Equivalence theorems for torsion-free groups. (English) Zbl 0801.20036

Hill, Paul; Megibben, Charles

Fuchs, Laszlo (ed.) et al., Abelian groups. Proceedings of the 1991 Curaçao conference. New York: Marcel Dekker, Inc.. Lect. Notes Pure Appl. Math. 146, 181-191 (1993).

Two pure subgroups \(H\) and \(H'\) of an abelian group \(G\) are said to be equivalent if there is an automorphism \(\alpha\) of \(G\) such that \(\alpha(H) = H'\). Clearly then \(H \cong H'\) and \(G/H \cong G/H'\). J. Erdős [Publ. Math. 5, 172-184 (1957; Zbl 0078.016)] showed that these conditions are sufficient for the equivalence of \(H\) and \(H'\) when \(G\) is a free abelian group. In this paper, the authors extend Erdős’ theorem to the case when \(G\) is a completely decomposable group, the isomorphism \(G/H \to G/H'\) respects heights (computed in \(G\)) and \(H\), \(H'\) satisfy an additional hypothesis that, for each type \(t\), both \(H(t) +G^*(t)\) and \(H'(t) + G^*(t)\) are pure in \(G\) and the ranks of \(H(t) / (H\cap G^*(t))\) and \(H'(t)/(H' \cap G^*(t))\) are equal.
For the entire collection see [Zbl 0778.00023].

Reviewer: K.M.Rangaswamy (Colorado Springs)

MSC:

20K20	Torsion-free groups, infinite rank
20K30	Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
20K27	Subgroups of abelian groups
20K15	Torsion-free groups, finite rank
20K25	Direct sums, direct products, etc. for abelian groups

Keywords:

equivalence theorems; pure subgroups; automorphism; completely decomposable group; isomorphism; heights; type

Citations:

Zbl 0078.016