Partially divisible completions of rigid metabelian pro-$p$-groups
N. S. Romanovskiiab a Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090 Russia
b Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090 Russia
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
$G=G_1\ge G_2\ge G_3=1$ such that the factor group
$A=G/G_2$ is torsion-free Abelian, and
$G_2$ being a
$\mathbb Z_pA$-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
$\mathbb Z_p$. A second factor, i.e., the group
$G_2$, is completed to a vector space over a field of fractions of a ring
$\mathbb Z_pA$, 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.
Keywords:
abstract rigid group, divisible group, coordinate group, irreducible algebraic set.
UDC:
512.5
Received: 05.03.2016
DOI:
10.17377/alglog.2016.55.504
Fulltext:
PDF file (180 kB)
