# Sh:605

- Shelah, S., & Truss, J. K. (1999).
*On distinguishing quotients of symmetric groups*. Ann. Pure Appl. Logic,**97**(1-3), 47–83. arXiv: math/9805147 DOI: 10.1016/S0168-0072(98)00023-2 MR: 1682068 -
Abstract:

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\le\mu^+. A many-sorted structure {\mathcal 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 {\mathcal 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)\le 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 {\mathcal N}^2_{\kappa \lambda \mu}, all of whose sorts have cardinality at most 2^{\aleph_0}. - Version 1998-05-19_10 (40p) published version (37p)

Bib entry

@article{Sh:605, author = {Shelah, Saharon and Truss, John Kenneth}, title = {{On distinguishing quotients of symmetric groups}}, journal = {Ann. Pure Appl. Logic}, fjournal = {Annals of Pure and Applied Logic}, volume = {97}, number = {1-3}, year = {1999}, pages = {47--83}, issn = {0168-0072}, mrnumber = {1682068}, mrclass = {03C60 (20B30)}, doi = {10.1016/S0168-0072(98)00023-2}, note = {\href{https://arxiv.org/abs/math/9805147}{arXiv: math/9805147}}, arxiv_number = {math/9805147} }