Showing papers on "Torsion-free abelian group published in 1974"

Almost completely decomposable torsion free abelian groups

Abstract: A finite rank torsion free abelian group G is almost completely decomposable if there exists a completely decomposable subgroup C with finite index in G The minimum of [G: C] over all completely decomposable subgroups C of G is denoted by i(G) An almost completely decomposable group G has, up to isomorphism, only finitely many summands If i(G) is a prime power, then the rank 1 summands in any decomposition of G as a direct sum of indecomposable groups are uniquely determined If G and H are almost completely decomposable groups, then the following statements are equivalent: (i) G Eb L t H Eb L for some finite rank torsion free abelian group L (ii) i(G) = i(H) and H contains a subgroup G' isomorphic to G such that [H: G ] is finite and prime to i(G) (iii) G ED L % H @ L where L is isomorphic to a completely decomposable subgroup with finite index in G A finite rank torsion free abelian group G is almost completely decomposable if there is a completely decomposable subgroup C having finite index in G It is well known that direct sum decompositions of such groups need not be unique In fact, this class of groups is the source of all the most familiar examples of nonunique decompositions of finite rank torsion free abelian groups This paper will show, however, that the situation is not completely unruly We show that in certain situations cancellation of direct summands is possible We show that a maximal completely decomposable summand is unique up to isomorphism We show ttat an almost completely decomposable group G has, up to isomorphism, only finite many summands We show that there are only finitely many groups H for which there exists a finite rank torsion free abelian group L such that G @ L ; H ( L Theorem 11 characterizes such groups H All groups in this paper, unless indicated otherwise, are finite rank torsion free abelian groups In general, we will follow the notation and convenPresented to the Society, February 2, 1973; received by the editors June 18, 1973 AMS (MOS) subject classifications (1970) Primary 20K15