Divisible Rigid Groups. Algebraic Closedness and Elementary Theory

Abstract

A group G is said to be rigid if it contains a normal series G = G₁ > G₂ > … > Gm > 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. We prove two theorems. Theorem 1 says that the following three conditions for a group G are equivalent: G is algebraically closed in the class Σ_m of all m-rigid groups; G is existentially closed in the class Σ_m; G is a divisible m-rigid group. Theorem 2 states that the elementary theory of a class of divisible m-rigid groups is complete.