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  • Algebraically determined topologies on permutation groups
    Banakh, Taras, 1968- ; Guran, Igor ; Protasov, Igorʹ Vladimirovič
    In this paper we answer several questions of Dikran Dikranjan about algebraically determined topologies on the groups ▫$S(X)$▫ (and ▫$S_\omega(X)$▫) of (finitely supported) bijections of a set ▫$X$▫. ... In particular, confirming conjecture of Dikranjan, we prove that the topology ▫${\mathcal T}_p$▫ of pointwise convergence on each subgroup ▫$G \supset S_\omega(X)$▫ 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 ▫${\mathcal T}_p$▫ coincides with the Zariski and Markov topologies ▫${\mathfrak Z}_G$▫ and ▫${\mathfrak M}_G$▫ on ▫$G$▫. Answering another question of Dikranjan, we prove that the centralizer topology ▫${\mathfrak T}_G$▫ on the symmetric group ▫$G = S(X)$▫ is discrete if and only if ▫$|X| \le \mathfrak{c}$▫. On the other hand, we prove that for a subgroup ▫$G \supset S_\omega(X)$▫ of ▫$S(X)$▫ the centralizer topology ▫${\mathfrak T}_G$▫ coincides with the topologies ▫${\mathcal T}_p = {\mathfrak M}_G = {\mathfrak Z}_G$▫ if and only if ▫$G = S_\omega(X)$▫. We also prove that the group ▫$S_\omega(X)$▫ is ▫$\sigma$▫-discrete in each Hausdorff shift-invariant topology.
    Source: Algebra meets Topology : special issue on Dikran Dikranjan's 60th bithday (Str. 2258-2268)
    Type of material - conference contribution
    Publish date - 2012
    Language - english
    COBISS.SI-ID - 16288601

    Link(s):

    http://dx.doi.org/10.1016/j.topol.2012.04.010

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