\begin{definition}[Definition:Group Action Induced on Stable Subset]
Let $G$ be a group.
Let $X$ be a set.
Let $\phi : G \times X \to X$ be a group action.
Let $S \subset X$ be a stable subset.
The '''group action induced on $S$''' is the restriction of $\phi$ to $G \times S$.
\end{definition}