On the Universal Theories of Generalized Rigid Metabelian Groups
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HE UNIVERSAL THEORIES OF GENERALIZED RIGID METABELIAN GROUPS N. S. Romanovskii
UDC 512.5; 510.6
Abstract: We prove that studying the universal theories of generalized rigid metabelian groups reduces to those of the pairs (A, R), where R is a commutative integral domain and A is a nontrivial torsion-free subgroup of the multiplicative group R∗ generating R. DOI: 10.1134/S0037446620050110 Keywords: solvable group, ring, universal theory
1. Introduction The author defined the generalized rigid groups (r-groups) and studied their general properties in [1], and began to study metabelian r-groups in [2, 3]. Recall the structure of an arbitrary length 2 solvable r-group G. It admits a unique normal series G = ρ1 (G) > ρ2 (G) > ρ3 (G) = 1, where the first quotient A = G/ρ2 (G) is an abelian group and either the order of A is some prime number p or A is torsion-free. The group ρ2 (G) amounts to the additive group of a right torsion-free module over a commutative integral domain R; furthermore, A embeds into the group R∗ of invertible elements and generates R as a ring, while the action of g ∈ G on ρ2 (G) by conjugation corresponds to the multiplication by the image of g in A in the module. The case that A is a cyclic group of order p is not too interesting, and here we consider the class R2 of solvable length 2 r-groups G for which G/ρ2 (G) = A is an (abelian) torsion-free group. In particular, this class includes solvable Baumslag–Solitar groups. We are interested in the universal theories of the groups of class R2 . Chapuis proved [4] in 1995 that the universal theory of the free metabelian group is solvable, and then characterized the groups with the same universal theory as the free metabelian group of rank ≥ 2 in [5]. In our terminology, these are precisely solvable rigid groups of length 2. As established in [6], the universal theory of the free solvable group of length ≥ 4 is not algorithmically solvable. As a perspective, we pose the problem of classifying the groups of class R2 according to universal properties and the question of algorithmic solvability of the universal theory of a finitely generated R2 -group. This article establishes that the universal theory of an arbitrary split group of class R2 is equivalent to the universal theory of the corresponding pair (A, R). 2. Statement and Proof of the Main Results 1. Let us discuss two general assertions. Theorem 3 of [1] points out a recursive system Λm of axioms for the class of length m solvable r-groups. With the exception of the ∃-axiom meaning that the solvability length of the group is at least m, the remaining axioms can be realized as ∀-formulas. This implies our first proposition: Proposition 1. Each group whose universal theory coincides with the universal theory of a length m solvable r-group G is itself a length m solvable r-group. The author was partially supported by the Mathematical Center in Akademgorodok under Agreement No. 075– 15–2019–1613 with the Ministry of Science and Higher Education of the Russian Federation. Original article submitted March
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