is an element n in N such that Pg = Pn. But then Pgn−1 = P, so h = gn−1 belongs to NG(P). Then g = hn as required.

Now we can prove the first theorem. Take N to be the Frattini subgroup of G: it is clearly normal in G. If P is any Sylow subgroup of N, then G = NG(PP.N. But because N consists of non-generators, we can delete its elements one by one and obtain G = NG(P). Thus P is a normal subgroup of G, and a fortiori of N. But a finite group is nilpotent if and only if all its Sylow subgroups are normal; so we are done.

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Happy new year

I hope the New Year is better than we deserve.

I will be spending most of the first six months in Cambridge, at a group theory programme at the Isaac Newton Institute. So hopefully I will have some time for catching up and tidying up; things have got in rather a mess, and it has been a couple of years since I had a holiday.

This also means that there might be a higher proportion of group theory in what I have to say.

Gull in Regents Park

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Sylow’s Theorem, from the book

The most important theorems of elementary group theory are those of Lagrange and Sylow. I want to describe here what I consider the most beautiful proof of the first part of Sylow’s Theorem, actually based on Sylow’s original proof. But first, some preliminaries.

Lagrange’s Theorem states that the order of a subgroup of the group G divides the order of G. This is closely connected with the theory of group actions, which is crucial to the modern way of looking at these results. If H is a subgroup of G, then G acts transitively on the set of right cosets of H by right multiplication, and every transitive action of G is of this form. So all that is required is to observe that all right cosets of H have the same cardinality, which is true because the map taking h to hx is a bijection from H to Hx. We learn additionally that the quotient |G|/|H| is equal to the size of a set on which G acts transitively.

The converse of this theorem is usually taken to be the assertion that, if n and m are positive integers such that m divides n, then a group of order n has a subgroup of order m. Of course things are not so simple. The smallest counterexample is the alternating group on 4 letters, which has order 12 but has no subgroup of order 6.

Sylow showed that this converse does hold in a special case, namely that when m is a prime power. His theorem is traditionally stated with four parts. Suppose that G is a finite group, and p a prime. We call a subgroup H of G a Sylow p-subgroup if the order of H is the p-part of the order of G, that is, the largest power of p dividing |G|. Now Sylow’s theorem states that Sylow subgroups exist for all choices of G and p; that they are all conjugate in G; that the number of Sylow p-subgroups is congruent to 1 (mod p); and that any subgroup of G of p-power order is contained in a Sylow p-subgroup. A very satisfactory state of affairs!

There are several very different proofs of the first part of the theorem; I will give my favourite shortly. But the other three parts are proved by very similar arguments, and almost no alternative approach to them is known. I outline the proof. It depends on the following claim:

Let P be a Sylow p-subgroup of G, and Q a p-subgroup normalising P. Then Q is contained in P.

For the assumption implies that PQ is a subgroup; its order is |P|.|Q|/|PQ| by a standard formula, so it is a p-group. By Lagrange’s Theorem its order is equal to that of P, so it is equal to P, and the claim is proved.

Now, assuming that Sylow p-subgroups exist, we prove the remaining parts of Sylow’s Theorem by considering the action of G on the set of all Sylow p-subgroups by conjugation.

We restrict the action to P. It clearly fixes itself, but by the above claim it does not fix any other Sylow p-subgroup. So all the orbits except for {P} have size divisible by p (using our observation on the connectiion between Lagrange’s Theorem and group actions). This shows that the number of Sylow p-subgroups is congruent to 1 (mod p). Moreover, the size of the G-orbit containing P is congruent to 1 (mod p), and the sizes of any other orbits are congruent to 0 (mod p). If Q is any other Sylow subgroup, then Q also lies in the (unique) orbit of size congruent to 1 (mod p); so P and Q are in the same orbit, and so are conjugate. (Incidentally, at this point we know that there is just one orbit.) Finally, if H is any p-subgroup, then restrict the action to H; it must fix a point, that is, normalise a Sylow p-subgroup R (say), and by our claim it is contained in R.

So to the first part of the theorem, the existence of Sylow p-subgroups.

Some time ago, I supervised the MSc project of a student who wanted to read and expound Sylow’s original proof of his theorem. This was non-trivial since three kinds of translation were required:

  • from French to English;
  • from the leisurely 19th century style to the terser modern style;
  • from the language of cosets and double cosets to that of group actions.

On the last point, I mentioned in connection with Lagrange’s Theorem that cosets are connected with group actions. It turns out that double cosets are connected with orbits of a group on ordered pairs of elements.

I am afraid that I don’t now even recall the student’s name, since my files from back then are long since lost. Much credit should go to him, altnough of course the main credit goes to Sylow. I claim a small amount; the very last step in the proof was my idea.

