Today (Wednesday 18 March) the Groups, Representations and Applicatons programme at the Isaac Newton Institute came to a premature end.

There are still some hopes that it can be revived later, but at the moment the only certainty is that it is closed now.

The director of the Institute, David Abrahams, drew our attention to the fact that while Isaac Newton was self-isolating, he did some very good research.

I have my grumbles about various things, especially the accommodation in the Weat Cambridge desert (as I hinted here), but now is not the time for that.

The third workshop was reduced to two and a half days, many of the talks presented remotely, and even for the live ones only ten of us allowed in the huge lecture room. (This despite the fact that most of us had been working together since January.)

So I will just mention one talk, a beautiful talk (as usual) by Tim Burness, on the length and depth of a group. The length of a group is the length of the longest chain of subgroups in the group, and has various applications (the question of determining the length of the symmetric group was raised by Babai and is fairly distantly connected with graph isomorphism). I believe I was the first person to find the beautiful formula

*l*(*S*_{n}) = ⌈3*n*/2⌉−*b*(*n*)−1

for the length of the symmetric group *S*_{n}, where *b*(*n*) is the number of ones in the base 2 expansion of *n*. This result depends on the classification of finite simple groups, but only rather weakly; I seem to remember that, at the time (early 1980s), I observed that using Babai’s “elementary” bound for the order of a primitive permutation group, the formula can be proved for all *n* greater than about 1000.

The depth of a group is the length of the shortest unrefinable chain of subgroups. There are some very appealing arguments concerning depth; for example, the proof that there is an absolute bound for the depth of *S*_{n} uses the proof by Helfgott of the ternary Goldbach conjecture. However, what I didn’t see was any application of depth comparable to the several applications of length. (If the depths of symmetric groups are bounded, it is not a very good measure of their size, methinks.)

Anyway, tomorrow we head back to Scotland and go to ground. There were several projects I had hoped to get finished here, and we have at least pushed them on a bit; also the two things from my February trips that I talked about earlier.

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## About Peter Cameron

I count all the things that need to be counted.