Q:

The number in the join of conjugate class and centralizer

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.

A:

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

