This article is cited in
3 papers
Rigid metabelian pro-$p$-groups
S. G. Afanas'evaa,
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:
A metabelian pro-
$p$-group
$G$ is rigid if it has 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
$C=G_2$ is torsion-free as a
$\mathbb Z_pA$-module. If
$G$ is a non-Abelian group, then the subgroup
$G_2$, as well as the given series, is uniquely defined by the properties mentioned. An Abelian pro-
$p$-group is rigid if it is torsion-free, and as
$G_2$ we can take either the trivial subgroup or the entire group. We prove that all rigid
$2$-step solvable
pro-
$p$-groups are mutually universally equivalent.
Rigid metabelian pro-
$p$-groups can be treated as
$2$-graded groups with possible gradings
$(1,1)$,
$(1,0)$, and
$(0,1)$. If a group is
$2$-step solvable, then its grading is
$(1,1)$. For an Abelian group, there are two options: namely, grading
$(1,0)$, if
$G_2=1$, and grading
$(0,1)$ if
$G_2=G$. A morphism between
$2$-graded rigid pro-
$p$-groups is a homomorphism
$\varphi\colon G\to H$ such that
$G_i\varphi\le H_i$.
It is shown that in the category of
$2$-graded rigid pro-
$p$-groups, a coproduct operation exists, and we establish its properties.
Keywords:
rigid metabelian pro-$p$-group, $2$-graded group.
UDC:
512.5
Received: 13.12.2013