Universal theories for rigid soluble groups

A group is said to be p-rigid, where p is a natural number, if it has a normal series of the form G = G ₁ > G ₂ > ... > G _p > G _p+1 = 1, whose quotients G _i/G _i+1 are Abelian and are torsion free when treated as ℤ[G/G _i]-modules. Examples of rigid groups are free soluble groups. We point out a recursive system of universal axioms distinguishing p-rigid groups in the class of p-soluble groups. It is proved that if F is a free p-soluble group, G is an arbitrary p-rigid group, and W is an iterated wreath product of p infinite cyclic groups, then ∀-theories for these groups satisfy the inclusions A(F) ⊇ A(G) ⊇ A(W). We construct an ∃-axiom distinguishing among p-rigid groups those that are universally equivalent to W. An arbitrary p-rigid group embeds in a divisible decomposed p-rigid group M = M(α ₁,..., α _p). The latter group factors into a semidirect product of Abelian groups A ₁A ₂... A _p, in which case every quotient M _i/M _i+1 of its rigid series is isomorphic to A _i and is a divisible module of rank α _i over a ring ℤ[M/M _i]. We specify a recursive system of axioms distinguishing among M-groups those that are Muniversally equivalent to M. As a consequence, it is stated that the universal theory of M with constants in M is decidable. By contrast, the universal theory of W with constants is undecidable.