Contractible space of maps between Eilenberg-Mac Lane spaces, 2

Let $G$ and $H$ be torsion abelian groups. Are the following are equivalent:

1. $\mathrm{Hom}(G, H) = 0$

2. $\mathrm{map}_*( K(G, m), K(H,n)) \sim *$ for all $n, m\geq 1$

?

Clearly (2) implies (1).

-

No, we have $Hom(\mathbb{Z}/n,\mathbb{Z})=0$ but there is a nontrivial Bockstein $K(\mathbb{Z}/n,1)\to K(\mathbb{Z},2)$.

For finite abelian groups we have (1) iff (2) iff ($|G|$ and $|H|$ are coprime).

-
Of course. I'll edit the question so that it makes more sense. –  Jeff Strom Jan 24 '11 at 20:35
I see that I haven't gotten around to editing the question, and I don't quite remember what I was driving at; and since you did answer the question, you get the cigar. –  Jeff Strom Apr 29 at 0:34