Divisible Rigid Groups. II. Stability, Saturation, and Elementary Submodels

Alexei Miasnikov, N. S. Romanovskii

Research output: Contribution to journalArticle

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Abstract

A group G is said to be rigid if it contains a normal series G = G1G2GmGm+1 = 1, whose quotients Gi/Gi+1 are Abelian and, treated as right ℤ[G/Gi]- modules, are torsion-free. A rigid group G is divisible if elements of the quotient Gi/Gi+1 are divisible by nonzero elements of the ring ℤ[G/Gi]. Every rigid group is embedded in a divisible one. Previously, it was stated that the theory m of divisible m-rigid groups is complete. Here, it is proved that this theory is ω-stable. Furthermore, we describe saturated models, study elementary submodels of an arbitrary model, and find a representation for a countable saturated model in the form of a limit group in the Fraïssé system of all finitely generated m-rigid groups. Also, it is proved that the theory m admits quantifier elimination down to a Boolean combination of ∀∃-formulas.

LanguageEnglish (US)
Pages29-38
Number of pages10
JournalAlgebra and Logic
Volume57
Issue number1
DOIs
StatePublished - Mar 1 2018

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Elementary Submodel
Divisible
Saturation
Quotient
Quantifier Elimination
Torsion-free
Finitely Generated
Countable
Model
Ring
Module
Series
Arbitrary

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Divisible Rigid Groups. II. Stability, Saturation, and Elementary Submodels. / Miasnikov, Alexei; Romanovskii, N. S.

In: Algebra and Logic, Vol. 57, No. 1, 01.03.2018, p. 29-38.

Research output: Contribution to journalArticle

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