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On pseudofree groups and sequential representations. (English) Zbl 0826.20051
The authors call an abelian group \(A\) pseudofree of rank \(k\) if for all primes \(p\) the \(p\)-localization \(A_p\) of \(A\) is isomorphic to the \(p\)- localization of \(\mathbb{Z}^k\). If \(A\) is pseudofree of rank \(k\) the natural embeddings \(f_0 : A \to \mathbb{Q}^k\) and \(f_p : A_p \to \mathbb{Q}^k\) can be used to describe \(A\). Fixing the isomorphism \(A_p \simeq \mathbb{Z}^k_p\) the map \(f_p\) can be given by a \(k \times k\)- matrix \(M_p\) with rational coefficients. The authors call the collection of the matrices \(M_p\) a sequential representation of \(A\).
An extension \(E\) of the pseudofree group \(A\) of rank \(k\) by the pseudofree group \(B\) of rank \(l\) is a pseudofree group of rank \(k + l\). Given sequential representations of \(A\) and \(B\) the authors construct a sequential representation of \(E\). Using this procedure an example of a pseudofree group of rank 2 is constructed which cannot be written as a direct sum of pseudofree groups of rank 1.

MSC:
20K15 Torsion-free groups, finite rank
20K35 Extensions of abelian groups
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