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
$\textrm{H}(G)$
. In this paper, we study
$\textrm{H}(G)$
for the case when G is primitive. Our main result asserts that either
$\textrm{H}(G)< 9\log t$
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
$\textrm{I}(G)$
, the maximum length of an irredundant base of G, in which case we prove that if G is primitive, then either
$\textrm{I}(G)<7\log t$
or else, again, G is in a particular family (which includes the primitive large–base groups as well as some others).
ON THE HEIGHT AND RELATIONAL COMPLEXITY OF A FINITE PERMUTATION GROUP