×

On the structure of Abelian p-groups. (English) Zbl 0573.20053

Let \(\mu\) be a limit ordinal, and the class \(A_{\mu}\) consists of those p-groups H for which there is a containing totally projective p-group G of length not exceeding \(\mu\) that satisfies the following conditions:
(a) H is isotype in G.
(b) \(p^{\lambda}(G/H)=<p^{\lambda}G,H>/H\) whenever \(\lambda <\mu.\)
(c) G/H is the direct sum of a totally projective group and a divisible group.
The members of the class \(A_{\mu}\) are called \(\mu\)-elementary A- groups. An A-group is a direct sum of \(\mu\)-elementary A-groups for various limit ordinals \(\mu\).
The author finds the invariants of A-groups (A-invariants) including the Ulm-Kaplansky invariants and proves a uniqueness theorem (theorem 3) and an existence theorem (theorem 4). Besides the author obtains the following results:
Theorem 1. Let \(\mu\) denote an arbitrary limit ordinal. The class \(A_{\mu}\) consists exclusively of totally projective groups if and only if \(\mu\) is cofinal with \(\omega\).
Theorem 7. If H is an A-group, then \(H=T\oplus K\), where K is an A-group and T is totally projective and has the same Ulm-Kaplansky invariants as H.
Theorem 9. Let H be an arbitrary reduced p-group and \(\alpha\) an arbitrary ordinal. Then H is an A-group if and only if both \(p^{\alpha}H\) and \(H/p^{\alpha}H\) are A-groups.
Theorem 11. Let \({\mathcal C}\) be a class of reduced p-groups closed with respect to direct sums (and such that membership is independent of notation). Suppose that the A-invariants determine the structure of all the members of \({\mathcal C}\). If \({\mathcal C}\) contains the A-groups then it is exactly the class of A-groups. Moreover the author obtains some other properties of A-groups.
Reviewer: A.M.Sebel’din

MSC:

20K10 Torsion groups, primary groups and generalized primary groups
20K21 Mixed groups
20K25 Direct sums, direct products, etc. for abelian groups
20K27 Subgroups of abelian groups
20K99 Abelian groups
PDFBibTeX XMLCite
Full Text: DOI