Algebraically determined topologies on permutation groups

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 T1-topology which turns G into a [semi]-topological group), and Tp coincides with the Zariski and Markov topologies ZG and MG on G. Answering another question of Dikranjan, we prove that the centralizer topology TG 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 TG coincides with the topologies Tp=MG=ZG if and only if G=Sω(X). We also prove that the group Sω(X) is σ-discrete in each Hausdorff shift-invariant topology.