On distinguishing quotients of symmetric groups
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 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} .
math.LO↗