Show that \(Z(G) \leq G\) is always characteristic.
Solution
Let \(\psi\in \Aut(G)\), \(g\in Z(G)\), and \(h\in G\). Since \(g\) commutes with \(\inverseof{\psi}(h)\), \[\psi(g)h=\psi(g)\,\psi(\inverseof{\psi}(h))=\psi(g\,\inverseof{\psi}(h))=\psi(\inverseof{\psi}(h)\,g)=h\,\psi(g).\] As \(h\) is arbitrary, \(\psi(g)\in Z(G)\), so \(\psi(Z(G)) \subseteq Z(G)\). Applying the same argument to \(\inverseof{\psi}\) yields \(\inverseof{\psi}(Z(G)) \subseteq Z(G)\). Since \(\psi\) is a bijection, \(\psi\inverseof{\psi}(A) = A\) for all \(A\leq G\), so \(Z(G) \subseteq \psi(Z(G))\).