Prove that $\displaystyle C_G(A)=\{g\in G:g^{-1}ag=a, \forall a\in A\}$
Is this just asking me to prove $\displaystyle g^{-1}ag$ iff ga = ag?
Presumably you are asked to prove that $\displaystyle \left\{g\in G:ga=ag,\;\forall a\in A\right\}=\left\{g\in G:g^{-1}ag=a,\; \forall a\in A\right\}$. Then, yes, it follows from the fact that $\displaystyle ag=ga$ if and only if $\displaystyle g^{-1}ag=a$.