Prove that group of symmetries is isomorphic to $S_n$

Question

In my algebra book the first section has the following exercise:

Prove that group of all symmetries(isometric bijections under composition) of a regular tetrahedral is isomorphic to $S_4$.

I did it by explicitly writing out all the symmetries of a regular tetrahedral and constructing the appropriate bijection.I would like to know a better(less tedious) way of doing this question.Also I cant imagine doing that for more complicated geometric objects like a octahedron.

What are the general proof strategies for proving isomorphism without explicitly constructing one?

Answer 1

You could use a counting argument: Each symmetry of the tetrahedron corresponds to a permutation of the four vertices of the tetrahedron. So the group of all these symmetries corresponds to a subgroup of $S_4$.

Then note that there are $24$ symmetries of the tetrahedron ($12$ rotations (orientation preserving) and $12$ reflections (orientation reversing)). So you have a subgroup of size $24$ in $S_4$, so the subgroup must be all of $S_4$.

Answer 2

Imagine a plane through one edge and passing through the midpoint of the opposite edge. This opposite edge will be perpendicular to the new plane. Now reflect the tetrahedron through this plane. Clearly, the tetrahedron remains unchanged, but you have interchanged two vertices. Combining all possible interchanges of two vertices generates $S_4$. Unfortunately, you really can't generalize this to the other regular polyhedrons.