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Lévy–Khinchin formula for the infinite symmetric group
Mathematische Zeitschrift, 2012A function \(f\) on a group \(G\) is said to be a positive type function if, for any \(g_1, g_2, \dotsc, g_n\in G\) and \(c_1, c_2, \dotsc, c_n\in \mathbb{C}\) \[ \sum_{i, j=1}^n c_i \overline{c_j} f(g_i^{-1}g_j)\geq 0. \] Also, a function \(\psi\) on \(G\) is said to be a negative type function if \(\psi(e)\geq 0\), \(\psi(g^{-1})=\overline{\psi(g)}\)
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Unbounded families and the cofinality of the infinite symmetric group
Archive for Mathematical Logic, 1995zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Groupwise density and the cofinality of the infinite symmetric group
Archive for Mathematical Logic, 1998For a group \(G\) which is not finitely generated, the cofinality of \(G\), written \(c(G)\), is defined to be the least cardinal \(\lambda\) such that \(G\) is the union of a chain of \(\lambda\) proper subgroups. For an infinite cardinal \(\kappa\), denote \(c({\mathfrak{Sym}}(\kappa))\) by \(c_\kappa\). \textit{H.~D.
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Implausible Subgroups of Infinite Symmetric Groups
Bulletin of the London Mathematical Society, 1988Let S denote the infinite symmetric group of all permutations of \(\omega\), the set of natural numbers. The authors study the possibilities for the induced action of subgroups \(G\subseteq S\) on the power set \({\mathcal P}(\omega)\). Assuming Martin's axiom (MA), they show, in particular, that for any infinite cardinal \(\kappa
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