Intersection of fixed point subspaces

Question

Let a finite group G act smoothly on a smooth manifold M of dimension n. Let $H_1$ and $H_2$ be subgroups of G and assume that dim$(M^{H_1})+$dim$(M^{H_2})=n$. I have two questions:

1. How to show that $M^{H_1}$ and $M^{H_2}$ are smooth submanifolds of $M$?
2. Assume additionally that $M^G$ is of dimension $0$ and that $H_1\cup H_2$ generates $G$. Is it true that $M^{H_1}$ and $M^{H_2}$ are transverse?

---