Exercise 1, § 3

Let $\alpha \mapsto f_\alpha$ be an action of $\Omega$ on $E$. Let $F$ be the image of $\Omega$ in $E^E$ and $G$ the stable subset of $E^E$ under the law $(f, g) \mapsto f \circ g$ generated by $F$ and the identity mapping of $E$.

(a) Show that every subset of $E$ which is stable under the action of $\Omega$ is also stable under the restriction to $G$ of the canonical action of $E^E$ on $E$ (no. 1, Example 3).

(b) Let $X$ be a subset of $E$; show that the stable subset of $E$ generated by $X$ is the set of $f(x)$ where $f$ runs through $G$ and $x$ runs through $X$.