# Outer automorphism group maps to automorphism group of center

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Suppose $G$ is a group. Denote by $\operatorname{Out}(G)$ the outer automorphism group of $G$ and by $Z(G)$ the center of $G$. Denote by $\operatorname{Aut}(Z(G))$ the automorphism group of $Z(G)$. Then, there is a canonical homomorphism:
$\operatorname{Out}(G) \to \operatorname{Aut}(Z(G))$
defined as follows: for any outer automorphism class $[\sigma]$ of $G$, pick a representative automorphism $\sigma$, and consider the restriction of $\sigma$ to $Z(G)$.
Proof that this is well defined: Since any inner automorphism restricts to the identity on $Z(G)$, the automorphism obtained by restriction is independent of the choice of representative.