
Quotient divisible groups and torsionfree groups corresponding to finite Abelian groups
E. I. Kompantseva^{ab}, A. A. Fomin^{a} ^{a} Moscow State Pedagogical University (Moscow)
^{b} Financial University under the Government of the Russian Federation
(Moscow)
Abstract:
The category of sequences $\mathcal{S}$ has been introduced in [1,
2, 3]. Objects of the category $\mathcal{S}$ are finite sequences
of the form $a_{1},\ldots,a_{n}$, where the elements
$a_{1},\ldots,a_{n}$ belong to a finitely presented module over
the ring of polyadic numbers $\widehat{{Z}}$. The ring of polyadic
numbers $\widehat{{Z}}=\prod\limits_{p}{\widehat{Z}}_{p}$ is the
product of the rings of $p$adic integers over all prime numbers
$p$. Morphisms of the category $\mathcal{S}$ from the object
$a_{1},\ldots,a_{n}$ to an object $b_{1},\ldots,b_{k}$ are all
possible pairs $(\varphi, T),$ where $\varphi: \langle
a_{1},\ldots,a_{n}\rangle_{\widehat{{Z}}} \rightarrow \langle
b_{1},\ldots,b_{k}\rangle_{\widehat{{Z}}}$ is a homomorphism of
$\widehat{{Z}}$modules, generated by given elements, and $T$ is a
matrix of dimension $k\times n$ with integer entries such that the
following matrix equality takes place
$$(\varphi a_{1},\ldots,\varphi a_{n})=(b_{1},\ldots,b_{k})T.$$
It is proved in [2] that the category $\mathcal{S}$ is equivalent
to the category $\mathcal{D}$ of mixed quotient divisible abelian
groups with marked bases. It is proved in [3] that the category
$\mathcal{S}$ is dual to the category $\mathcal{F}$ of
torsionfree finiterank abelian groups with marked bases, a basis
means here a maximal linearly independent set of elements. The
composition of these equivalence and duality is the duality
introduced in [1] and in [4], which can be considered as a version
of the duality introduced in [5].
If an object of the category $\mathcal{S}$ consists of one
element, then it corresponds to rank1 groups of the categories
$\mathcal{\mathcal{D}}$ and $\mathcal{F}$. This case is considered
in [6] and we obtain the following. The duality
$\mathcal{S}\leftrightarrow\mathcal{F}$ gives us the classical
description by R. Baer [7] of rank$1$ torsionfree groups. The
equivalence $\mathcal{S}\leftrightarrow\mathcal{D}$ coincides with
the description by O.I. Davydova [8] of rank$1$ quotient divisible
groups.
We consider another marginal case in the present paper. Every
torsion abelian group can be considered as a module over the ring
of polyadic numbers. Moreover, a torsion group is a finitely
presented $\widehat{{Z}}$module if and only if it is finite.
Thus, for every set of generators $g_{1},\ldots,g_{n}$ of every
finite abelian group $G$ the sequence $g_{1},\ldots,g_{n}$ is an
object of the category $\mathcal{S}$. Such objects determine a
complete subcategory of the category $\mathcal{S}$.
We show in the present paper that the object $g_{1},\ldots,g_{n}$
of the category $\mathcal{S}$ corresponds to an object of the
category $\mathcal{D}$, which is of the form $G\oplus Q^{n}$ with
the marked basis $g_{1}+e_{1},\ldots,g_{n}+e_{n}$, where
$e_{1},\ldots,e_{n}$ is the standard basis of the vector space
$Q^{n}$ over the field of rational numbers $Q$. The same object
$g_{1},\ldots,g_{n}$ corresponds to an object of the category
$\mathcal{F}$, which is a free group $A$, satisfying the
conditions $Z^{n}\subset A\subset Q^{n}$ and $A/Z^{n}\cong
G^{\ast}$, where $ G^{\ast}=Hom(G,Q/Z)$ is the dual finite group.
We consider also the group homomorphisms corresponding to
morphisms of the category $\mathcal{S}$.
Keywords:
abelian groups, modules, dual categories.
DOI:
https://doi.org/10.22405/222683832018202221233
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UDC:
517 Received: 13.02.2019 Accepted:12.07.2019
Citation:
E. I. Kompantseva, A. A. Fomin, “Quotient divisible groups and torsionfree groups corresponding to finite Abelian groups”, Chebyshevskii Sb., 20:2 (2019), 221–233
Citation in format AMSBIB
\Bibitem{KomFom19}
\by E.~I.~Kompantseva, A.~A.~Fomin
\paper Quotient divisible groups and torsionfree groups corresponding to finite Abelian groups
\jour Chebyshevskii Sb.
\yr 2019
\vol 20
\issue 2
\pages 221233
\mathnet{http://mi.mathnet.ru/cheb765}
\crossref{https://doi.org/10.22405/222683832018202221233}
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