\(\omega\)-elongations and Crawley's problem. The solution to Crawley's problem
All groups in this review are abelian p-groups. An \(\omega\)-elongation of a group B by a group A is a group H such that \(p^{\omega}H\cong A\) and H/A\(\cong B\). The group B is a Crawley group if every two elongations of B by Z(p) are isomorphic. It is well-known that \(\Sigma\)-cyclic groups are Crawley groups. Crawley's problem is to determine whether the converse is true. \textit{C. Megibben} [Pac. J. Math. 107, 205-212 (1983; Zbl 0521.20035)] showed that the converse holds for groups of cardinality \(\aleph_ 1\) assuming \((V=L)\) but that it fails assuming \(MA+\neg CH.\) In the papers under review it is shown that under the assumption (V\(=L)\) every Crawley group (of arbitrary cardinality) is \(\Sigma\)-cyclic. Further it is shown assuming \((V=L)\) that a non-\(\Sigma\)-cyclic (separable) group B of cardinality \(\aleph_ 1\) fails to be a Crawley group in a spectacular fashion: There exist \(2^{\aleph_ 1}\) \(\omega\)- elongations of B by Z(p) such that for any two different groups H and G of this family and every \(f\in Hom(H,G)\) necessarily \(f(p^{\omega}H)=0\). It is shown further that it is consistent that there exists a Crawley group of cardinality \(\aleph_ 1\) which is not \(\aleph_ 1\)-separable. The latter means that the group is separable and every countable subset is imbedded in a countable direct summand. The papers contain several other incisive theorems and interesting methods but the proofs are sketchy at times and require background knowledge about elongations, filtrations, and, of course, set-theoretic principles. \(\{\) In the definition of ''maximal direct summand'' on p. 125 of the first paper it is probably ment that ''any Ulm invariant of B/A is either 0 or \(>\omega\), or else Lemma 1.5 is incorrect as it stands. At any rate the lemma only affects a new proof of a result of Megibben. In the proof of Lemma 2 of the second paper \(A^*\) must be chosen such that \(A^*/B_ 1=p^{\omega}(A/B_ 1).\}\)
scientific article; zbMATH DE number 3941804