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7.22 (1980)

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Suppose that a finite group $G$ is realized as the automorphism group of some torsion-free abelian group. Is it true that for every infinite cardinal $\mathfrak{m}$ there exist $2^{\mathfrak{m}}$ non-isomorphic torsion-free abelian groups of cardinality $\mathfrak{m}$ whose automorphism groups are isomorphic to $G$?

Contributor: S. F. Kozhukhov

Progress

Yes, this is true in the Zermelo–Frenkel system with axioms of choice and 'weak diamond' (M. Dugas, R. Göbel, Proc. London Math. Soc. (3), 45, no. 2 (1982), 319–336), or if $\mathfrak{m}$ is smaller than the first measurable cardinal (V. A. Nikiforov, Mat. Zametki, 39, no. 5 (1986), 641–646 (Russian)).

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