Chapter 17 Constructive abelian groups

Abstract

This chapter focuses on exposition of basic results of the theory of Constructive Abelian Groups. Algorithmic problems in group theory arose before the appearance of a precise concept of an algorithm. At the beginning of the 1930's, this concept was made more precise, and the first results of the theory of algorithms were obtained. On the basis of these results, Novikov proved the undecidability of the word problem for groups. The chapter introduces two important operations over constructive groups. It shows that the existence problem of constructivizations for the class of abelian groups can be reduced to the same problem for the classes of periodic groups and torsion-free groups. The concept of a “p”-basis introduced by Kulikov has important meaning in the theory of abelian groups. A connection between the constructivizability of groups and the existence of a recursive “p”-basis has been discussed. From this connection, a criterion for the strong constructivizability of an abelian “p”-group is obtained. The chapter discusses model theoretic method in the theory of constructivizable abelian groups, connections between constructivizability and strong constructivizability, constructivizability of subgroups and factor groups, and the arithmetic hierarchy of abelian groups.