L.S. Pontryagin [1], A.G. Kurosh [2], A.I. Mal'cev [3], D. Derry
[4], R. Baer [5], R. Beaumont and R. Pierce [6,7] began research
on abelian torsion free groups of finite rank. In particular, R.
Beaumont and R. Pierce [6] introduced the notion of the quotient
divisible torsion free group. The notion of quotient divisible
group was extended to the case of mixed groups in the work [8]. It
was also proved in [8] that the category of quotient divisible
mixed groups with quasi-homomorphisms was dual to the category of
torsion-free finite-rank groups with quasi-homomorphisms. A
modern version of the duality [8] was obtained in [9, 10]. The
categories of groups with quasi-homomorphisms were replaced by the
categories of groups with marked bases and with usual
homomorphisms such that their matrices with respect to the marked
bases consisted of integers. The duality [8] was also extended by
S. Breaz and P. Schultz [11] on the class of self-small groups.
The mixed quotient divisible groups as well as the self-small
groups are in the focus of attention now [12-35].In the present paper we prove two theorems about homogeneous
completely decomposable quotient divisible mixed groups. In the
first theorem we show that for every basis of such group there
exists a decomposition of this group into a direct sum of rank-1
groups such that the elements of the basis are the bases of the
corresponding rank-1 groups. Moreover, for every two bases such
decompositions are isomorphic. In the second theorem we show that
every exact sequence of quotient divisible groups $0\rightarrow
B\rightarrow A\rightarrow C\rightarrow 0$ is splitting, if the
group $A$ is homogeneous completely decomposable. This theorem is
a dual version of the following classic result by R. Baer. Every
pure subgroup of a homogeneous completely decomposable torsion
free group of finite rank is a direct summand of this group.
