Ischia group theory conference 2024

The island of Ischia, about a 45-minute hydrofoil ride from Napoli, is rich in history. The museum has a Homeric drinking cup with an inscription in ancient Greek from the 8th century BCE. At the other end of history, the British composer
Sir William Walton and his Argentine wife Susana bought a desolate hilltop producing only myrtle scrub, and turned it into one of the finest gardens in Italy.

More relevant is the fact that, for the last two decades, it has hosted the biennial Ischia group theory conference, organised by mathematicians from the University of Salerno, just down the coast a bit beyond Amalfi.

Salerno, incidentally, may have a claim to be the oldest university in Italy. Around the end of the first millennium, the Schola Medica Salernitana flourished, specialising in patching up pilgrims and crusaders on the way back from Palestine, and sending them on their way with good medical advice.

The conference was dedicated to Otto Kegel, a regular at Ischia, whose 90th birthday approaches this summer. (Otto was unable to be at the meeting for health reasons, but attended many sessions on Zoom.) But as a sign of the times, half a dozen sessions were dedicated to mathematicians who have died in the last couple of years, including Bertram Huppert (d. 1 October 2023), Nikolai Vavilov (d. 14 September 2023), Zvonimir Janko (d. 12 April 2022), and Rex Dark (d. late 2022). Of course, there are others, including Richard Parker (d. 16 January 2024), Anatoly Vershik (d. 14 February 2024) and Colin Mallows (d. 4 November 2023), though perhaps the last two would not be regarded as “group theorists”.

Among many great lectures at the conference, I would nominate as my favourite the talk in memory of Zvonimir Janko by Gernot Stroth. This brought back the heady days of the 1960s and 1970s, when I was just beginning as a mathematician, and new sporadic simple groups were being found under every lamp-post. I remember on one occasion arriving at the old DPMMS building in Cambridge, when the team there had just finished the construction of J₄; the notice on Conway’s door said, simply, ∃J₄. (Janko’s fourth sporadic simple group, for the uninitiated.)

Apart from the conference dinner, the memorable social events were a recital of traditional Neapolitan songs by a singer/guitarist and a mandolin player (it struck me that there is a parallel between this music and Portuguese fado, especially Coimbra fado, the mandolin substituting for the Portuguese guitar); a concert of baroque flute and cello music in Chieso Santa Maria di Portosalvo on the harbour (my favourite, of course, the first movement of Bach’s cello suite in G); and a visit to the Waltons’ Giardin La Mortella, on its hilltop at the north of the island, with stunning flowers at this time of year.

The main thing detracting from my pleasure was the fact that I was suffering from a miserable cold the whole time, in common with my family, students, and probably a good part of the population of Britain.

West Virginia University

Anthony Hilton, my former colleague at Queen Mary, spent some time as an Eberly Professor at West Virginia University. Now he has passed on to me the news that the University has decided to close down Pure Mathematics research and put the money into “computer related activities”, whatever that means.

One of the last mathematicians standing, John Goldwasser, who has been there since 1988, is organising a small conference (I suppose I could say a wake) to mark the demise of pure mathematics at the university. As I expected, when I looked at the web page of the School of Mathematics and Data Sciences (slogan: The Revolution Starts Here), I found no mention of this conference, or indeed of the closure.

I am afraid I do not have the address of anyone you could write to if you want to express an opinion on this (which in any case seems to be a done deal). If you have such an address, please let us know.

Cauchy numbers: job done

After recruiting Scott Harper to the team, we have finished the job of determining all the Cauchy numbers (these are the positive integers n for which there exists a finite list F of finite groups so that a finite group has order divisible by n if and only if some member of F is embeddable in it).

The answer is: n is a Cauchy number if and only if one of the following holds:

• n is a prime power;
• n = 6;
• n = 2p^a, where p is a Fermat prime greater than 3 and a is a positive integer.

In the second and third cases, we can tell you the list F: for example, for n = 6, the list consists of the cyclic and dihedral groups of order 6 and the alternating group A₄. (Of course, in the first case, the list consists of all groups of order n.)

The paper will appear on the arXiv on Tuesday, with the same number as before.

Anatoly Vershik

Anatoly Vershik died the day before yesterday.

As I have told here, he was the person who told me about the Urysohn space. I had given a talk at the ECM in Barcelona on the countable random graph, and after it he approached me and asked “Do you know about the Urysohn space?” We wrote a paper on it, extending some of my results on the random graph. Indeed, the Urysohn space has many different Abelian group structures.

He also made my two trips to St Petersburg possible, by having birthdays in 2004 and 2014 (actually his birthdays were in late December the previous year). I learned so much at these conferences, and had a small adventure when I came to leave after the first one.

Richard Parker

Richard Parker died last month. Now only two of the authors of the ATLAS of finite groups remain, the two Robs.

I knew Richard, but perhaps not well enough to write anything appropriate as a tribute. But I recommend you take a look at Rob Wilson’s account on his blog.

More on Cauchy numbers

Following on from the earlier post, the new version of the paper has just gone on the arXiv: 2311.15652 (version 2).

If we say that n is a Cauchy number if there is a finite set F of finite groups, all with orders divisible by n, such that every group with order divisible by n must contain a group in F as a subgroup, then our result is as follows:

Let n be the product of two distinct primes q and r. Then n is a Cauchy number if and only if one of the primes is 2 and the other is a Fermat prime.

