# Definitions ###### Group action axioms Let $G$ be a [[Group]] and $X$ a [[Sets]]. 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 $\phi$ ###### As subgroup of symmetry group An action of $G$ on $X$ is a [[Homomorphism|homomorphism]] $\phi:G \to \text{Sym}(X)$.