# Chapter 14 Free Solvable Groups

Identifiers
DOI: 10.1016/s0049-237x(08)70551-2
Disciplines
• Logic

## Abstract

Publisher Summary This chapter presents the derivation of some properties of free solvable groups, and based on these it is shown that the elementary theory of any free solvable non-commutative group is not recursively decidable. When considering classes of groups, it is assumed that the signature consists of predicate symbols for multiplication and inversion, although operation notation for conciseness is used in the chapter. With groups or classes of groups with fixed elements, the individual constants for the chosen fixed elements appear in the signatures of the corresponding models. Subgroups that are elementary in classes of groups whose signatures have no individual constants are characteristic subgroups, and the search for them appears interesting not only for this or that axiomatizable class, but also for the more important individual groups.

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