Can someone help me with this proof? It was an exam problem, but I can't get it.

D_2n = < r, s | r^n = s^2 = 1, rs = sr^(-1) >

If n = 2k is even and n>= 4, show that z = r^k is an element of order 2 which commutes with all elements of D_2n. Show also that z is the only nonidentity element of D_2n which commutes with all elemtns of D_2n.