We have already shown that any pair of transpositions of vertices is equivalent to a rotation about the center of a regular tetrahedron. Now we will expand this to include the full set of symmetries, including both rotations and reflections.
A single transposition is equivalent to a geometric reflection through the normal plane bisecting the edge between the affected vertices. Two such reflections, (12) and (34) are shown at the right. Any single transposition reverses the orientation of the tetrahedron. I've already shown that a second transposition will revert the orientation back to the original, so it follows that a third transposition will reverse the orientation again. This continues, so that any odd number of transpositions of the vertices will be equivalent to a series of rotations and a reflection. But I also proved that any pair of rotations is equivalent to another single rotation.
From this, any odd number of transpositions of the vertices is equivalent one rotation and one reflection. Since the set of rotations of the tetrahedra is equivalent to the even-permutation group of four elements, the addition of odd permutations would comprise the full symmetry group, S(4), but I have not yet show that all odd permutations represent symmetries.
To show that all odd permutations are in fact symmetries of regular tetrahedra, it will suffice to demonstrate that rotations and reflections can produce all 24 possible permutations of four elements. We know there are 24 symmetries of a tetrahedron, so if all symmetries can be mapped to a permutation, the groups will be isomorphic. The table at left shows the permutationsn equivalent to a rotation followed by a reflection described by the transpositon (34). This restriction on using a single reflection, which I have designated S, does not prevent the full permutation group from being demonstrated.
Since there are only 24 possible symmetries of regular tetrahedra, any other reflection must be equivalent to one of the 12 in the table. This could be shown in a group table, but the full symmetry table is rather large and not very enlightening.
Though rather trivial for the tetrahedron, it is important to note that by specifying the order of vertices by each permutation, the orientation of each vertex is not explicitly chosen. It is implied by the configuration of the remaining vertices in such a manner as to reflect a rotation or reflection of the original tetrahedron. Thus mapping any vertex to another vertex in the image through such a permutation specifies which three edges will meet at the vertex in the image, but it remains for the other vertices to determine which edge is mapped to which edge in the image. This will be more important for other symmetry groups, where the objects being permuted are lines or solids that we tend to think of in terms of location and orientation, instead of points. As a simple analogy, are you more likely to find yourself thinking that a tennis ball or a toaster is upside-down?