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)

@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}
}