Algebra and Logic

, Volume 57, Issue 6, pp 478–489

# Divisible Rigid Groups. III. Homogeneity and Quantifier Elimination

• N. S. Romanovskii
Article

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.

## Keywords

rigid group divisible group strongly ℵ0-homogeneous group quantifier elimination

## References

1. 1.
N. S. Romanovskii, “Divisible rigid groups. Algebraic closedness and elementary theory,” Algebra and Logic, 56, No. 5, 395-408 (2017).
2. 2.
A. G. Myasnikov and N. S. Romanovskii, “Divisible rigid groups. II. Stability, saturation, and elementary submodels,” Algebra and Logic, 57, No. 1, 29-38 (2018).
3. 3.
C. Perin and R. Sklinos, “Homogeneity in the free groups,” Duke Math. J., 161, No. 13, 2635-2668 (2012).
4. 4.
B. Poizat, A Course in Model Theory. An Introduction to Contemporary Mathematical Logic (Universitext), Springer, New York (2000).
5. 5.
Yu. L. Ershov and E. A. Palyutin, Mathematical Logic [in Russian], 6th edn., Fizmatlit, Moscow (2011).Google Scholar
6. 6.
A. G. Myasnikov and N. S. Romanovskii, “Universal theories for rigid soluble groups,” Algebra and Logic, 50, No. 6, 539-552 (2011).
7. 7.
N. S. Romanovskii, “Coproducts of rigid groups,” Algebra and Logic, 49, No. 6, 539-550 (2010).
8. 8.
N. S. Romanovskii, “Divisible rigid groups,” Algebra and Logic, 47, No. 6, 426-434 (2008).