Divisible Rigid Groups. III. Homogeneity and Quantifier Elimination
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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.
Keywordsrigid group divisible group strongly ℵ0-homogeneous group quantifier elimination
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- 5.Yu. L. Ershov and E. A. Palyutin, Mathematical Logic [in Russian], 6th edn., Fizmatlit, Moscow (2011).Google Scholar