Anyway, here we go. The proof of the existence of Sylow p-subgroups proceeds in three steps.

Step 1: Translation. We show the following:

A group G has a Sylow p-subgroup if and only if it has a transitive action on a set X, such that the size of X is coprime to p but the order of any point stabiliser is a power of p.

If P is a Sylow p-subgroup, then the action on the cosets of P has the required property. In the other direction, given such an action, the point stabiliser is a Sylow p-subgroup, since its order is a p-power and its index is coprime to p.

Step 2: The heart of the proof. We show the following:

Suppose that G has a Sylow p-subgroup. Then any subgroup of G has a Sylow p-subgroup.

Take an action of G satisfying the conditions of Step 1. Let H be a subgroup of G, and restrict the action to H. Clearly the number of points is still coprime to p and the orders of the point stabilisers are p-powers. The small difficulty is that the action may not be transitive. But at least some orbit has size coprime to p, since if all sizes were divisible by p then their sum would be as well. Now the action of H on this orbit gives the required conclusion.

Step 3: Conclusion.

Given an arbitrary group, all we have to do is to embed it in a group which has a Sylow p-subgroup.

The first candidate, if you have seen Cayley’s theorem, is a symmetric group. (Cayley’s Theorem asserts that a group of order n is isomorphic to a subgroup of the symmetric group Sn.) It is possible to construct with bare hands a Sylow p-subgroup of Sn. (Write n in base p; the required subgroup is constructed as a direct product of wreath products of copies of the cyclic group of order p.)

But there is an easier way. For any field F, the symmetric group Sn is embeddable in the group GL(n,F) of invertible n×n matrices over F: just take all the permutation matrices, the matrices with entries 0 and 1 having a single 1 in any row or column. Now take F to be the field of integers mod p. A short calculation shows that its order is

(pn−1)(pnp)…(pnpn−1),

and so the exponent of p in its p-part is

0+1+…+(n−1) = n(n−1)/2.

Now consider the group of upper unitriangular matrices (1 on the diagonal, 0 below, and arbitrary entries above). The number in the last display is just the number of arbitrary entries; so the order of this group is the p-part of the order of the general linear group.

We are done.

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Some good news among the gloom

The University of St Andrews has signed up to DORA, a mere seven years after the declaration was drawn up.

We are committed to fair assessment of research rather than reliance on bibliometrics.

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Not dark yet

I had a bit of a shock this morning, when David Wallace sent sincere condolences to Rosemary.

The December issue of the IMA magazine Mathematics Today records the death of Peter Cameron. But, I am afraid, it is not me.

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Election musings

We have a general election in just over a week, the third in less than five years.

Like every election, there are multiple issues. I recommend Diamond Geezer’s post today on the Conservative manifesto; he has heroically read the whole thing and discovered some rather disturbing promises buried at the back. But in this post (I don’t usually do politics so I will be very circumscribed) I will pretend that there is just one thing (or possibly two things) on our mind: Brexit, and a second Scottish Independence Referendum (Indyref2).

Here is how the parties stand.

  • Conservative: Brexit YES, Indyref2 NO.
  • Labour: undecided (but they could be persuaded to support Indyref2 if it would get their leader into Downing Street)
  • Liberal Democrat: Brexit NO, Indyref2 NO.
  • Scottish National Party: Brexit NO, Indyref2 YES.

There is no need to consider others, since we have candidates only from these parties standing here.

In our constituency, North-East Fife, fortunately, the party led by the lying hypocrite and the party led by the indecisive bully seem to have very little chance of getting in, so it is between the LibDems and the SNP. So, under our blanket assumption, my view on Indyref2 should be decisive.

But two things complicate matters.

First, one’s attitude to Indyref2 depends very much on whether and how Brexit happens. If the UK leaves the EU, the best thing for Scotland might be to leave the UK and rejoin the EU; but of course this might mean that Hadrian’s Wall might have to be re-fortified …

Second, and possibly more relevant. The SNP push No Brexit very hard in their campaign literature, but say less about Indyref2 (though it is clear in their leader’s pronouncements). For example,

It is important to remember that Scotland, and North-East Fife in particular, voted decisively to remain in the European Union. It is also important to forget that Scotland, and North-East Fife in particular, voted decisively to remain in the United Kingdom.

Actually they didn’t say that. The first sentence is in their campaign literature, while the second was written by me but seems a fairly clear reflection of the statements made by their leader.

Our current MP, Stephen Gethins, who won by two votes at the third recount in hte last election, is a very good constituency member. But it is worrying that he signs up for this small hypocrisy.

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