On distinguishing quotients of symmetric groups

S. Shelah, J. K. Truss*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

A study of the elementary theory of quotients of symmetric groups is carried out in a similar spirit to Shelah (1973). Apart from the trivial and alternating subgroups, the normal subgroups of the full symmetric group S(μ) on an infinite cardinal μ are all of the form Sκ(μ)= the subgroup consisting of elements whose support has cardinality < κ, for some κ≤μ. A many-sorted structure Mκλμ is defined which, it is shown, encapsulates the first order properties of the group Sλ(μ)/Sκ(μ). Specifically, these two structures are (uniformly) bi-interpretable, where the interpretation of Mκλμ in Sλ(μ)/Sκ(μ) 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(κ)>2Ν0, cf(κ)≤2Ν0 <κ, Ν0 <κ <2Ν0, and κ = Ν0, we make a further analysis of the first order theory of Sλ(μ)/Sκ(μ), introducing many-sorted second order structures script N2 κλμ, all of whose sorts have cardinality at most 2Ν0, and in terms of which we can completely characterize the elementary theory of the groups Sλ(μ)/Sκ(μ).

Original languageEnglish
Pages (from-to)47-83
Number of pages37
JournalAnnals of Pure and Applied Logic
Volume97
Issue number1-3
DOIs
StatePublished - 21 Mar 1999

Keywords

  • Elementary theory
  • Infinite symmetric group
  • Many sorted structure
  • Quotient

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