Algebraically determined topologies on permutation groups. (Q429307)

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.NEWLINENEWLINE The authors unify a result of \textit{E. D. Gaughan} [Proc. Natl. Acad. Sci. USA 58, 907-910 (1967; Zbl 0153.04301)] with a result of \textit{S. Dierolf} and \textit{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 \(G\).NEWLINENEWLINE 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.