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Valuated Butler groups of finite rank

Part of: Abelian groups

Published online by Cambridge University Press:  09 April 2009

L. Fuchs  and
K. M. Rangaswamy
L. Fuchs
Affiliation:
Department of Mathematics, Tulane University, New Orleans, Louisiana 70118, USA, e-mail: fuchs@tulane.edu
K. M. Rangaswamy
Affiliation:
Department of Mathematics, University of Colorado, Colorado Springs, Colorado 80933, USA, e-mail: ranga@math.uccs.edu
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Abstract

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Valuated Butler groups of finite rank are investigated. The valuated B2-groups are both epic images and pure subgroups of completely decomposable valuated groups of finite rank (Theorem 3.1). However, there are fundamental changes in the theory of Butler groups when valuations are involved. We introduce valuated B1-groups and show that they are valuated B2-groups. Surprisingly, valuated B2-groups of rank greater than 1 need not be valuated B1 -groups, unless they carry a special kind valuation, see Theorem 7.1. Theorem 6.5 gives a full characterization of valuated B1 -groups of finite rank, generalizing Bican's characterization of Butler groups.

Keywords

valuated groupButler groupcompletely decomposable valuated groupvaluated

MSC classification

Secondary: 20K15: Torsion-free groups, finite rank
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 80 , Issue 3 , June 2006 , pp. 335 - 350
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Arnold, D. M., Abelian groups and representations of finite partially ordered sets, CMS Books in Mathematics/Ouvrages de Mathematiques de la SMC, (Springer, New York, 2000).CrossRefGoogle Scholar
[2]Arnold, D. M., Hunter, R. and Richman, F. ‘Direct sums of cyclic valuated groups’, Sympos. Math. 23 (1979), 77–84.Google Scholar
[3]Arnold, D. M. and Richman, F., ‘Subgroups of finite direct sums of valuated cyclic groups’,J.Algebra 114 (1998),1–15CrossRefGoogle Scholar
[4]Bican, L., ‘Purely finitely generated abelian groups’, Comment. Math. Univ. Carolin. 21 (1980), 209–218.Google Scholar
[5]Bican, L. and Salce, L., ‘Butler groups of infinite rank’, in: Abelian group theory. Lecture Notes Math. 1006 (Springer, 1983) pp. 171–189.CrossRefGoogle Scholar
[6]Butler, M., ‘A class of torsion-free abelian groups of finite rank’, Proc. London Math. Soc. 15 (1965), 680–698.Google Scholar
[7]Fuchs, L., Infinite abelian groups, vol. 1 and 2 (Academic Press, New York, 1970 and 1973).Google Scholar
[8]Fuchs, L. and Viljoen, G., ‘Valuated Butler groups of special type’, Czechoslovak Math. J. 49 (1999), 507–516.Google Scholar
[9]Richman, F. and Walker, E., ‘Ext in pre-abelian categories’, Pacific J. Math. 71 (1977), 521–535.Google Scholar