Centralizer of element in group PSL(2,F_p)

Question

Is it true, that $\forall g\in PSL(2,F_p)\setminus\{e\}$, $Z(g)$ is Abelian?

I think that this is true, but i can't find a simple proof.

Answer 1

This is false if $-1$ is a square in $\mathbb{F}_p$ (i.e. $p\equiv 1$ (mod. 4)). Then the homography $z\mapsto -z$ is in $\mathrm{PSL}(2,\mathbb{F}_p)$, and its centralizer contains the homographies $z\mapsto \lambda z$ for $\lambda \in \mathbb{F}_p^{*2}$ and $z\mapsto 1/z$, which do not commute.