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**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.

(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 |