# Definition(s) Let $G$ be a [[Finite Group|finite group]] and $X$ a [[Finite Set|finite set]]. > [!NOTE] Definition 1 (Group Action) > The map $\phi:G\times X\to X$ is an action of $G$ on $X$ iff it satisfies the following axioms > > (A1) For all $x\in X$, $\phi(1_{G},x)=x$ > > (A2) For all $x \in X$ and $g,h\in G$, $\phi(g,\phi(h,x))=\phi(gh,x)$. > > We also say $G$ acts on $X$ if there exists such a map $\phi$ > [!Example] Example > Contents > [!NOTE] Definition 2 (Group Action) > An action of $G$ on $X$ is a [[Homomorphism|homomorphism]] $\phi:G \to \text{Sym}(X)$. # Properties(s) # Application(s) **More examples**: # Bibliography