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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