Abstract algebra: prove that the center is a characteristic group

Prove that for any group G, the center Z(G) is a characteristic group.

Thanks for the help!

Let f be an automorphism of G; we need to show that f(h) is in Z(G) for any h in Z(G).

It is characteristic because the property of commuting with all elements does not change upon performing automorphisms:

Since gh = hg for all g in G, applying f yields f(gh) = f(hg)
==> f(g) f(h) = f(h) f(g) for all f(g) in G (which is an arbitrary element in G, because f is a bijection of G with itself).
==> f(h) is in Z(G).

I hope this helps!