Subject: Abelian group question Date: Sat, 15 Sep 2001 21:21:04 +0200 (MET DST) From: Marek Golasinski Don Davis wrote: ===================== Now for my question about abelian groups, which was asked by a student in my algebraic topology class. If G and G' are abelian groups such that for all abelian groups H there is an isomorphism of abelian groups Hom(G,H) iso Hom(G',H), can one conclude that G and G' are isomorphic? No naturality conditions are assumed about the isomorphisms of Hom-groups. Perhaps the result follows from Pontryagin duality, but the hypothesis does not assert an isomorphism of topological groups. =================== The question of your student is exactly Problem 34 of Fuchs which has been answered negatively by Hill, Paul, "Two problems of Fuchs concerning $Tor$ and $Hom$", J. Algebra 19 (1971), 379-383. Two abelian non-isomorphic groups $G,G'$ have been constructed with $Hom(G,H)$ isomorphic to $Hom(G',H)$ for all abelian groups $H$. Marek Golasinski