Every automorphism preserves the center.Suggest a slogan
For every group \(G\), the center \(Z(G)\) is a characteristic subgroup of \(G\), \(Z(G) \ch G\).
Let \(\phi \in \Aut(G)\), \(z \in Z(G)\), and \(g \in G\). Then \(\phi(z)\,g = \phi\big(z\,\inverseof{\phi}(g)\big) = \phi\big(\inverseof{\phi}(g)\,z\big) = g\,\phi(z)\), so \(\phi(z) \in Z(G)\).