Balanced Subgroups of Finite Rank Completely Decomposable Abelian Groups

Abstract

ABSTRACT. It is proved that, if a finite rank completely decomposable group has extractable typeset of cardinality at most 5, all its balanced subgroups are also completely decomposable. Balanced Butler groups with extractable typeset of size at most 3 are almost completely decomposable and decompose into rank 1 and/or rank 3 indecomposable summands. We also construct an indecomposable balanced Butler group whose extractable typeset is of size 4 which fails to be almost completely decomposable.In this note we wish to establish a few results on the structure of balanced subgroups of finite rank completely decomposable groups. There are a number of similarities between these results and some published results on pure subgroups of finite rank completely decomposable groups and we employ some techniques developed by Butler in [4] and Arnold in [1]. Specifically, we show that if a finite rank completely decomposable group has an extractable typeset of cardinality at most five, then all its balanced subgroups are completely decomposable. In [6] we showed that there exists a finite rank completely decomposable group with extractable typeset of size six which contains an indecomposable balanced subgroup. We also showed that if H is a balanced subgroup of a finite rank completely decomposable group and the extractable typeset of H contains at most two elements, then H is completely decomposable. We give an alternative proof of this theorem. We also show that if H is balanced in a finite rank completely decomposable group and the extractable typeset of H is of size three, then H is almost completely decomposable and decomposes into rank 1 and/or rank 3 indecomposable summands. We construct an indecomposable balanced subgroup H of a finite rank completely decomposable group, such that the extractable typset of H is of size 4 and H fails to be almost completely decomposable.All the groups we consider here are assumed to be abelian and for general notation, terminology, and results we refer the reader to [5].Let G be a torsion-free group and let 9 E G. We shall denote by Xc(g), the height-sequence or characteristic of 9 in G and typeC (g) will mean the type of 9 in G.T(G) = {typec(g): 0"# 9 E G} will be called the typeset of G. If 8 = {Xl, X2, ... , Xn} is a finite set of heightsequences then inf(8) and sup(8) are height-sequences given by component minimums and maximums of the height-sequences in 8. If 8' = {71' 72, ... ,7 n} is a finite set of types with Xi E 7i, 1 :::; i :::; n, then inf(8') and sup(8') are the types