Let \(\Omega\) be an infinite set having cardinality \(n:=| \Omega |\), and let \(S:=Sym(\Omega)\) be its symmetric group having then cardinality \(2^ n\). Moreover, let G be a subgroup of S and let \(S_{\{\Delta \}}\) or \(G_{\{\Delta \}}\) be the setwise stabilizer of \(\Delta\) in S or in G, respectively, for each subset \(\Delta\subseteq \Omega\). Similarly, let \(S_{(\Delta)}\) and \(G_{(\Delta)}\) be pointwise stabilizers of \(\Delta\subseteq \Omega\). And finally, let \({\mathcal P}(\Omega)\) denote the power set of \(\Omega\).
The authors, working in set theory with Axiom of Choice, investigate those subgroups G having the property \(| S:G| <2^ n\) and prove the following three main results: (1) if \(n=\aleph_ 0\) and \(| S:G| <2^{\aleph_ 0}\) then there is a finite subset \(\Delta_ 0\subseteq \Omega\) such that \(S_{(\Delta_ 0)}\leq G\leq S_{\{\Delta_ 0\}}\); (2) assuming the generalized continuum hypothesis it holds that if \(| S:G| <2^ n\) then there is a subset \(\Delta\subseteq \Omega\) such that \(| \Delta | <n\) and \(S_{(\Delta)}<G\); and (3) if \(G\leq S\) and \(| S:G| <2^ n\) then by setting \[ {\mathcal G}:=\{(\Gamma \subseteq \Omega | \quad \exists \Delta \subseteq \Omega: | \Omega -\Delta | =n\quad and\quad S_{(\Delta)}\leq G\} \] we get a filter in \({\mathcal P}(\Omega)\) such that \(S_{({\mathcal G})}\leq G\leq S_{\{{\mathcal G}\}}\). Some other questions and open problems relating these results are discussed, as well.