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 S ω ( X ) ) of (finitely supported) bijections of a set X. In particular, confirming conjecture of Dikranjan, we prove that the topology T p of pointwise convergence on each subgroup G ⊃ S ω ( 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 T p 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 ⊃ S ω ( X ) of S ( X ) the centralizer topology T G coincides with the topologies T p = M G = Z G if and only if G = S ω ( X ) . We also prove that the group S ω ( X ) is σ-discrete in each Hausdorff shift-invariant topology.