##
**Generalized Dieudonné and Hill criteria.**
*(English)*
Zbl 1146.20034

Abelian \(p\)-groups are considered. The author tries to carry a theorem of Dieudonné’s type to the classes of the simply presented groups, \(C_\lambda\)-groups, pillared groups, torsion complete groups, thick groups, weakly \(\omega_1\)-separable groups and \(Q\)-groups. He succeeds to do it with the exception of the main results of three sections: for \(C_\lambda\)-groups, weakly \(\omega_1\)-separable groups and \(Q\)-groups. In the proofs of these results the author makes crude mistakes which we shall see further down.

The author claims in vain that he generalizes the Hill criteria for simply presented groups [P. Hill, Trends in Mathematics. 7-22 (1999; Zbl 0938.16022)]. Namely, in Theorem 1.1 of the article there are restrictive conditions for the group \(A\) while in the Hill criteria there are no restrictions.

In detail we shall indicate the following mistakes in the proofs of the results and defects of the article.

1) In the proof of Lemma 2.1 the author writes absolutely ungrounded that \[ (p^\alpha A+p^\delta(G+p^\alpha A))/p^\alpha A\subseteq p^\delta((p^\alpha A+G)/p^\alpha A) \] (page 124, lines 8-7 from below). Since Lemma 2.1 is used in the proof of the main result of Section 2 for \(C_\lambda\)-groups, namely for Theorem 2.1, this result remains unproved.

2) In the proof of the necessity of Theorem 7.1 for weakly \(\omega_1\)-separable groups (page 136, lines 3-4 from above) the author writes \[ \aleph_0=\cdots=|\bigcap_{i<\omega}[(p^iA+T)/G]|=|[\bigcap_{i<\omega} (p^iA+T)]/T|\leq\cdots\tag{*} \] The last equality does not hold. We can give the following counterexample. Let \(A\) be a direct sum \[ A=\sum_{i=1}^{.\infty} (a_i) \] of cyclic groups \((a_i)\) of order \(p^i\) and \[ T=\sum_{i=2}^{.\infty} (a_i),\quad G=\sum_{i\in M}^{.\infty} (a_i),\quad M=\{2n+1\mid n\in\mathbb{N}\}. \] Obviously, \(|T/G|=\aleph_0\) and \(|T|=\aleph_0\). Then \(|(p^2A+T)/T|=1\). Hence \(|[\bigcap_{i<\omega}(p^iA+T)]/T|=1\neq\aleph_0\), which is in a contradiction with the author’s assertion (*).

Consequently, the main result of Section 7 for weakly \(\omega_1\)-separable groups remains unproved.

3) An analogous mistake as in 2) is made in the proof of the necessity of Theorem 8.1 for \(Q\)-groups (we can give the same counterexample as in 2) since in the proof the author writes the same equality \[ |\bigcap_{i<\omega}[(p^iA+T)/G]|=|[\bigcap _{i<\omega}(p^iA+T)]/T|. \] Consequently, the main result of Section 8 for \(Q\)-groups remains unproved.

4) In Section 9 the author’s concept of a “nice basis” contradicts the concept of a basis of an Abelian group, i.e. the concept of a “nice basis” has nothing to do with the concept of a basis of an Abelian group.

5) The proof of (c) in Lemma 2.1, as the author notes (page 125, line 17 from above) follows “directly from [13], vol. II, property (c), p. 78”, i.e. from [L. Fuchs, Infinite Abelian groups. Vol. II, Academic Press, New York (1973; Zbl 0257.20035)]. Therefore, the indicated proof of (c) is superfluous.

6) In the proof of Proposition 6.2, namely of \((b)\Leftrightarrow (c)\), the last five lines are superfluous.

7) The author incorrectly notes (page 128, lines 5-4 from below) that P. Hill and C. Meggiben give a negative answer to Problem 54 of [L. Fuchs, loc. cit.] “thirty years ago” the appearance of this book. It is true that they solve this problem in the year 1974, i.e. one year after the appearance of the above book.

8) In Section 6 Danchev uses a theorem of J. Dieudonné [Port. Math. 11, 1-5 (1952; Zbl 0046.02001). However he cites it scarcely in the end of his paper.

9) In Section 3 the author uses without a definition the concept strongly \(\Sigma\)-cyclic subgroup. He writes “\(G/p^\omega G\) is strongly \(\Sigma\)-cyclic in \(A/p^\omega G\) (for the terminology see [3] and [8])” (page 126, lines 7-8 from above). This is incorrect since the mentioned concept is away in the paper [3] [P. V. Danchev, Acta Math. Univ. Comen., New Ser. 74, No. 1, 15-24 (2005; Zbl 1111.20045)] and the article [8] is “to appear”.

The author claims in vain that he generalizes the Hill criteria for simply presented groups [P. Hill, Trends in Mathematics. 7-22 (1999; Zbl 0938.16022)]. Namely, in Theorem 1.1 of the article there are restrictive conditions for the group \(A\) while in the Hill criteria there are no restrictions.

In detail we shall indicate the following mistakes in the proofs of the results and defects of the article.

1) In the proof of Lemma 2.1 the author writes absolutely ungrounded that \[ (p^\alpha A+p^\delta(G+p^\alpha A))/p^\alpha A\subseteq p^\delta((p^\alpha A+G)/p^\alpha A) \] (page 124, lines 8-7 from below). Since Lemma 2.1 is used in the proof of the main result of Section 2 for \(C_\lambda\)-groups, namely for Theorem 2.1, this result remains unproved.

