## 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

### MSC:

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