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 ..read more

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 Scie ..read more

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 = 2pa, 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, th ..read more

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 th ..read more

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 ..read more

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 ..read more

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 ..read more

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 ..read more

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 ..read more

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 T ..read more