Divisible Rigid Groups. III. Homogeneity and Quantifier Elimination
- Authors: Romanovskii N.S.1,2
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Affiliations:
- Sobolev Institute of Mathematics
- Novosibirsk State University
- Issue: Vol 57, No 6 (2019)
- Pages: 478-489
- Section: Article
- URL: https://journal-vniispk.ru/0002-5232/article/view/234113
- DOI: https://doi.org/10.1007/s10469-019-09518-2
- ID: 234113
Cite item
Abstract
A group G is said to be rigid if it contains a normal series G = G1 > G2 > . . . > Gm > Gm+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. Our main result is the theorem which reads as follows. Let G be a divisible rigid group. Then the coincidence of ∃-types of same-length tuples of elements of the group G implies that these tuples are conjugate via an automorphism of G. As corollaries we state that divisible rigid groups are strongly ℵ0-homogeneous and that the theory of divisible m-rigid groups admits quantifier elimination down to a Boolean combination of ∃-formulas.
About the authors
N. S. Romanovskii
Sobolev Institute of Mathematics; Novosibirsk State University
Author for correspondence.
Email: rmnvski@math.nsc.ru
Russian Federation, pr. Akad. Koptyuga 4, Novosibirsk, 630090; ul. Pirogova 1, Novosibirsk, 630090
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