Prove that: the center of any group is characteristic subgroup . Let $G$ be any group , $Z(G)$ is the center of the group $G$ , prove that :
$\forall \tau \in Aut(G) , \tau [(Z(G)] = Z(G)$

My first trial was to prove that the center of any group is the unique subgroup of its order hence is a characteristic . but i found a counterexample easily which is $D_8$ the Dihedrial group of order $8$ ( i use the notation $D_{2n}$ )

So , any hints ?
Can you show that for any $x\in Z(G)$ and $y\in G$, $\tau(x)\tau(y)=\tau(y)\tau(x)$? This will say $\tau(x)$ commutes with every element of $G$ (since $y$ was arbitrary), thus $\tau(x)\in Z(G)$. This shows containment in one direction, can you show containment in the other direction now?
consider that any Automorphism preserves the order, means order of Z(G) = order of τ(Z(G)).