Transitive group action in cube and icosahedral group There are two claims:

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*A group of symmetries of a cube acting on vertices have only 1 orbit, so vertices of a cube has transitive action

*The action of icosahedral group is transitive on the vertices

Shouldn't the orbit be the number of vertices of cube or icosahedron? So the action of group of symmetries of a cube on its vertex has 8 orbits and the action of icosahedral group on its vertex has 12 orbits. I think I probably misunderstand something here. When will the # of orbit be 1 and when will the # of orbit be the number of vertices?

The orbit of a given vertex, say $v\in V$, is by definition the set of vertices $v$ can be brought to upon the action of the group of symmetries. If the action is transitive, then any vertex can be brought to any other one (by possibly more than one symmetry): there is one orbit, only, whose size is $|V|$.
