### 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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