Define the centralizer of an element g in a group G to be the set
C(g)={x∈G:xg=gx}.
Show that C(g) is a subgroup of G. If g generates a normal subgroup of G, prove that C(g) is normal in G.
Hint. Suppose that ⟨g⟩ is normal in G and let y be an arbitrary element of G. If x∈C(g), we must show that yxy−1 is also in C(g). Show that (yxy−1)g=g(yxy−1).