 Solution

A group action assigns to each $g\in G$ and $x\in \Omega$ a point $g\cdot x$ so that $1\cdot x=x$ and $(gh)\cdot x=g\cdot (h\cdot x)$. The orbit of $x$ is $Gx=\{g\cdot x:g\in G\}$. The action is transitive if there is exactly one orbit, equivalently any point can be carried to any other by some group element.