DE VIVO, CLORINDA

METELLI, CLAUDIA

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Zentralblatt MATH Database 1931 – 2012c￿ 2012 European Mathematical Society, FIZ Karlsruhe & Springer-VerlagZbl 0867.20045de Vivo, Clorinda; Metelli, Claudia (Vivo, C. de)B(1)-groups: Some counterexamples. (English)Arnold, David M. (ed.) et al., Abelian groups and modules. Proceedings of the inter-national conference at Colorado Springs, CO, USA, August 7–12, 1995. New York, NY:Marcel Dekker. Lect. Notes Pure Appl. Math. 182, 227-232 (1996). ISBN 9-8247-9789-2/pbkA B(1)-group is a finite rank torsion-free abelian group that can be realized as thequotient of a finite direct sum of subgroups of the additive rationals factored by a rank-one subgroup. This class of groups has been well-studied, in particular by Fuchs andMetelli, who used 0-1 matrices to obtain a characterization. Out of their work arose aconjecture that B(1)-groups might in some sense be determined by their typesets. Thispaper produces some clever, hard-won examples that show the conjecture is false.The first example gives B(1)-groups G, H of rank five and having the same typeset, withG strongly indecomposable and H not. In the second example, G and H have rank 8and the same typeset, are both strongly indecomposable, but not quasi-isomorphic. Thethird example is similar, with groups of rank 11. These examples can also be used torefute various conjectures on B(1)-groups made by other authors.C.Vinsonhaler (Storrs)Keywords : strongly indecomposable groups; finite rank torsion-free Abelian groups;direct sums of subgroups of the additive rationals; B(1)-groups; typesetsClassification :∗20K15 Torsion free abelian groups, finite rank20K99 Abelian groups1

B(1)-groups: some counterexamples

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Archivio della ricerca - Università degli studi di Napoli Federico II