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We construct an embedding G of the category of graphs into the category ofabelian groups such that for graphs X and Y we have Hom(GX,GY)=Z[Hom(X,Y)], thefree abelian group whose basis is the set Hom(X,Y). The isomorphism isfunctorial in X and Y. The existence of such an embedding implies that,contrary to a common belief, the category of abelian groups is as complex andcomprehensive as any other concrete category. We use this embedding to settlean old problem of Isbell whether every full subcategory of the category ofabelian groups, which is closed under limits, is reflective. A positive answerturns out to be equivalent to weak Vopenka's principle, a large cardinal axiomwhich is not provable but believed to be consistent with standard set theory.Several known constructions in the category of abelian groups are obtained asquick applications of the embedding. In the revised version we add someconsequences to the Hovey-Palmieri-Stricland problem about existence ofarbitrary localizations in a stable homotopy category

An almost full embedding of the category of graphs into the category of abelian groups