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 $t^{1/2}$.

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 $\Omega$ of size $t$. Then one of the following holds:

  1. G is a subgroup of $S_m \wr S_t$ containing $(A_m)^t$, where the action of $S_m$ is on $k$-subsets of ${1,\dots, m}$ and the wreath product has the product action of degree $t=\choose{m}{k}^t$;
  2. The height of the action of $G$ on $\Omega$ is at most $22\log_2 t$.

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 $\Omega$ is at most $22\log_2 t +1$.

The main theorem will appear as a joint paper, authored by Bianca, Pablo and I, in the Nagoya Mathematical Journal.