Solution
= 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 $g$.
$$
\boxed{x\in Z(G)\iff\operatorname{Stab}_G(x)=G\iff C(x)=\{x\}.}
$$
In particular, the singleton classes are in one-to-one correspondence with the elements of the <centre>.