Almost completely decomposable torsion free abelian groups

Abstract

A finite rank torsion free abelian group G G is almost completely decomposable if there exists a completely decomposable subgroup C C with finite index in G G . The minimum of [ G : C ] [G:C] over all completely decomposable subgroups C C of G G is denoted by i ( G ) i(G) . An almost completely decomposable group G G has, up to isomorphism, only finitely many summands. If i ( G ) i(G) is a prime power, then the rank 1 summands in any decomposition of G G as a direct sum of indecomposable groups are uniquely determined. If G G and H H are almost completely decomposable groups, then the following statements are equivalent: (i) G ⊕ L ≈ H ⊕ L G \oplus L \approx H \oplus L for some finite rank torsion free abelian group L L . (ii) i ( G ) = i ( H ) i(G) = i(H) and H H contains a subgroup G ′ G’ isomorphic to G G such that [ H : G ′ ] [H:G’] is finite and prime to i ( G ) i(G) . (iii) G ⊕ L ≈ H ⊕ L G \oplus L \approx H \oplus L where L L is isomorphic to a completely decomposable subgroup with finite index in G G .