# ag.algebraic geometry – Maximal closed subscheme stable under the action of a finite connected group scheme

Let $$k$$ be a field of characteristic $$p>0$$, $$X$$ a smooth projective $$k$$-variety and $$Ysubseteq X$$ a closed irreducible subvariety. Let $$G$$ be a connected finite $$k$$-group scheme acting on $$X$$.

Does there exist a maximal closed subscheme $$T$$ of $$Y$$ stable under the action of $$G$$?

If $$G$$ is étale, then I think one can define $$T:=bigcap_{gin G}g(Y)$$. In case $$T$$ exists, is there a similar description for it?