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
torsion-free finite-rank 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 rank-1 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-$1$ torsion-free 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 $\mathcal{S}$.
