Well, having already been drenched on the way into work today, I am now stuck in my office as I wait for a lull in the rainstorm. So I guess I should put the time to use, and a blog post is as good as any right now.

This continues from the previous post. I’ll also continue the experiment of writing things up in an anecdotal style, rather than in the mode of a presentation or paper — the aim is not to present the outcome as quickly or cleanly as possible, but to give an account of its evolution. Apologies if it proves tedious, irksome, or outright Pooterish.

This is also my first attempt at using the LaTeX-to-Wordpress script written by Luca Trevisan, which has saved me a certain amount of teeth-grinding when fighting WP’s limited LaTeX support.

A few weeks after writing the previous post, I was at a rather enjoyable conference in Granada where I spent some time, on and off during lulls in the conference schedule, reconstructing the proof of the lower bound 4/3, from memory while away from my books. Annoyingly, the 4/3 comes as the limiting value of a weak lower bound, in one branch of a case-by-case analysis, but in a branch where known results suggest the true value of the amenability constant should be at least 2. (In the other branch of the analysis, one is led to 7/4 and knows that this is sharp.)

Without going into details about how the case-by-case analysis works, let’s just say that for all finite groups G in a certain class ${{\mathcal C}}$, the method in question yields a lower bound $\displaystyle {\rm AM}_{\rm Z}(G) \geq \frac{4}{3} \quad\hbox{for all }G\in{\mathcal C}.$

However, for various speculative reasons (well, guesswork) it seemed likely that ${{\rm AM}_{\rm Z}(G)\geq 2}$ for all ${G\in{\mathcal C}}.$

In fact, the existing method already implied that ${{\rm AM}_{\rm Z}(G)\geq 3/2}$ whenever ${G\in{\mathcal C}}$ and all non-central conjugacy classes have size at least 3. This got me thinking about what one could say about those ${G\in{\mathcal C}}$ which had a conjugacy class of size 2 in the group, since — as part of the small amount of finite group theory that I remembered from my undergraduate days — this ensures G has a normal subgroup of index 2, from which one might try to play games with induction or restriction of characters.

Fast-forward to June. About a week ago, shortly after getting back to Saskatoon, some idle tweaking done while waiting for food showed that the method used to get the 4/3 bound could be modestly improved by a slightly more careful use of various inequalities. The improved method still only gave $\displaystyle {\rm AM}_{\rm Z}(G) \geq \frac{13}{9} \quad\hbox{for all }G\in{\mathcal C}$

but it did now show that ${{\rm AM}_{\rm Z}(G)\geq 7/4}$ whenever ${G\in{\mathcal C}}$ and all its non-central conjugacy classes have size at least 3.

The upshot? To prove that ${{\rm AM}_{\rm Z}(G)\geq 7/4}$ for all ${G\in{\mathcal C}}$, one now only needed to do it for those ${G\in{\mathcal C}}$ which had a conjugacy class of size 2. Moreover, the calculations I’d been doing in Granada hinted strongly that any such group would be forced to be something like a dihedral group of order 2 mod 4, and in my earlier paper with Alaghmandan and Samei, we had already calculated the exact values of ${{\rm AM}_{\rm Z}}$ for dihedral groups, showing that in the 2 mod 4 case this constant is at least 7/3.

So, at this point, you naturally go and ask a group theorist. On the other hand, since none were at hand, and since the question seemed a little basic to throw on MathOverflow without having a more serious go on one’s own…

… and it works. To be precise for a moment, here is what I was able to show last week.

Theorem 1 Let G be a finite group which has trivial centre, and where every proper quotient is abelian. If G has a conjugacy class of size 2, then it is isomorphic to the dihedral group of order 2p for some odd prime p.

Well, assuming there isn’t a mistake, for many of you this is probably like watching a young child come up to you proudly to show you their latest kindergarten handiwork. I have a strong suspicion that this result is already known, perhaps implicitly, or as one of those folklore exercises that is set in Proper Courses for Proper Students. Certainly, in the argument I have right now, one shows en route that G is solvable, whereupon one can invoke a classification of “just non-abelian, metabelian groups” due to M. F. Newman.

On the other hand, since it turns out that one can prove the theorem using only what one learns in a first course on finite group theory, my plan is to write up the proof in the next blog post in this series.

(The rain has stopped.)