Definition 3

Let $\alpha \mapsto f_\alpha$ be an action of $\Omega$ on $E$ and $A$ a stable subset of $E$. The mapping which associates with an element $\alpha \in \Omega$ the restriction of $f_\alpha$ to $A$ (considered as a mapping of $A$ into itself) is an action of $\Omega$ on $A$ said to be induced by the given action.