The classification of N-groups. (English) Zbl 0545.20042

Let G be a direct sum of countable abelian groups (d.s.c.). We call a subgroup H an N-group if it satisfies the conditions: (i) H is isotype in G. (ii) \(p^{\alpha}(G/H)=<p^{\alpha}G,H>/H\) if \(\alpha<\Omega,and p^{\Omega +1}(G/H)=0.\) (iii) G/H is totally projective.
The following are the main results: Theorem 1. An N-group is determined by its Ulm invariants together with its \(\Omega\)-number. Theorem 2. There exists an N-group H with \(\Omega\)-number \({\mathcal M}\) that satisfies, for each \(\alpha\), \(f_{\alpha}(H)=f(\alpha)\) if and only if \(f(\alpha)\) is an admissible function of length not exceeding \(\Omega\) such that \(\sum_{\alpha>\lambda}f(\alpha)\leq {\mathcal M}\) for all countable \(\lambda\). Theorem 3. A summand of an N-group is an N-group. Theorem 4. If A and B are N-groups, then Tor(A,B) is a d.s.c.
Reviewer: A.M.Sebel’din


20K25 Direct sums, direct products, etc. for abelian groups
20K99 Abelian groups
20K27 Subgroups of abelian groups