Congratulations to Bianca Loda
Congratulations to Bianca Loda who has successfully defended her PhD thesis entitled The height and the relational complexity of finite primitive permutation groups. I supervised Bianca with Pablo Spiga from Milan; Bianca’s PhD examiners were Colva Roney-Dougal (St Andrews) and Dugald Macpherson (Leeds).
Bianca’s thesis was a beautiful piece of work, of which she should be very proud. She ended up addressing a question of Cherlin, Martin and Saracino, in their 1996 paper Arities of permutation groups. They suggested that it should be possible to classify those permutation groups which act primitively on a set of size t and which have relational complexity at most t1/2.
A corollary of the main result of Bianca’s thesis yields this classification in a very strong form. Her main result is this:
Theorem Let G be a finite primitive group on a set of Ω of size t. Then one of the following holds:
- G is a subgroup of Sm≀St containing (Am)t, where the action of Sm is on k-subsets of 1,…,m and the wreath product has the product action of degree (t=mkt);
- The height of the action of G on Ω is at most 22log2t.
An immediate corollary of this theorem is that if G is a finite primitive group of degree t, then either the first listed possibility of the theorem holds or else the relational complexity of the action of G on Ω is at most 22log2t+1.
The main theorem will appear as a joint paper, authored by Bianca, Pablo and I, in the Nagoya Mathematical Journal.