Prove new group action is properly defined.

Question

In continuation with part (b) of the problem here. Show that there is injective map from the sets of actions on set $X$ to set of new-action on set $Y.$

How to proceed?

Answer 1

We have a pseudo-action $\star$ of $G$ on $X$. Given $g\in G,\,x\in X$, we need to find $y\in Y$ (where $Y=\{e\star x\mid x\in X\}$ is the orbit of $e$), such that $g\star x=g\star y$.

Let $y=e\star x$. Then $g\star y=g\star(e\star x)=ge\star x=g\star x$, as required.