### Abstract

A group G is said to be rigid if it contains a normal series G = G_{1}G_{2}G_{m}G_{m+1} = 1, whose quotients G_{i}/G_{i+1} are Abelian and, treated as right ℤ[G/G_{i}]- modules, are torsion-free. A rigid group G is divisible if elements of the quotient G_{i}/G_{i+1} are divisible by nonzero elements of the ring ℤ[G/G_{i}]. 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.

Original language | English (US) |
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Pages (from-to) | 29-38 |

Number of pages | 10 |

Journal | Algebra and Logic |

Volume | 57 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 2018 |

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*Algebra and Logic*,

*57*(1), 29-38. https://doi.org/10.1007/s10469-018-9476-7

}

*Algebra and Logic*, vol. 57, no. 1, pp. 29-38. https://doi.org/10.1007/s10469-018-9476-7

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

Research output: Contribution to journal › Article

TY - JOUR

T1 - Divisible Rigid Groups. II. Stability, Saturation, and Elementary Submodels

AU - Miasnikov, Alexei

AU - Romanovskii, N. S.

PY - 2018/3/1

Y1 - 2018/3/1

N2 - 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.

AB - 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.

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UR - http://www.scopus.com/inward/citedby.url?scp=85047136866&partnerID=8YFLogxK

U2 - https://doi.org/10.1007/s10469-018-9476-7

DO - https://doi.org/10.1007/s10469-018-9476-7

M3 - Article

VL - 57

SP - 29

EP - 38

JO - Algebra and Logic

JF - Algebra and Logic

SN - 0002-5232

IS - 1

ER -