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. 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 ℵ 0 -homogeneous and that the theory of divisible m-rigid groups admits quantifier elimination down to a Boolean combination of ∃-formulas.
- divisible group
- quantifier elimination
- rigid group
- strongly ℵ -homogeneous group
- strongly (0)-homogeneous group