Primitive permutation groups satisfying the small orbit property and a problem of Bourgain and Kalai

Abstract

We classify the primitive but not affine permutation subgroups of S_n with the following property: There exists some unbounded subset X of size at most hbox {clog}n whose orbit is polynomially bounded (in terms of n). This class of subgroups arises naturally in the study of threshold behavior of G-symmetric properties of Boolean functions. Our result answers a problem of Bourgain and Kalai. Contrary to a suggestion in their paper, we give an example for a primitive permutation group with this property, which contains {A_{m}} as a section for m rightarrow infty .