On decomposable pseudofree groups. (Q1587822)

The author considers so-called pseudofree Abelian groups of finite rank. The notions and methods used were partly motivated by results on the ``Mislin genus'' and the ``extended genus'' of (finitely generated) nilpotent groups. The ``extended genus'' of a (Abelian) group \(A\), \(\mathcal{EG}(A)\), consists of all groups \(B\), respectively of all corresponding isomorphism classes, such that the localizations \(A_{(p)},B_{(p)}\) are isomorphic for every prime \(p\). It was shown by \textit{C. Casacuberta} and \textit{P. Hilton} [Bull. Soc. Math. Belg., Sér. A 41, No. 1, 51-72 (1989; Zbl 0672.20024)] that in order to study the extended genus of any finitely generated Abelian group it is enough to consider \(\mathcal{EG}(\mathbb{Z}^l)\) for positive integers \(l\); \(A\in\mathcal{EG}(\mathbb{Z}^l)\) is called ``pseudofree'' of rank \(l\). With any such pseudofree group \(A\) one can associate a sequence of matrices \(M_p\) (\(p\) prime), depending on isomorphisms \(f_p\colon A_{(p)}\to\mathbb{Z}_{(p)}^l\), called a ``sequential representation'' of \(A\). Using an isomorphism criterion for pseudofree groups in terms of their sequential representations the author provides a new proof of the well-known result (due to Baer) that any two decompositions of a completely decomposable group into rank-1 groups are isomorphic, although the proof is restricted to the class of pseudofree completely decomposable groups of finite rank. Also, using these sequential representations, examples of pseudofree groups with non-isomorphic decompositions into indecomposable subgroups are given. Moreover, a criterion for pseudofree groups of finite rank to be nearly isomorphic is proved, again in terms of their sequential representations.