Perhaps there is a simple answer, but I'm very puzzled by the following question:


**Question**: Does there exist a (smooth, connected) algebraic group $G$ such that the general centralizer (i.e. the centralizer in a Zariski open set) is finite?


I'm pretty sure the answer is no (maybe some extra assumptions such as reductive, algebraically closed field... must be added), but I have no clue about how to prove it.


Of course, typical groups such as $GL\_n, SL\_n, O\_n...$ are not counterexamples.


Thank you!



General centralizer of algebraic group
Every element $g\in G$ is contained in a Borel subgroup $B\subseteq G$. The quotient $B^{ab}:=B/(B,B)$ has positive dimension since $B$ is solvable. Moreover, the $B$-conjugacy class of $g$ maps to a point in $B^{ab}$. Hence $\dim B/C\_B(g)\le\dim (B,B)$ and therefore $\dim C\_G(g)\ge\dim C\_B(g)\ge \dim B^{ab}>0$.