2) In the proof of the necessity of Theorem 7.1 for weakly \(\omega_1\)-separable groups (page 136, lines 3-4 from above) the author writes \[ \aleph_0=\cdots=|\bigcap_{i<\omega}[(p^iA+T)/G]|=|[\bigcap_{i<\omega} (p^iA+T)]/T|\leq\cdots\tag{*} \] The last equality does not hold. We can give the following counterexample. Let \(A\) be a direct sum \[ A=\sum_{i=1}^{.\infty} (a_i) \] of cyclic groups \((a_i)\) of order \(p^i\) and \[ T=\sum_{i=2}^{.\infty} (a_i),\quad G=\sum_{i\in M}^{.\infty} (a_i),\quad M=\{2n+1\mid n\in\mathbb{N}\}. \] Obviously, \(|T/G|=\aleph_0\) and \(|T|=\aleph_0\). Then \(|(p^2A+T)/T|=1\). Hence \(|[\bigcap_{i<\omega}(p^iA+T)]/T|=1\neq\aleph_0\), which is in a contradiction with the author’s assertion (*).

Consequently, the main result of Section 7 for weakly \(\omega_1\)-separable groups remains unproved.

3) An analogous mistake as in 2) is made in the proof of the necessity of Theorem 8.1 for \(Q\)-groups (we can give the same counterexample as in 2) since in the proof the author writes the same equality \[ |\bigcap_{i<\omega}[(p^iA+T)/G]|=|[\bigcap _{i<\omega}(p^iA+T)]/T|. \] Consequently, the main result of Section 8 for \(Q\)-groups remains unproved.

4) In Section 9 the author’s concept of a “nice basis” contradicts the concept of a basis of an Abelian group, i.e. the concept of a “nice basis” has nothing to do with the concept of a basis of an Abelian group.

5) The proof of (c) in Lemma 2.1, as the author notes (page 125, line 17 from above) follows “directly from [13], vol. II, property (c), p. 78”, i.e. from [L. Fuchs, Infinite Abelian groups. Vol. II, Academic Press, New York (1973; Zbl 0257.20035)]. Therefore, the indicated proof of (c) is superfluous.

6) In the proof of Proposition 6.2, namely of \((b)\Leftrightarrow (c)\), the last five lines are superfluous.

7) The author incorrectly notes (page 128, lines 5-4 from below) that P. Hill and C. Meggiben give a negative answer to Problem 54 of [L. Fuchs, loc. cit.] “thirty years ago” the appearance of this book. It is true that they solve this problem in the year 1974, i.e. one year after the appearance of the above book.

8) In Section 6 Danchev uses a theorem of J. Dieudonné [Port. Math. 11, 1-5 (1952; Zbl 0046.02001). However he cites it scarcely in the end of his paper.

9) In Section 3 the author uses without a definition the concept strongly \(\Sigma\)-cyclic subgroup. He writes “\(G/p^\omega G\) is strongly \(\Sigma\)-cyclic in \(A/p^\omega G\) (for the terminology see [3] and [8])” (page 126, lines 7-8 from above). This is incorrect since the mentioned concept is away in the paper [3] [P. V. Danchev, Acta Math. Univ. Comen., New Ser. 74, No. 1, 15-24 (2005; Zbl 1111.20045)] and the article [8] is “to appear”.

Reviewer: Todor Mollov (Plovdiv)

### MSC:

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

### Keywords:

simply presented Abelian \(p\)-groups; \(C_\lambda\)-groups; pillared groups; torsion complete groups; thick groups; weakly \(\omega_1\)-separable groups; \(Q\)-groups### References:

[1] | K. Benabdallah and R. Wilson, Thick groups and essentially finitely indecomposable groups. Canad. J. Math. 30 (1978), 650-654. · Zbl 0399.20049 · doi:10.4153/CJM-1978-056-9 |

[2] | P. V. Danchev, A nice basis for S\eth FGÞ=Gp. Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 53 (2005), 3-11. · Zbl 1199.16084 |

[3] | P. V. Danchev, Generalized Dieudonnećriterion. Acta Math. Univ. Comenian. (N.S.) 74 (2005), 15-24. · Zbl 1111.20045 |

[4] | P. V. Danchev, The generalized criterion of Dieudonne\' for valuated abelian groups. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 49 (2006), 149-155. · Zbl 1180.20045 |

[5] | P. V. Danchev, The generalized criterion of Dieudonne\' for valuated p-groups. Acta Math. Univ. Ostrav. 14 (2006), 17-19. · Zbl 1119.20047 |

[6] | P. V. Danchev, On the coproducts of cyclics in commutative modular and semisimple group rings. Bul. Acad. S\?tiint\?e Repub. Mold. Mat. 51 (2006), 45-52. · Zbl 1109.16028 |

[7] | P. V. Danchev, Nice bases for primary abelian groups. Ann. Univ. Ferrara Sez. VII Sci. Mat. 53 (2007), 39-50. · Zbl 1130.20039 · doi:10.1007/s11565-007-0004-2 |

[8] | P. V. Danchev, Generalized Dieudonneánd Honda criteria. Algebra Colloq., to appear. · Zbl 1166.20047 · doi:10.1142/S1005386709000170 |

[9] | P. V. Danchev, The generalized criterion of Dieudonne\' for primary valuated groups. To appear. · Zbl 1197.20047 |

[10] | P. V. Danchev, Theorems of the type of Cutler for abelian p-groups. To appear. · Zbl 1179.20046 |

[11] | P. V. Danchev and P. W. Keef, Generalized Wallace theorems. Math. Scand., to appear. · Zbl 1169.20029 |

[12] | J. Dieudonne\', Sur les p-groupes abeĺiens infinis. Portugaliae Math. 11 (1952), 1-5. · Zbl 0046.02001 |

[13] | L. Fuchs, Infinite abelian groups, vol. I, II. Pure Appl. Math. 36, Academic Press, New York 1970, 1973. · Zbl 0209.05503 |

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