zbMATH — the first resource for mathematics

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.

20K15 Torsion-free groups, finite rank
20K35 Extensions of abelian groups
PDF BibTeX Cite