International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 3, Pages 617-628
On decomposable pseudofree groups
Centre de Recerca Matemàtica, Apartat 50, Bellaterra E-08193, Spain
Received 16 October 1997
Copyright © 1999 Dirk Scevenels. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
An Abelian group is pseudofree of rank if it belongs to the extended genus of , i.e., its localization at every prime is isomorphic to . A pseudofree group can be studied through a sequence of rational matrices, the so-called sequential representation. Here, we use these sequential representations to study the relation between the product of extended genera of free Abelian groups and the extended genus of their direct sum. In particular, using sequential representations, we give a new proof of a result by Baer, stating
that two direct sum decompositions into rank one groups of a completely decomposable pseudofree Abelian group are necessarily equivalent. On the other hand, sequential representations can also be used to exhibit examples of pseudofree groups having nonequivalent direct sum decompositions into indecomposable groups. However, since this cannot occur when using the notion of near-isomorphism rather than isomorphism, we conclude our work by giving a characterization of near-isomorphism for pseudofree groups in terms of their sequential representations.