How do you prove you've found all symmetries of an object in 3D?

Question

In group theory class we studied the example of rotational symmetries of the regular tetrahedon. The teacher showed us 12 symmetries and then said "if you stare long and hard you can convince yourself that those are all symmetries". Is there a way to rigorously prove this?

Answer 1

A linear transformation in $\mathbb R^3$ is uniquely determined by its action on any set of $3$ linearly independent vectors, or a superset of such a set. In particular, it is determined by its action on the vertices of the tetrahedron. So it suffices to consider permutations of the vertices.

There are $4!=24$ different permutations. Half of these reverse the orientation of the tetrahedron, which is not done by rotations. That leaves just $12$ permutations that could possibly be realized by rotations.