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).

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).