What does the order of an element in a group tell you about its centralizer? So if the order is lets say 5 what does that say about the centralizer?
Remember that $\displaystyle \left< a \right> \subseteq C(a)$.
Therefore if $\displaystyle \text{ord}(a)=5$ then it means $\displaystyle |C(a)|$ is a multiple of 5.