Generalized Rigid Groups: Definitions, Basic Properties, and Problems

We find a natural generalization of the concept of rigid group. The generalized rigid groups are also called r-groups. The terms of the corresponding rigid series of every r-group can be characterized by both ∃-formulas and ∀-formulas. We find a recursive system of axioms for the class of r-groups of fixed solubility length. We define divisible r-groups and give an appropriate system of axioms. Several fundamental problems are stated.