  We construct an embedding G of the category of graphs into the category of
abelian groups such that for graphs X and Y we have Hom(GX,GY)=Z[Hom(X,Y)], the
free abelian group whose basis is the set Hom(X,Y). The isomorphism is
functorial 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 and
comprehensive as any other concrete category. We use this embedding to settle
an old problem of Isbell whether every full subcategory of the category of
abelian groups, which is closed under limits, is reflective. A positive answer
turns out to be equivalent to weak Vopenka's principle, a large cardinal axiom
which is not provable but believed to be consistent with standard set theory.
Several known constructions in the category of abelian groups are obtained as
quick applications of the embedding. In the revised version we add some
consequences to the Hovey-Palmieri-Stricland problem about existence of
arbitrary localizations in a stable homotopy category
