I've been trying to prove this (probably very simple) result that is stated in a paper that I'm reading:


Let $G$ and $H$ be connected semisimple algebraic groups defined over a field $F$, and let $f: G \longrightarrow H$ be a separable isogeny. Then $f$ is a central isogeny.


I'm not seeing why the fact that $\mathrm{ker}(f)$ is smooth and finite implies that it is contained in the center of $G$. Any hints would be much appreciated.



How do I show that a separable isogeny is central?
Consider the action of $G$ on $\ker(f)$ by conjugation. We have that $\ker(f)$ is finite and smooth, and $G$ is connected, so the action of $G$ on $\ker(f)$ is trivial. Since this conjugaction action is trivial, we have that $\ker(f)$ is central.


