Example

Let G be the group S₃. Then G consists of six elements:

e, y = (1,2,3), z = (1,3,2), a = (1,2), b = (1,3), c = (2,3).

By the theorem, the left and right regular representations are injective.

What about C? By the theorem, C_g is the identity if and only if it commutes with every element of S₃. Since each conjugacy class distinct from {e} consists of more than one element, we find Z(G) = Ker(C) = {e}.