Abstract:
In the paper, representations of torsion-free Abelian groups of rank $2$ using torsion-free groups of rank $1$ are studied. Necessary and sufficient conditions are found under which a group given by such a representation is quotient divisible. A criterion is obtained for two $p$-minimal quotient divisible torsion-free groups of rank $2$ to be isomorphic to each other. An example is constructed showing that two such groups can be embedded in each other but be not isomorphic. A series of properties of fundamental systems of elements of quotient divisible groups of arbitrary finite rank is established.
Keywords:Abelian group, quotient divisible group, quotient divisible envelope, group of rank $2$.
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