# On the height and relational complexity of a finite permutation group

Gill, Nick; Lodá, Bianca and Spiga, Pablo (2021). On the height and relational complexity of a finite permutation group. Nagoya Mathematical Journal (Early Access).

## Abstract

Let be a permutation group on a set of size . We say that is an if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of . We define the of to be the maximum size of an independent set, and we denote this quantity .

In this paper we study for the case when is primitive. Our main result asserts that either , or else 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 , the maximum length of an irredundant base of , in which case we prove that if is primitive, then either or else, again, is in a particular family (which includes the primitive large--base groups as well as some others).

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