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

Corner, A. L. S.; Göbel, R.; Goldsmith, B.

Q. J. Math. 57, No. 2, 183-192 (2006).

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).

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

Keywords:

Crawley groups; almost free groups; pure subgroups; torsion-free Abelian groups; completely decomposable groups; direct sums

Citations:

Zbl 0521.20035