Hi Jason Bourne.
Here is a detailed proof of the class equation posted here not so long ago: http://www.mathhelpforum.com/math-he...al-center.html
To show that $\displaystyle G/Z(G)$ cylic, say $\displaystyle G/Z(G)=\left<gZ(G)\right>$ for some $\displaystyle g\in G,$ implies $\displaystyle G$ Abelian, let $\displaystyle x,y\in G.$ Then $\displaystyle x,y$ belong to some left cosets of $\displaystyle Z(G)$ say $\displaystyle x=g^iz_1,\,y=g^jz^2$ for some $\displaystyle z_1,z_2\in Z(G)$ and $\displaystyle i,j\in\mathbb Z.$ It is then a straightforward matter to verify that $\displaystyle xy=yx.$