1998/05/15 by J. K. Truss, John Truss, Saharon Shelah +2
Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Rings, Modules, and Algebras #math.GR #math.LO
paper · pdf · doi:10.48550/arxiv.math/9805147
published as Ann. Pure Appl. Logic 97 No. 1-3 (1999) 47--83
arxiv created 1998/05/15 · openalex publication_date 1998/05/15 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Skappa(mu)= the subgroup consisting of elements whose support has cardinality <kappa for some kappa <= mu+. A many-sorted structure Mkappa lambda mu is defined which, it is shown, encapsulates the first order properties of the group Slambda (mu)/Skappa (mu). Specifically, these two structures are (uniformly) bi-interpretable, where the interpretation of Mkappa lambda mu in Slambda(mu)/Skappa(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) > 2aleph0, cf(kappa) <= 2aleph0< kappa, aleph0< kappa < 2aleph0, and kappa = aleph0, we make a further analysis of the first order theory of Slambda(mu)/Skappa(mu), introducing many-sorted second order structures N2kappa lambda mu, all of whose sorts have cardinality at most 2aleph0 .