Possible Duplicate:
Does the action of an affine group scheme preserve the nilradical of an algebra?
Let the group scheme $G$ act on the scheme $X$. I labored for a time under the misapprehension that the reduced subscheme $X_\text{red}$ would be automatically $G$-invariant -- it's such a canonical subscheme, how could it leave? But if $X=G$ is a nonreduced group scheme (e.g. {$g \in {\mathbb G}_m : (g-1)^p = 0$}), over a field of characteristic $p$), then this obviously fails.
What is a good condition to ensure that $G$ preserves $X_\text{red}$? Is it enough to assume $G$ is itself reduced?
EDIT: As Angelo points out in his comment, this is nearly an exact duplicate. And of a question I evidently read, since I commented on it. Sigh.