##
**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.

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.

Reviewer: Todor Mollov (Plovdiv)

### MSC:

20K10 | Torsion groups, primary groups and generalized primary groups |