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Divisible Rigid Groups. III. Homogeneity and Quantifier Elimination
Algebra i logika, Tome 57 (2018) no. 6, pp. 733-748 Cet article a éte moissonné depuis la source Math-Net.Ru

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A group $G$ is said to be rigid if it contains a normal series $$ G=G_1>G_2>\dots>G_m>G_{m+1}=1, $$ whose quotients $G_i/G_{i+1}$ are Abelian and, treated as right $\mathbb{Z}[G/G_i]$-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 $\mathbb{Z}[G/G_i]$. Every rigid group is embedded in a divisible one. THEOREM. Let $G$ be a divisible rigid group. Then the coincedence of $\exists$-types of same-length tuples of elements of the group $G$ implies that these tuples are conjugate via an authomorphism of $G$. As corollaries we state that divisible rigid groups are strongly $\aleph_0$-homogeneous and that the theory of divisible $m$-rigid groups admits quantifier elimination down to a Boolean combination of $\exists$-formulas.
Mots-clés : rigid group, divisible group, quantifier elimination.
Keywords: strongly в„μ0-homogeneous group 
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     author = {N. S. Romanovskii},
     title = {Divisible {Rigid} {Groups.} {III.} {Homogeneity} and {Quantifier} {Elimination}},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2018_57_6_a5/}
}
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N. S. Romanovskii. Divisible Rigid Groups. III. Homogeneity and Quantifier Elimination. Algebra i logika, Tome 57 (2018) no. 6, pp. 733-748. http://geodesic.mathdoc.fr/item/AL_2018_57_6_a5/

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