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Showing papers on "Torsion-free abelian group published in 1976"


Journal ArticleDOI
01 Jan 1976
TL;DR: In this paper, the authors defined a special group to be a strongly homogeneous group such that G/pG Z/pZ for some prime p and qG = G for all primes q =# p and characterized these groups as the additive groups of certain valuation rings in algebraic number fields.
Abstract: An abelian group is strongly homogeneous if for any two pure rank 1 subgroups there is an automorphism sending one onto the other. Finite rank torsion free strongly homogeneous groups are characterized as the tensor product of certain subrings of algebraic number fields with finite direct sums of isomorphic subgroups of Q, the additive group of rationals. If G is a finite direct sum of finite rank torsion free strongly homogeneous groups, then any two decompositions of G into a direct sum of indecomposable subgroups are equivalent. D. K. Harrison, in an unpublished note, defined a p-special group to be a strongly homogeneous group such that G/pG Z/pZ for some prime p and qG = G for all primes q =# p and characterized these groups as the additive groups of certain valuation rings in algebraic number fields. Richman [7] provided a global version of this result. Call G special if G is strongly homogeneous, G/pG = 0 or Z/pZ for all primes p, and G contains a pure rank 1 subgroup isomorphic to a subring of Q. Special groups are then characterized as additive subgroups of the intersection of certain valuation rings in an algebraic number field (also see Murley [5]). Strongly homogeneous groups of rank 2 are characterized in [2]. All of the above-mentioned characterizations can be derived from the more general (notation and terminology are as in Fuchs [3]): THEOREM 1. Let G be a torsion free abelian group of finite rank. Then G is strongly homogeneous iff G R 0 z H where H is a finite direct sum of isomorphic torsion free abelian groups of rank 1, R is a subring of an algebraic number field K (with IK E R), and every element of R is an integral multiple of a unit in R. PROOF. (e) First of all, the additive group of R, denoted by R +, is strongly homogeneous. Let X and Y be pure rank 1 subgroups of R +. There are units u and v of R in X and Y, respectively. Left multiplication by vu 1 induces an automorphism g of R + with g(X) = Y. Secondly, R 0zA is strongly homogeneous, where A is a torsion free abelian group of rank 1 with H Sk= 1 0 A. Choose 0 7# a e A and define 8: R + R 0z A by 8(r) = r 0 a. Then S is a monomorphism. LetX and Y Received by the editors January 20, 1975. AMS (MOS) subject classifications (1970). Primary 20K15.

23 citations