On the height and relational complexity of a finite permutation group

Abstract

Let $G$ be a permutation group on a set $\Omega$ of size $t$. We say that $\Lambda\subseteq\Omega$ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $\Lambda$. We define the height of $G$ to be the maximum size of an independent set, and we denote this quantity $\mathrm{H}(G)$. In this paper we study $\mathrm{H}(G)$ for the case when $G$ is primitive. Our main result asserts that either $\mathrm{H}(G)< 9\log t$, or else $G$ is in a particular well-studied family (the "primitive large--base groups"). An immediate corollary of this result is a characterization of primitive permutation groups with large "relational complexity", the latter quantity being a statistic introduced by Cherlin in his study of the model theory of permutation groups. We also study $\mathrm{I}(G)$, the maximum length of an irredundant base of $G$, in which case we prove that if $G$ is primitive, then either $\mathrm{I}(G)<7\log t$ or else, again, $G$ is in a particular family (which includes the primitive large--base groups as well as some others).