Quasi-co-local subgroups of abelian groups

Let {0}≠K{0}≠K be a subgroup of the abelian group GG. In [J. Buckner, M. Dugas, Co-local subgroups of abelian groups, in: Abelian Groups, Rings, Modules, and Homological Algebra, in: Lect. Notes Pure and Appl. Math., vol. 249, Chapman & Hall/CRC, Boca Raton, FL, 2006, pp. 29–37], KK was called a co-local (cl) subgroup of GG if Hom(G,G) is naturally isomorphic to Hom(G,G/K). We generalize this notion to the quasi-category of abelian groups and call the subgroup K≠{0}K≠{0} of GG a quasi-co-local (qcl) subgroup of GG if Q⊗ZHom(G,G) is naturally isomorphic to Q⊗ZHom(G,G/K). We show that qcl subgroups behave quite differently from cl subgroups. For example, while cl subgroups KK are pure in GG, i.e. G/KG/K is torsion-free if GG is torsion-free, any reduced torsion group TT can be the torsion subgroup t(G/K)t(G/K) of G/KG/K where GG is torsion-free and KK is a qcl subgroup of GG.