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It is a well-known homological fact that every Abelian group G has the property that Hom(G,−) commutes with direct products. Here we investigate the ‘dual’ property: an Abelian group G is said to be cosmall if Hom(−,G) commutes with direct products. We show that cosmall groups are cotorsion-free and that no group of cardinality less than a strongly compact cardinal can be cosmall. In particular, if there is a proper class of strongly compact cardinals, then there are no cosmall groups.
Goldsmith, B. & Kolman, O. (2007). On Cosmall Abelian Groups. Journal of Algebra, vol. 317, no. 2, pg. 510-518. doi:10.1016/j.jalgebra.2007.05.009