Homogeneity of infinite permutation groups
Let \(\Omega\) be an infinite set, \(\lambda <| \Omega |\) an arbitrary cardinal, and G a group of permutations of \(\Omega\). G is said to be \(\lambda\)-homogeneous if for each pair A,B\(\subseteq \Omega\) with \(| A| =| B| =\lambda\), there exists \(g\in G\) with \(A^ g=B\). It is known that if G is m-homogeneous for some \(m\in \omega\), then G is also n-homogeneous for all \(n<m\). Recently, \textit{P. M. Neumann} [Bull. Lond. Math. Soc. 20, 305-312 (1988; Zbl 0644.20007)] proved that if G is \(\lambda\)-homogeneous for some \(\lambda\geq \omega\), then G is n- homogeneous (even n-transitive) for all \(n\in \omega\). Using Martin's Axiom (MA) and combinatorial arguments, the authors show: Theorem. Assume (MA) and \(\omega_ 2<2^{\omega}\). Let \(\omega_ 1<\kappa <2^{\omega}\). Then there exists a group G of permutations of \(\kappa\) such that G is \(\lambda\)-homogeneous for all \(\omega <\lambda <\kappa\), but not \(\omega\)-homogeneous. - This answers a question of P. M. Neumann [loc. cit.] in the negative. The authors remark that it remains open whether it is consistent with (ZFC) that every \(\omega_ 1\)-homogeneous permutation group is \(\omega\)-homogeneous.
\(\omega _ 1\)-homogeneous permutation group
scientific article; zbMATH DE number 4095736
scientific article