This means that, in all other cases, there are infinitely many groups which are minimal with respect to containing the cyclic groups of orders q and r. The arguments fall into two quite different cases. Apart from the pair {3,5}, there are infinitely many simple groups with this property. But for {3,5}, there is only one simple group, namely A₅. So instead we use a theorem of Gaschütz which guarantees that we can build an infinite sequence of extensions, each a Frattini extension of the one before, starting with A₅. (This means that each group after the first has a normal 2-subgroup contained in its Frattini subgroup such that the quotient is the preceding group.) These do the job.

I learned of this theorem from my co-authors. So collaboration, and putting a paper on the arXiv, have very positive results!

Cauchy’s theorem for the prime 6

Before you think I have gone totally crackers:

Cauchy’s theorem says that a finite group whose order is divisible by a prime number p contains a subgroup which is cyclic of order p.

My co-authors and I have proved some similar results, of which the one referred to in the title is the following:

A finite grup whose order is divisible by 6 contains a subgroup which is either cyclic of order 6, dihedral of order 6, or isomorphic to the alternating group of degree 4 (with order 12).

When the more general theorem is proved and the paper written, I hope to elaborate on this. But my question for now is: have you seen this before?

Simon Norton lecture

I have been honoured by an invitation to give the inaugural Simon Norton lecture at the London Institute for Mathematical Sciences on 12 February. The webpage is here.

Of course there are other people who knew Simon better than I did. Nevertheless, I think I have something to say. It was a paper of mine, with Jean-Marie Goethals and Jaap Seidel, which (as far as I know) introduced the term Norton algebra for a commutative but non-associative algebra of the general type which Bob Griess later used to construct the Monster. Our reference for this is to a personal communication from J. H. Conway. Nearly half a century later, I cannot remember exactly what Conway said to me.

Anyway, I will be very glad to see you there if you can come.

New Year

The youth could not help breaking a rule of courtesy towards this heavily burdened and yet, as he felt, noble man by asking: “But tell me, I beseech you, why do you carry on such wars on your star? Who is to blame for them? Are you yourself in part responsible?”

The King seemed angered at this audacity and for a time stared at the messenger. But he could not continue to meet with his dark gaze the bright and guileless eyes of the stranger.

“You are a child,” said the King, “and there are things you cannot understand. War is no one’s fault, it occurs of itself, like storm and lightning, and all of us who have to fight wars, we are not their originators, we are only their victims.”

Hermann Hesse, Strange News from Another Star

Programming and typesetting

Is computer typesetting a kind of programming?

One of the pioneers, Donald Knuth, clearly thought so. In The TeXBook, he gives TeX code for computing and typesetting the first thirty primes; apart from anything else, this demonstrates that TeX has the capacity to act as an all-purpose program.

But there is a rather significant difference.

A programmer requires the program to deliver the right answer: exactly, for a discrete mathematician, and to with a specified approximation, for a continuous mathematician. Computer typesetting doesn’t deliver an answer as such. Knuth realises this when The TeXBook ends with the admonition “GO FORTH and create masterpieces of the publishing art“. Despite Keats, truth and beauty are not quite the same.

It is still true, though, that computer typesetting can pose problems similar to those faced by programmers. This summer, I was editing a book with four substantial chapters written by different teams of authors. When I put the chapters together and compiled the book, I discovered that the diagrams in one chapter were wrong. After a lot of searching, I discovered that this was caused by a conflict between two packages used by different authors. The standard remedy is to load the packages in the other order. But this brought LaTeX to a screaming halt without typesetting anything.

The two conflicting packages were curves and tikz. The authors who had loaded the second of these made extensive use of it, but for the first package only two commands had actually been used. The solution wasn’t going to be to delve inside the packages and make changes, since the LaTeX file had to go off to the publisher.

I used plain TeX for several years before being (more or less forcibly) converted to LaTeX. (If you have looked at my Combinatorics book for CUP, you will have noticed that it doesn’t look like most textbooks produced with LaTeX – that is the reason.) So I delved into my knowledge of TeX. I loaded curves, renamed the two required commands with the TeX \let command, and then loaded tikz, then held my breath and compiled the book: it worked!

So no pretence that this was the “correct” programming to use here, but it happened to work to produce the book: perhaps not a masterpiece of the publishing art, but at least a beautiful book. (It should appear next year.)

I now have to face a somewhat similar problem. A coauthor just sent me a copy of the paper, so that it is now my responsibility to make some edits. I was quite horrified, when I looked through it, to find the string lcmp. (I am showing you just an approximation in HTML.) I checked the input: lcm was correctly defined as a math operator, so the output should have looked something like lcm p. (A math operator is designed to leave a certain amount of space between itself and its operand; this looks better and is easier to read and understand. All this is based on a combination of centuries of typesetters’ experience and research in neuropsychology.)

I now have to track down the problem.

I also noticed that the preamble to the paper loads several dozen packages, many of which I have never heard of. I suspect that one of these is causing the trouble, and my next task will be to find which one. At the same time, I also suspect that most of these packages are not necessary. I know that some authors start with a LaTeX template which loads every package they have ever heard of, just in case it is needed. So I might do some judicious filtering …