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**On torsion-free Crawley groups.**
*(English)*
Zbl 1116.20035

In the present paper, the authors extend the theory of Crawley \(p\)-groups to torsion-free groups. This extension is based on the following characterization, enunciated in this form by C. Megibben [in Pac. J. Math. 107, 205-212 (1983; Zbl 0521.20035)]: The separable \(p\)-group \(G\) is a Crawley group if and only if \(\operatorname{Aut}(G)\), the automorphism group of \(G\), acts transitively on the dense subsocles of codimension one. Therefore the authors define the ‘Crawley torsion-free groups’ as those groups \(G\) with the property that for every pair of pure and dense subgroups of corank 1 in \(G\) there exists an automorphism of \(G\) mapping one onto the other.

In Section 2 it is shown that the class of Crawley torsion-free groups contains groups of every finite rank (Proposition 2.1), every completely decomposable torsion-free group of rank \(\leq 2\) is Crawley (Proposition 2.5), and there exists a completely decomposable groups of rank 3 which is not Crawley.

In Section 3 infinite rank groups are studied. It is proved that every homogeneous completely decomposable group is Crawley. Then Crawley torsion-free groups of the form \(G=F\oplus N\), where \(F\) is a free group and \(N\) has trivial dual, are characterized: if \(F\) is of infinite rank, \(G\) is Crawley if and only if \(N=0\) (Proposition 3.5); if \(F\) is of finite rank, \(G\) is Crawley if and only if \(N\) is a Crawley group (Proposition 3.6).

The main results of the paper are about almost free Crawley groups. It is shown that, as in the case of \(p\)-groups, the structure of \(\aleph_1\)-free Crawley groups depends on axioms of set theory. More exactly, if we assume Jensen’s diamond principle (e.g., in Gödel’s constructible universe \(V=L\)) then every \(\aleph_1\)-free Crawley group is free (Theorem 4.3), while if we assume Martin’s Axiom and the negation of the Continuum Hypothesis, there exists an \(\aleph_1\)-free group which is not free (Theorem 4.5).

In Section 2 it is shown that the class of Crawley torsion-free groups contains groups of every finite rank (Proposition 2.1), every completely decomposable torsion-free group of rank \(\leq 2\) is Crawley (Proposition 2.5), and there exists a completely decomposable groups of rank 3 which is not Crawley.

In Section 3 infinite rank groups are studied. It is proved that every homogeneous completely decomposable group is Crawley. Then Crawley torsion-free groups of the form \(G=F\oplus N\), where \(F\) is a free group and \(N\) has trivial dual, are characterized: if \(F\) is of infinite rank, \(G\) is Crawley if and only if \(N=0\) (Proposition 3.5); if \(F\) is of finite rank, \(G\) is Crawley if and only if \(N\) is a Crawley group (Proposition 3.6).

The main results of the paper are about almost free Crawley groups. It is shown that, as in the case of \(p\)-groups, the structure of \(\aleph_1\)-free Crawley groups depends on axioms of set theory. More exactly, if we assume Jensen’s diamond principle (e.g., in Gödel’s constructible universe \(V=L\)) then every \(\aleph_1\)-free Crawley group is free (Theorem 4.3), while if we assume Martin’s Axiom and the negation of the Continuum Hypothesis, there exists an \(\aleph_1\)-free group which is not free (Theorem 4.5).

Reviewer: Simion Sorin Breaz (Cluj-Napoca)

### MSC:

20K20 | Torsion-free groups, infinite rank |

20K15 | Torsion-free groups, finite rank |

20K25 | Direct sums, direct products, etc. for abelian groups |

20K27 | Subgroups of abelian groups |

20K30 | Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups |

03E50 | Continuum hypothesis and Martin’s axiom |