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.



The number in the join of conjugate class and centralizer
Take $G$ to be a symmetric group of degree at least 5, and $a$ a transposition.


