Algebraically determined topologies on permutation groups

Abstract

In this paper we answer several questions of Dikran Dikranjan about algebraically determined topologies on the groups S(X) (and Sω(X) ) of (finitely supported) bijections of a set X. In particular, confirming conjecture of Dikranjan, we prove that the topology Tp of pointwise convergence on each subgroup G ⊃ Sω(X) of S(X) is the coarsest Hausdorff group topology on G (more generally, the coarsest T 1 -topology which turns G into a [semi]topological group), and Tp coincides with the Zariski and Markov topologies Z G and M G on G. Answering another question of Dikranjan, we prove that the centralizer topology T G on the symmetric group G = S(X) is discrete if and only if |X| ≤ c. On the other hand, we prove that for a subgroup G ⊃ Sω(X) of S(X) the centralizer topology T G coincides with the topologies Tp = M G = Z G if and only of G = Sω(X). We also prove that the group Sω(X) is σ-discrete in each Hausdorff shift-invariant topology.