Here is a problem that I really like. I found it in an algebra book.

(It is nice, because if approached properly the proof is a one liner).

"Show that if $\displaystyle p$ divides $\displaystyle |Z(G)|$ then every Sylow $\displaystyle p$-subgroup of $\displaystyle G$ contains the Sylow $\displaystyle p$-subgroup of $\displaystyle Z(G)$".