The following is an exercise from Humphrey's Linear Algebraic Groups (page 108):
Let $G$ be an algebraic group, $ \displaystyle H= \cap_{ \chi \in X (G)} \ker ( \chi) $. Prove that:
(a) $H$ is a closed, normal subgroup
(b) $G/H$ is diagonalizable
(c) $X(G) \cong X(G/H)$
(d) Describe $X(GL_n(K))$
For (a): $H$ is closed (normal) as intersection of closed (normal reps.) sets.
For (b): I am trying to find a morphism of algebraic groups $G/H \to G_m \times \cdots \times G_m $ but I can't. Is there any other approach that will work.
But for the others I have no idea.
I would really appreciate some help.
Thanking in advance.