(Solved): 4. Let (G,G) and (H,H) be two groups. We denote the set of group homomorphisms from G t ...

4. Let $\left(G,{\ast }_G\right)$ and $\left(H,{\ast }_H\right)$ be two groups. We denote the set of group homomorphisms from $G$ to $H$ by $\mathrm{Hom}(G,H)$. (a) Let $\phi ,\psi \in \mathrm{Hom}(G,\mathbb{Z})$ and define $\phi +\psi$ to be the function such that $(\phi +\psi )(g)=$ $\phi (g)+\psi (g)$ for all $g\in G$. Show that $\mathrm{Hom}(G,\mathbb{Z})$ with addition is a group. (b) Let $n$ be some fixed element of $\mathbb{Z}$. Let ${\phi }_n:\mathbb{Z}\to \mathbb{Z}$ by ${\phi }_n(a)=na$ for all $a\in \mathbb{Z}$. Verify that ${\phi }_n$ is a group homomorphism for all $n\in \mathbb{Z}$. (c) Show the every group homomorphism from $\mathbb{Z}\to \mathbb{Z}$ is of the form ${\phi }_n(a)=$ na for all $a\in \mathbb{Z}$ for some $n\in \mathbb{Z}$. (d) Show that the map $\Psi :\mathrm{Hom}(\mathbb{Z},\mathbb{Z})\to \mathbb{Z}$ given by $\Psi (\phi )=\phi (1)$ is a group isomorphism.