Two new preprints in group theory


Assoc Prof Nick Gill has recently uploaded two papers to the mathematics preprint archive. Both are the result of work with PhD students.

The first, entitled "On the height and relational complexity of a finite permutation group" includes the main result from Bianca Loda's thesis. The paper is joint with Bianca, and with Pablo Spiga, Bianca's other supervisor. This main result of the paper gives a strong upper bound on the height and relational complexity of a primitive permutation group, thereby answering a question of  Cherlin, Martin and Saracino. The result also strengthens a well-known result of Liebeck on the minimum base-size of a primitive permutation group. The paper is here:

The second, entitled "Nilpotent covers of symmetric groups" is joint work with Kimeu Arphaxad Ngwava and Ian Short. Kimeu is a PhD student based at Moi University in Kenya; he is being supervised by Nick and Ian, as part of the London Mathematical Society's "Mentoring African Research Mathematicians" scheme. The main result of this paper completely answers the question of how to minimally cover a finite symmetric group, S_n, with maximal nilpotent subgroups. It turns out that there is a unique minimal cover, and this cover is made up of k conjugacy classes of nilpotent subgroup, where k is the number of partitions of n into distinct parts. The paper is here:

Congratulations are in order for both Bianca and Kimeu: both have produced some beautiful mathematics as part of their PhD research. Bianca has already submitted and successfully defended her thesis; Kimeu expects to submit his thesis in the next couple of months -- as soon as universities in Kenya reopen after the COVID19 crisis.