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Coproducts of rigid groups
N. S. Romanovskiiab a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
b Novosibirsk State University, Novosibirsk, Russia
Abstract:
Let
$\varepsilon=(\varepsilon_1,\dots,\varepsilon_m)$ be a tuple consisting of zeros and ones. Suppose that a group
$G$ has a normal series of the form
$$
G=G_1\ge G_2\ge\dots\ge G_m\ge G_{m+1}=1,
$$
in which
$G_i>G_{i+1}$ for
$\varepsilon_i=1$,
$G_i=G_{i+1}$ for
$\varepsilon_i=0$, and all factors
$G_i/G_{i+1}$ of the series are Abelian and are torsion free as right
$\mathbb Z[G/G_i]$-modules. Such a series, if it exists, is defined by the group
$G$ and by the tuple
$\varepsilon$ uniquely. We call
$G$ with the specified series a rigid
$m$-graded group with grading
$\varepsilon$. In a free solvable group of derived length
$m$, the above-formulated condition is satisfied by a series of derived subgroups. We define the concept of a morphism of rigid
$m$-graded groups.
It is proved that the category of rigid
$m$-graded groups contains coproducts, and we show how to construct a coproduct
$G\circ H$ of two given rigid
$m$-graded groups. Also it is stated that if
$G$ is a rigid
$m$-graded group with grading
$(1,1,\dots,1)$, and
$F$ is a free solvable group of derived length
$m$ with basis
$\{x_1,\dots,x_n\}$, then
$G\circ F$ is the coordinate group of an affine space
$G^n$ in variables
$x_1,\dots,x_n$ and this space is irreducible in the Zariski topology.
Keywords:
rigid $m$-graded group, coproduct, coordinate group of affine space, Zariski topology.
UDC:
512.5
Received: 02.08.2010
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