Almost completely decomposable torsion free abelian groups

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 ⊕ L ≈ H ⊕ L for some finite rank torsion free abelian group L L . (ii) i ( G ) = i ( H ) i(G) = i(H)