Invariant affine coverings for diagonizable groups

Let $K$ be an algebraic closed field and $X$ a normal variety over $K$. Consider a torus $T$ over $K$ which acts on $X$. Then by a theorem of Sumihiro, $X$ is covered by $T$-invariant affine open subsets.

Is this still true if one allows $T$ to be an arbitrary diagonalizable group?

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What do you mean by "diagonalizable"? Have you looked at Hironaka's example of a $\mathbb{Z}/2\mathbb{Z}$-action on a proper scheme in the appendix to Hartshorne's text? –  Jason Starr Mar 29 '13 at 22:03
By diagonalizable, I mean something of the form $\mathrm{Spec}\ K[G]$, where $G$ is a finitely generated abelian group. I assume that you are referring to example 3.4.1 in appendix 3. Where is the $\mathbb{Z}/2\mathbb{Z}$-action? –  Nathan Ilten Mar 29 '13 at 22:46
@Nathan: Turn the page. –  Jason Starr Mar 29 '13 at 23:49