Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / B / B2 / i / Solution

A left action is a map $(g,\omega )\mapsto g\omega$ satisfying $1\omega =\omega$ and $(gh)\omega =g(h\omega )$. The orbit of $\omega$ is $G\omega =\{g\omega :g\in G\}$. The action is transitive when there is exactly one orbit on the nonempty acted-on set, equivalently when any point can be moved to any other by a group element. An action is equivalently a homomorphism from $G$ to the permutation group of the set.