## Generalized Wallace theorems.(English)Zbl 1169.20029

In the paper Abelian $$p$$-groups are considered. A homomorphism $$f\colon G\to A$$ is said to be $$\omega_1$$-bijective if $$|\ker f|<\aleph_1$$ and $$|A/f(G)|<\aleph_1$$. In addition, the authors introduce the concepts $$\omega_1$$-$$*$$-group, $$k$$-$$Q$$-group and $$n$$-Honda group and prove that any totally projective group is an $$\omega_1$$-$$*$$-group.
The main results are the following. Let $$f\colon G\to A$$ be an $$\omega_1$$-bijective homomorphism. Then 1) if $$G$$ is simply presented, then $$A$$ is simply presented (Theorem 2.4); 2) if $$G$$ is an $$\omega_1$$-$$*$$-group, then $$A$$ is an $$\omega_1$$-$$*$$-group (Proposition 2.6); 3) if $$\alpha\leq\omega_1$$ and $$G$$ is a $$C_\alpha$$-group, then $$A$$ is also a $$C_\alpha$$-group (Theorem 3.5); 4) if $$G$$ and $$A$$ are separable groups, then $$G$$ is $$p^{\omega+n}$$-projective iff $$A$$ is $$p^{\omega+n}$$-projective (Theorem 4.2) and 5) if $$G$$ and $$A$$ are separable groups and $$k$$ is an uncountable cardinal, then $$G$$ is a $$k$$-$$Q$$-group iff $$A$$ is a $$k$$-$$Q$$-group (Theorem 5.1).
Besides, in the paper the following statements are proved: (a) if $$0<n<\omega$$, then a group $$A$$ is an $$n$$-$$\Sigma$$-group iff $$A$$ is a $$C_{\omega+n}$$-group (Proposition 3.2) and (b) if $$G$$ is an isotype subgroup of a reduced group $$A$$ and $$A/G$$ is countable, then $$G$$ is $$n$$-Honda iff $$A$$ is $$n$$-Honda (Proposition 6.7).
Remarks: Following Linton and Megibben, the authors give the following definition of a $$\sigma$$-summable group (page 46, line 1 after Corollary 6.2). “A group $$A$$ of length $$\lambda$$ is called $$\sigma$$-summable if $$A[p]=\bigcup_{i<\omega}A_i$$, where for all $$i<\omega$$, $$A_i\subseteq A_{i+1}$$ and $$A_i\cap p^{\alpha_i}A=0$$ for some $$\alpha_i<\lambda$$”.
We will make the following notes. (i) The statement that “a separable group is $$\sigma$$-summable iff it is a direct sum of cyclics” (page 46, line 15-16 from above) is not true. It is not hard to see (and the authors have not marked) the following: the definition of a $$\sigma$$-summable group $$A$$ implies that the length $$\lambda$$ of $$A$$ must be a limit ordinal. Therefore, if $$A$$ is a direct sum of cyclics and $$A$$ is bounded, then $$A$$ is not $$\sigma$$-summable which contradicts the above authors’ assertion.
(ii) The authors cite and use P. Hill’s criterion in the paper [P. Hill, Proc. Am. Math. Soc. 126, No. 11, 3133-3135 (1998; Zbl 0907.20048)] for $$\sigma$$-summable groups in non-equivalent form. Foremost, in the same article, Paul Hill gives the following definition of a $$\sigma$$-summable group. “A $$p$$-primary Abelian group $$G$$ is $$\sigma$$-summable if $$G[p]$$ is the union of an ascending sequence of subsocles $$S_n$$ having the property that the heights of the nonzero elements of $$S_n$$, computed in $$G$$, are bounded by some ordinal $$\lambda_n$$ less than the length $$\lambda$$ of $$G$$.”
The definitions of the authors and of Hill are not equivalent. The class of $$\sigma$$-summable groups, introduced from Hill, is larger, since it includes all direct sums of cyclic groups.
Besides, Paul Hill proves in his paper the following assertion. “Proposition 1. An Abelian $$p$$-group $$G$$ is $$\sigma$$-summable if and only if $$G$$ is the union of an ascending sequence of subgroups $$H_n$$ with the property that the heights of the nonzero elements of $$H_n$$, computed in $$G$$, are bounded by some ordinal $$\lambda_n$$ less than the length $$\lambda$$ of $$G$$.” – In Section 6 the authors make the following non-equivalent quotation (page 46) of Hill’s Proposition 1. “Hill’s criterion 6.3. ([14]). A group $$A$$ of length $$\lambda$$ is $$\sigma$$-summable iff $$A=\bigcup _{i<\omega}\Gamma_i$$, where for all $$i<\omega$$, $$\Gamma_i\subseteq\Gamma_{i+1}$$ and there is an ordinal $$\alpha_1<\lambda$$ such that $$\Gamma_i\cap p^{\alpha_i}A=\{0\}$$.” – The cited “Hill’s criterion 6.3” is weaker than Hill’s Proposition 1, since it holds only for limit ordinals.

### MSC:

 20K10 Torsion groups, primary groups and generalized primary groups

Zbl 0907.20048
Full Text: