After my trip to Florence in February, I wrote about the work I did there with Carlo Casolo and Francesco Matucci.
After Carlo’s untimely death the following month, we were left with many pages of notes from him about the work he had done on the problem during my visit and subsequently.
Our job, then, was to distil these notes into a paper and publish it in his memory.
It has taken a while to do this, but the resulting paper has just appeared on the arXiv. I won’t go through it blow by blow, but I would like to draw attention to two things about it, one major and the other rather smaller.
Let me remind you that we use the term “integral of a group G” to mean a group H whose derived group (generated by all commutators) is isomorphic to G. Every abelian group is integrable, but we were concerned in our first paper with the smallest integral of a given abelian group. In particular, if G is an infinite abelian group, does it have an integral in which it has finite index?
Carlo had put a lot of effort into this question, and had come up with, not a complete solution, but something close to it, at least in the case of a countable torsion abelian group. Unfortunately, his argument depended quite heavily on an “obvious” result from linear algebra. We had trouble with this, and spent a long time trying to come up with a proof. In the end, trying to track down where the proof attempts failed, I came up with a counterexample. Then, of course, the failure of this lemma brought down a lot of the beautiful arguments that followed from it.
I think that much of what Carlo did will prove to be correct, but a different approach needs to be found. So we simply took out everything depending on the lemma. (The paper is still a very solid 40 pages without this material.)
So I will state a very special case of this analysis. A proof of this would hopefully open the door to restoring much of the rest. I state it as a conjecture, though it was a corollary of a big theorem in Carlo’s notes.
Conjecture Let G be a countable group which is a direct product of finite cyclic groups. Then G has finite index in some integral if and only if the number of even positive integers n for which the cyclic group of order n occurs with multiplicity 1 in the product is finite.
The other matter is the solution of one of the problems in our earlier paper by Efthymios Sofos, from Glasgow. We had characterised the set of positive integers n for which every finite group of order n is integrable, and asked for the asymptotics of the number of such n up to x, for any x. It turns out that the leading asymptotics of this number is the same as for various similar-looking sets, “every group of order n is cyclic” (this was done by Paul Erdős in 1948), “every group of order n is abelian”, or “every group of order n is nilpotent”. The answer is
e−γ x/log log log x,
where γ is the Euler–Mascheroni constant.
There is much more in the paper; take a look.