# Question on geometric invariant theory

I'm reading Geometric Invariant Theory by Mumford, and confuse about the Proposition 2.4 on P54.

It states that:

Let $$G$$ be a reductive group, act on an algebraic scheme. Then the action of $$G$$ on $$X$$ is proper if and only if for every non-trivial 1-PS $$\lambda: \mathbb{G}_m\to G$$, the induced action of $$\mathbb{G}_m$$ on $$X$$ is proper.

Here the action is proper means $$(\sigma, p_2):G\times X\to X\times X$$ is proper as morphism.

I get confused since in the proof of this proposition, the book use a theorem of Iwahori on P52, which is stated for $$G$$ semisimple of adjoint type... Mumford use a similar argument on P53, which also just require $$G$$ to be reductive....

I wonder that how to reduce these proofs to the case when $$G$$ is semisimple of adjoint type, so I can use Iwahori's theorem. I try to pass to the case when $$G= G/C(G)$$, but I can't find a natural action of $$G/C(G)$$ on $$X$$.

Thanks for any help.

I believe this is addressed on page 52, underneath the statement of Iwahori's theorem. He provides an argument for why Iwahori's result can be strengthened to G reductive by considering $$G \rightarrow G'$$ where $$G$$ is reductive and $$G'$$ is the associated adjoint group.