On the model theory of free metabelian groups of bounded exponent

Abstract

Publisher Summary Every stable solvable group of bounded exponent is nilpotent by finite. This can be proved by considering F μ (q, p), the free group with μ free generators. In the elementary theory of F μ with χ 0 ≤ μ , the chapter interprets the monadic second order theory of an infinite vectorspace over the field K P with p elements, where the second order variables range over finite subspaces. It is noted that for this interpretation G/H gives the desired vectorspace and the elements of H code the finite subspaces. One subspace is given by several points in H. Magnus-embedding is the main tool from algebra that can also be used for F μ (q,p) as proved by N. Blackburn. Using the methods of W. Baur and the undecidability of the universal theory of finite groups, the undecidability of the elementary theory Th (F μ ) of F μ (χ 0 ≤ μ) is obtained. Furthermore it follows that Th (F μ ) is unstable and has strict order and independence property.