Relation between Centraliser of a and b

Question

Let $a$ and $b$ are elements belongs to group $G$ such that $b= a^k$ i.e $b$ belongs to subgroup generated by $a$. We can easily prove that Centraliser of $a$ is subset of Centraliser of $b$ ,where $b$ belongs to subgroup generated by $a$. But I want to know is Centraliser of $a =$ Centraliser of $b$ in any condition? What happen if I choose such $b$ which is generator of subgroup $\langle a\rangle$?

Answer 1

In general, this is false: take $G=D_8$, the dihedral group on four vertices, where $\sigma$ is the rotation and $\tau$ the reflection. then $\sigma^2$ commutes with every element of $D_8$, but $\sigma$ does not, thus their centralisers are different.

However, as I noted in my comment, the centralisers would be equal if $b$ generates $\left< a \right>$, that is $gcd(|a|, l) = 1$ where $l$ is the integer such that $a^l = b$