Solution
= 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.