# Sh:605

• Shelah, S., & Truss, J. K. (1999). On distinguishing quotients of symmetric groups. Ann. Pure Appl. Logic, 97(1-3), 47–83.
• 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}
}