Existence of group homomorphisms

Question

Given groups $G$, $H$ with $e_G \neq g_1 \in G, e_H \neq h_1 \in H$. Does there exist a homomorphism $G \to H$ with $g_1 \mapsto h_1$ ?

Answer 1

In general, no. Take $G=(\mathbb{Z}_2,+)$, $H=(\mathbb{Z},+)$, $g_1=1$, and $h_1=1$. There is no group homomorphism $f$ from $G$ into $H$ such that $f(1)=1$, because $1+1=0$ (in $\mathbb{Z}_2$) and $1+1\neq0$ (in $\mathbb{Z}$).