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4. Let $(G,∗_{G})$ and $(H,∗_{H})$ be two groups. We denote the set of group homomorphisms from $G$ to $H$ by $Hom(G,H)$. (a) Let $φ,ψ∈Hom(G,Z)$ and define $φ+ψ$ to be the function such that $(φ+ψ)(g)=$ $φ(g)+ψ(g)$ for all $g∈G$. Show that $Hom(G,Z)$ with addition is a group. (b) Let $n$ be some fixed element of $Z$. Let $φ_{n}:Z→Z$ by $φ_{n}(a)=na$ for all $a∈Z$. Verify that $φ_{n}$ is a group homomorphism for all $n∈Z$. (c) Show the every group homomorphism from $Z→Z$ is of the form $φ_{n}(a)=$ na for all $a∈Z$ for some $n∈Z$. (d) Show that the map $Ψ:Hom(Z,Z)→Z$ given by $Ψ(φ)=φ(1)$ is a group isomorphism.

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