On socles of Abelian p-groups in L
The subject of this paper is the relationship of a separable abelian p- group A to its socle A[p] viewed as a valuated vector space. If, for example, A is a direct sum of cyclics, then A is determined by A[p]. In Section 1 the authors generalize a result of Hill and Megibben by proving the following Theorem (V\(=L\)). Let A be a separable, abelian p- group of cardinality \(\aleph_ 1\). If A is neither a direct sum of cyclics nor torsion-complete, then there exists a separable p-group \(A'\) such that \(A'\) and A are not isomorphic but \(A'[p]\) and A[p] are isometric. In Section 2, \(\omega_ 1\)-separable p-groups A of cardinality \(\aleph_ 1\) are examined. Any group in this class has an \(\omega_ 1\)- filtration \(A=\cup_{\nu <\omega_ 1}A_{\nu}\) into pure countable subgroups \(A_{\nu}\) such that \(A_{\nu +1}\) is a summand of \(A_{\mu}\) for all \(\nu <\mu <\omega_ 1\). If \(A'=\cup_{\nu <\omega_ 1}A'_{\nu}\) is another such group, then A and \(A'\) are called filtration-equivalent if, for suitable filtrations of A and \(A'\), there are isomorphisms \(f_{\nu}: A_{\nu}\to A'_{\nu}\), for all \(\nu <\omega_ 1\), such that \(f_{\nu}(A_{\mu})=A'_{\mu}\) for all \(\mu\leq \nu.\) Theorem \((2^{\aleph_ 0}<2^{\aleph_ 1})\). There exist \(\omega_ 1\)-separable p-groups which are filtration-equivalent, have isometric socles and isomorphic basic subgroups, but are not themselves isomorphic. Denote by C the class of all \(\omega_ 1\)-separable p-groups A which have a filtration \(\cup A_{\nu}\) where either \(A_{\lambda +1}/A_{\lambda}=Z(p^{\infty})\oplus C_{\lambda}\) with \(C_{\lambda}^ a \)direct sum of cyclics, or \(A_{\lambda}\) is a summand of \(A_{\lambda +1}\) for all \(\lambda <\omega_ 1\). Theorem. If A and \(A'\) in C have isometric socles, then they are filtration equivalent. Theorem (MA\(+\neg CH\)). If A and \(A'\) belong to C, then \(A\simeq A'\) if and only if A[p] and \(A'[p]\) are isometric. As a consequence, it is undecidable in ZFC if groups in C are determined by their socles.
scientific article; zbMATH DE number 4134334