I don’t know how things have got so busy. I had two interesting trips in February; I worked hard, and some interesting mathematics resulted; but I don’t seem to have found the time to describe it. So here goes. This is, by a long way, not all that I managed to do, just a little sample from each trip.
I was meant to be in Firenze, working with Carlo Casolo and Francesco Matucci, from 10 to 14 February. Unfortunately Storm Ciara had other ideas, and my flight out was cancelled; I had to re-book on a flight the next day. So our work on the first day was briefer than intended.
We were working on integrals of groups. We call the group H an integral of the group G if the derived subgroup of H is isomorphic to G. When João Araújo and Francesco invited me to work on this topic, my first thought was that it was a linguistic joke, or possibly just light relief from the hard stuff. But we recruited Carlo to the team, and published a 30-page paper in the Israel Journal of Mathematics, which I described here.
So this visit was to work on the follow-up. It was not entirely clear what this should be: other topics in “inverse group theory”, or further results on integrals? It turned out to be the latter. Here is one of the results, a three-part theorem to which we all contributed, but Carlo was certainly in the driving seat. The context is the question: if G belongs to a certain class of groups, and G has an integral, does it have an integral in that class? This is what we did, for the class of profinite groups.
- Let G be a profinite group which has an integral H such that G has finite index in H. Then G has a profinite integral.
- Let G be a finitely generated profinite group which has an integral. Then G has a profinite integral.
- There is a profinite group which has an integral but has no profinite integral.
So things can be very interesting!
I had never been to Firenze before, but we were so busy that I onlly got to see the Duomo at night. But above is a picture of it; I have a new camera which is better in low light than my old one.
Two weeks later I was off to Portugal. The intention had been to work on the Road Closure Conjecture with João and Pablo Spiga. But the coronavirus had other ideas, and Pablo wasn’t able to make it. So João and I thought about complete mappings instead. These are usually defined on quasigroups (they are essentially the same thing as transversals of Latin squares), and especially groups; but I was now in semigroup land, so what about complete mappings of semigroups?
I should mention that the post linked above concerns the Hall–Paige conjecture, now proved (using the Classification of Finite Simple Groups): a finite group has a complete mapping if and only if either it has odd order or its Sylow 2-subgroups are cyclic.
A complete mapping on a multiplicative structure S is a bijection f from S to itself such that the map g given by g(x) = x·f(x) is also a bijection. (At least for groups, g is called an orthomorphism. João had already convinced himself that the critical case for the theory consists of Rees matrix semigroups together with their variants with a zero element.
I’m not going to give a definition here. Suffice to say that such a semigroup is defined by a group G and a rectangular matrix P with entries from G, which can be assumed to be normalised (that is, all entries in the first row and column are the identity of G).
With a combination of computer experiment and hand calculation we came up with a conjecture:
A Rees matrix semigroup defined by a group G and normalised matrix P has a complete mapping if and only if one of the following holds:
- the group G has a complete mapping;
- the number of entries in P is even;
- some entry in P is a non-square.
We were able to show that the disjunction of the three conditions is necessary; the first two are sufficient, and the third is sufficient if the group has twice odd order.
I was planning to talk about this at the NBSAN meeting in Hull next Friday (20 March), but things are changing so fast that I am more than half expecting the meeting to be cancelled. If so, this is part of what you missed!
Again, we were busy all week, but I had one extra day, and João took me to Mira de Aire, the largest caves in Portugal, with one gallery 11 kilometres long (we didn’t get to see it all). We saw much more during the day, including a fossil of a Jurassic starfish: