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On reconstructing separable reducedp-groups with a given socle

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Let\(\bar B^* \) be a separable reduced (abelian)p-group which is torsion complete. We ask whether for\(G \subseteq \bar B^* \) there is\(H \subseteq _{pr} \bar B^* ,H[p] = G[p]\),H[p]=G[p],H not isomorphic toG. IfG is the sum of cyclic groups or is torsion complete, the answer is easily no. For otherG, we prove that the answer is yes assuming G.C.H. Even without G.C.H. the answer is yes if the density character ofG is equal to Min n|p nG|, i.e.,

$$\mathop {Min}\limits_{n< \omega } |p^n G| = \mathop {Min}\limits_m \mathop \Sigma \limits_{n > m} |(p^n G)[p]/(p^{n + 1} G)[p]|$$

Of course, instead of two non-isomorphic we can get many, but we do not deal much with this.

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Partially supported by NSERC Grant A3040.

Part A arose from the conference in Udine, Italy, April 1984, in answer to a question of Cutler. Part B was written in vancouver, summer 1985. For complementary consistency results, see Mekler and Shelah [Mk-Sh].

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Shelah, S. On reconstructing separable reducedp-groups with a given socle. Israel J. Math. 60, 146–166 (1987).

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