Definition:Group Action Induced on Stable Subset

Definition
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 $G\times S$.

Also see

 * Definition:Group Action Induced on Subgroup