On distinguishing quotients of symmetric groups (Q1302299)

If \(\Omega\) is a set of infinite cardinality \(\mu\), and \(S(\mu)\) is the full symmetric group on \(\Omega\), then for every infinite cardinal \(\lambda \leq \mu^+\) there is a normal subgroup \(S_{\lambda}(\mu)\) of \(S(\mu)\) consisting of permutations whose support has cardinality less than \(\lambda\). Furthermore, apart from the finitary alternating groups, the \(S_{\lambda}(\mu)\) are the only non-trivial normal subgroups of \(S(\mu)\). This paper concerns the elementary theories in the language of groups of the quotient groups \(S_{\lambda}(\mu)/S_{\kappa}(\mu)\) (where \(\kappa<\lambda\leq\mu^+\) are infinite cardinals), and the extent to which these depend on the triple \((\kappa,\lambda,\mu)\). This extends earlier work of \textit{S. Shelah} [``First order theory of permutation groups'', Israel J. Math. 14, 149-162 (1973; Zbl 0284.20003); errata: ibid. 15, 437-441 (1973; Zbl 0285.20001)] on the elementary theories of infinite symmetric groups, and also some work of Rubin. Given cardinals \(\kappa<\lambda\leq \mu^+\), the authors construct a many-sorted first order structure \({\mathcal M}_{\kappa\lambda\mu}\), and show that it is interpretable in \(S_{\lambda}(\mu)/S_{\kappa}(\mu)\); furthermore, they show the latter group is `uniformly semi-interpreted' in \({\mathcal M}_{\kappa\lambda\mu}\). The structure \({\mathcal M}_{\kappa\lambda\mu}\) can have arbitrarily large size. However, the authors also introduce second order structures \({\mathcal N}_{\kappa\lambda\mu}^2\), of cardinality at most \(2^{\aleph_0}\). They prove that if \(\kappa_1<\lambda_1\leq \mu_1^+\) and \(\kappa_2<\lambda_2\leq \mu_2^+\), then the groups \(S_{\lambda_i}(\mu_i)/S_{\kappa_i}(\mu_i)\) (for \(i=1,2\)) are elementarily equivalent if and only if the structures \({\mathcal M}_{\kappa_i\lambda_i\mu_i}\) are elementarily equivalent, if and only if the second order structures \({\mathcal N}_{\kappa_i\lambda_i\mu_i}^2\) are elementarily equivalent. In addition, the authors show that if \({\mathcal P}_{\kappa}(\mu)\) is the ring of subsets of \(\Omega\) of size less than \(\kappa\), then the ring \({\mathcal P}_{\lambda}(\mu)/{\mathcal P}_{\kappa}(\mu)\), and the natural action of \(S_{\lambda}(\mu)/S_{\kappa}(\mu)\) on it, are interpretable in this quotient group.