Stabilizers of Trivial Ideals

In papers by Semmes [6], Macpherson and Neumann [4] and Brazil, Covington, Penttila, Praeger and Woods [2], it has been shown that stabilizers of filters (equivalently, stabilizers of ideals) play a crucial role in the study of maximal subgroups of infinite symmetric groups. Semmes proved that if if is a maximal subgroup of S = Sym (Q), where Q is a set of infinite cardinality K and H contains the pointwise stabilizer of some set A with |A| < K, then H is the stabilizer of a filter. This was investigated further in [2], where it was shown that if H is a maximal subgroup containing the pointwise stabilizer of some set A with |A| = K, then H is the stabilizer of a quasiideal. This leads to the result that any such subgroup is either the almost stabilizer of a partition of Q into finitely many parts or the stabilizer of an ideal. The ideals obtained in these papers are all nontrivial ideals, that is, they contain some set of cardinality K. In [2] it was shown that if a maximal subgroup of S is the stabilizer of a nontrivial ideal, then this is the unique such nontrivial ideal.