Abstract
Previously, the author defined the concept of a rigid (abstract) group. By analogy, a metabelian pro-p-group G is said to be rigid if it contains a normal series of the form G = G1 ≥ G2 ≥ G3 = 1 such that the factor group A = G/G2 is torsion-free Abelian, and G2 being a ZpA-module is torsion-free. An abstract rigid group can be completed and made divisible. Here we do something similar for finitely generated rigid metabelian pro-p-groups. In so doing, we need to exit the class of pro-p-groups, since even the completion of a torsion-free nontrivial Abelian pro-p-group is not a pro-p-group. In order to not complicate the situation, we do not complete a first factor, i.e., the group A. Indeed, A is simply structured: it is isomorphic to a direct sum of copies of Zp. A second factor, i.e., the group G2, is completed to a vector space over a field of fractions of a ring ZpA, in which case the field and the space are endowed with suitable topologies. The main result is giving a description of coordinate groups of irreducible algebraic sets over such a partially divisible topological group.
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Supported by RFBR, project No. 15-01-01485.
Translated from Algebra i Logika, Vol. 55, No. 5, pp. 571-586, September-October, 2016.
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Romanovskii, N.S. Partially Divisible Completions of Rigid Metabelian Pro-p-groups. Algebra Logic 55, 376–386 (2016). https://doi.org/10.1007/s10469-016-9409-2
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DOI: https://doi.org/10.1007/s10469-016-9409-2