I want to know that whether follow equality holds: $ N_G(C_G(a)):C_G(a)=a^G\cap C_G(a).$ It is easy to see that the left hand is no more than the right hand. I think this equality does not hold, but I can't find a counter example.

Take $G$ to be a symmetric group of degree at least 5, and $a$ a transposition. 


Write $C$ for $C_G(a)$ and $N$ for $N_G(C_G(a))$. Although, as Jeremy has demonstrated, the equality $N:C = a^G\cap C$ does not hold in general, it seems to me that there is a chance of characterising when it is true. Consider, for instance, the situation when $C$ is a TIsubgroup. That is to say, $C\cap C^g$ is equal to $\{1\}$ or $G$ for every $g\in G$. I think it's fairly clear that in this case one does indeed have $$N:C = a^G\cap C.$$ It seems to me that something like a converse might also be true. I don't have the wherewithal to work this through right now but I will try and return to it in due course... Edit: As Mark has pointed out in comments below, the TIcondition is too strong if we want to characterise when $$N:C = a^G\cap C.$$ The condition that we want is this: $$x\in G\backslash N \Longrightarrow h^x \not\in H.$$ The contrapositive might be clearer: $$h^x \in H \Longrightarrow x\in N.$$ Clearly the TIcondition implies this, but it also accounts for the example given by Mark... And it's quite easy to see that it precisely characterizes when $N:C = a^G\cap C.$ 

