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Algebraically determined topologies on permutation groups. (English) Zbl 1277.20006

Let \(X\) be a set, \(S(X)\) the symmetric group of \(X\) and \(S_\omega(X)\) be the subgroup of \(S(X)\) consisting of permutations with finite support.
The authors unify a result of E. D. Gaughan [Proc. Natl. Acad. Sci. USA 58, 907-910 (1967; Zbl 0153.04301)] with a result of S. Dierolf and U. Schwanengel [Bull. Sci. Math., II. Ser. 101, 265-269 (1977; Zbl 0375.22001)]. Namely, they prove that if \(G\) is a subgroup of \(S(X)\) and \(S_\omega(X)\subset G\), then the topology of pointwise convergence on \(G\) is the coarsest Hausdorff group topology on \(G\).
Various group topologies on \(G\) and relations between them are studied: Zariski topology, Markov topology, the centralizer topology of Taimanov. Two group topologies on \(S(X)\) associated with the Alexandrov and the Stone-Čech compactifications of the discrete space \(X\) are introduced. Using the topology on \(S(X)\) associated with the Stone-Čech compactification of \(X\), a non-discrete Hausdorff group topology on the factor group \(S(X)/S_\omega(X)\) is constructed. At the end of the paper a list of open problems is given.

MSC:

20B35 Subgroups of symmetric groups
22A05 Structure of general topological groups
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
54H11 Topological groups (topological aspects)
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