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Gill, Nick; Lodá, Bianca and Spiga, Pablo
(2021).
DOI: https://doi.org/10.1017/nmj.2021.6
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).