> [!NOTE] Definition (Transitive Group Action) > A [[Group action|group action]] $*:G\times X\to X$ is transitive on $X$ if for all $x,y\in X$, there exists $g\in G$ such that $g*x=y$. > [!Example] Symmetric group acts transitively. > The [[Symmetric Group|symmetric group]] $S_{n}$ acts transitively on $\{ 1,2,\dots,n \}$. Too see this, let $x,y\in \{ 1,2,\dots,n \}$. If $x=y$ then $e*x=x=y$ where $e\in S_{n}$ is the identity element. Suppose $x\neq y$. Let $g=(x,y)$ be the transposition that swaps $x$ and $y$. Then $g*x=y$. Or equivalently, the [[Fixed Points for Group Action|group fixed point]] is given by $\text{Fix}(G)=\emptyset$.