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The following problem of Fuchs is considered: relate the abelian groups A and B assuming Tor(^4, G) = Tor(fl, G) for all reduced abelian groups G . A complete characterization is obtained in any set-theoretic universe in which E(k) is valid for a proper class of regular cardinals k .doi:10.2307/2160114 fatcat:ikoialh4cvh23bvb4mb2oelibm