On distinguishing quotients of symmetric groups

by Shelah and Truss. [ShTr:605]
Annals Pure and Applied Logic, 1999
A study is carried out of the elementary theory of quotients of symmetric groups in a similar spirit to [Sh:24]. Apart from the trivial and alternating subgroups, the normal subgroups of the full symmetric group S(mu) on an infinite cardinal mu are all of the form S_kappa (mu)= the subgroup consisting of elements whose support has cardinality < kappa for some kappa <= mu^+ . A many-sorted structure M_{kappa lambda mu} is defined which, it is shown, encapsulates the first order properties of the group S_lambda (mu)/S_kappa (mu) . Specifically, these two structures are (uniformly) bi-interpretable, where the interpretation of M_{kappa lambda mu} in S_lambda (mu)/S_kappa (mu) is in the usual sense, but in the other direction is in a weaker sense, which is nevertheless sufficient to transfer elementary equivalence. By considering separately the cases cf(kappa) > 2^{aleph_0}, cf(kappa) <= 2^{aleph_0}< kappa, aleph_0< kappa < 2^{aleph_0}, and kappa = aleph_0, we make a further analysis of the first order theory of S_lambda (mu)/S_kappa (mu), introducing many-sorted second order structures N^2_{kappa lambda mu}, all of whose sorts have cardinality at most 2^{aleph_0} .

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