Q:

Singleton conjugacy classes

How does one prove that the conjugacy class of an element $x$ in a group $G$ is a singleton if and only if $x$ belongs to the centre of $G$?

A:

If $x$ is the only element in its own conjugacy class i.e. $C_x=\{x\}$ then clearly $gxg^{-1}=x \iff gx=xg$ for all $g \in G$ thus by definition of the centre we have $x \in Z(G)$. 
On the other hand if $x \in Z(G)$ then $x$ commutes with every element in the group by definition of the centre of a group so the conjugacy class of $x$: 
$C_x=\{gxg^{-1}:g \in G\}=\{gg^{-1}x:g \in G\}=\{x\}$. (The second equality is due to the fact that $x$ commutes with every element of $G$ in particular $g^{-1}$.)
So it is a singleton as required.

