Algebraic and set-theoretical properties of some subsets of families of convergent and divergent permutations (Q2843892)

A permutation \(p\) of \(\mathbb N\) is called divergent (in symbols, \(p\in \mathfrak {D}\)) if there is a convergent series \(\sum a_n\) of real numbers such that the series \(\sum a_{p(n)}\) diverges. A permutation \(p\) is called convergent if \(p\in \mathfrak {C}:=\mathfrak {P}\setminus \mathfrak {D}\), where \(\mathfrak {P}\) denotes the group of all permutations of \(\mathbb N\). The following classes are investigated: \(\mathfrak {DC}:=\{p: p\in \mathfrak {D},\;p^{-1}\in \mathfrak {C}\}\) and, analogously defined, \(\mathfrak {CD}\), \(\mathfrak {DD}\), \(\mathfrak {CC}\). The following two facts are among the main four results: \begin{itemize} \item\item [(i)] \(\mathfrak {P}\setminus \mathfrak {DD}\subsetneq (\mathfrak {CD}\circ \mathfrak {DC})\cap (\mathfrak {DC}\circ \mathfrak {CD})\);\ \item \item [(ii)] \(p\mathfrak {D}p^{-1}\subset \mathfrak {D}\) if and only if \(p\in \mathfrak {CC}\).\end{itemize}