Group_Action_determines_Bijection

Let $*$ be a group action of $G$ on $X$. Then each $g \in G$ determines a bijection $\phi_g: X \to X$ given by $\map {\phi_g} x = g * x$ Its inverse is $\phi_{g^{-1} }: X \to X$. These bijection are sometimes called '''transformations''' of $X$.