On Torsion-Free Abelian k-Groups (original) (raw)

It is shown that a knice subgroup with cardinality Ni, of a torsion-free completely decomposable abelian group, is again completely decomposable. Any torsion-free abelian fc-group of cardinality H" has balanced projective dimension < n. Introduction. Recently, Hill and Megibben introduced the concept of a knice subgroup in their study of abelian fc-groups [6] and also while considering the local Warfield groups in [5]. In this paper, we introduce a modified definition of a knice subgroup of a torsion-free abelian group. This helps us to extend the results of Hill and Megibben [6] and also simplify the proofs of their main theorems. Specifically we show that a knice subgroup with cardinality < Ni, of a torsion-free completely decomposable abelian group, is again completely decomposable. This enables us to prove that any torsion-free abelian fc-group (in particular, a separable group) of cardinality < Nn has balanced projective dimension < n. All the groups that we consider here are torsion-free and abelian. We generally follow the notation and terminology of L. Fuchs [3]. Let P denote the set of all primes. By a height sequence we mean a sequence s = (sp), p G P, where each 5p is a nonnegative integer or the symbol oo. If G is a torsion-free group and x G G, then |x| denotes the height sequence of x where, for each p G P, \x\p denotes the height of x at the prime p. For any height sequence s = (sp), ps is the height sequence (tp), where tp-sp + 1 and tq = sq for all q / p. G(s) denotes the subgroup {x G G: \x\> s}. G(s*) is the subgroup generated by the set {x G G(s): J2pepi\x\p ~ sp) 's unbounded}. Two height sequences (sp) and (tp) are said to be equivalent if YlPep \sp ~ ¿p\ 18 finite.