SET IDEALS WITH MAXIMAL SYMMETRY GROUP AND MINIMAL DYNAMICAL SYSTEMS

Abstract

In answer to a question of Macpherson and Neumann related to the classification of maximal subgroups of infinite symmetric groups, we characterize set ideals on an infinite set X whose stabilizers in the symmetric group Sym(X) are maximal subgroups. Specifically, for an ideal I containing all subsets of cardinality <[mid ]X[mid ], the stabilizer S_{I} is maximal if and only if the corresponding Boolean dynamical system (X, I, S_{I}) is minimal, that in turn can be expressed as an intrinsic structural property of the Boolean space (X, I). The general case is reduced to a minimal dynamical system on a subset. This characterization clarifies many earlier results, and reveals an interesting link between the theory of infinite permutation groups, topological dynamics and ergodic theory.