Vol. 40, No. 3, 1972

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The classification of certain classes of torsion free Abelian groups

Charles Estep Murley

Vol. 40 (1972), No. 3, 647–665

Let 𝒜 denote the class of torsion free Abelian groups of finite rank. It is shown that for A ∈𝒜, there is a quotient divisible subgroup QD(A) such that A∕QD(A) is a reduced torsion group. Furthermore, QD(A) and A∕QD(A) are unique up to quasi-isomorphism. Let denote the subclass of 𝒜 of groups A such that for almost all primes p, the p-primary component of A∕QD(A) is the direct sum of rp(A) isomorphic cyclic groups where rp(A) denotes the p-rank of A. The groups in are classified up to quasi-isomorphism, which generalizes the Beaumont-Pierce classification of quotient divisible groups.

The main results of this paper concern the subclass g of 𝒜 of groups A such that rp(A) 1 for all primes p. The class may be profitably treated as a generalization of the class of rank one groups in 𝒜.

Mathematical Subject Classification 2000
Primary: 20K15
Revised: 28 October 1971
Published: 1 March 1972
Charles Estep Murley