 Solution

An element $x$ is central exactly when $gx=xg$ for every $g\in G$. By part (b), this is exactly the assertion that every $g$ stabilizes $x$. That means every conjugate of $x$ is $x$; conversely, a singleton conjugacy class has this property for every $g$.In particular, the singleton classes are in one-to-one correspondence with the elements of the centre.