Let $G$ be the group of rotational symmetries of a regular tetrahedron. I'm trying to think of an argument proving that since $G\cong H$, where $H\leq S_4$, $H=A_4$.
There are 12 rotational symmetries in $G$ and thus $|H|=12$ (since the homomorphism induced by $G$ acting of the vertices of a tetrahedron is injective). But the only group of index $2$ in $S_4$ is $A_4$. Hence the isomorphism.
However, the difficult part for me here is to prove that the above action is faithful. I've used a geometric argument, but I don't think it's rigorous enough. I've also thought about the orbits of this action, but I don't seem to come up with the idea of how an orbit of $x\in X$ (the set of the vertices) can be of size 12 (to conclude that the stabilizer has size 1). Some hints would be appreciated.