J. Aust. Math. Soc.  80 (2006), 335-350
Valuated Butler groups of finite rank

L. Fuchs
  Department of Mathematics
  Tulane University
  New Orleans
  Louisiana 70118
  USA
  fuchs@tulane.edu
and
K. M. Rangaswamy
  Department of Mathematics
  University of Colorado
  Colorado Springs
  Colorado 80933
  USA
  ranga@math.uccs.edu


Abstract
Valuated Butler groups of finite rank are investigated. The valuated $B_2$-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 $B_1$-groups and show that they are valuated $B_2$-groups. Surprisingly, valuated $B_2$-groups of rank greater than $1$ need not be valuated $B_1$-groups, unless they carry a special kind valuation, see Theorem 7.1. Theorem 6.5 gives a full characterization of valuated $B_1$ -groups of finite rank, generalizing Bican's characterization of Butler groups.
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