Center/Normal subgroup/Relation to conjugation/Exercise

Let ${}G$ be a group. Show that the center ${}Z\subseteq G$ is a normal subgroup in ${}G$. Relate the center to the group homomorphism

$\kappa \colon G\longrightarrow \operatorname {Aut} \,(G),g\longmapsto \kappa _{g}.$

Was is the image of this homomorphism, and what is the content of the homomorphism theorems in this situation?

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