A4 Subset of S4: Artin 6.2 Question

[Juneu436]

Homework Statement

A question from artin 6.2:
Two tetrahedra can be inscribed into a cube C, each one using half the vertices. Relate this to
the inclusion A4 is a subset of S4.

The Attempt at a Solution

I can only think that the tetrahedral group is isomorphic to A4, and the cube is isomorphic to S4. And since you can fit two tetrahedra in a cube, this would imply that A4 is a subset of S4.

Is this correct?

Thanks

 

[Juneu436]

If I can show that the group of rotational symmetries of a tetrahedron is A4 and the full group of symmetries of a tetrahedron is S4, then I can conclude that A4 is a subset of S4.

Does this approach satisfy the question?

 

[murmillo]

I guess I don't understand what the question is even really asking. Isn't A4 defined to be a subset of S4?

 

[Juneu436]

Yeah, but I have to link A4 subset of S4 with the tetrahedra somehow.

 

[Deveno]

consider all the symmetries that map a cube to itself. these either map the two tetrahedra to themselves, or to each other.

identify A4 with the symmetries of the cube that map the 2 tetrahedra back into themselves.

for example, a 90 rotation along the x,y or z-axis (assuming the cube is aligned with these), swaps the 2 tetrahedra, and a 180 degree rotation preserves them. the "corner diagonal" rotations all preserve the 2 tetrahedra (they just rotate around a vertex from each of the 2 tetrahedra), while the "midpoint diagonal" rotations swap the tetrahedra.

one can view a symmetry of the cube as a permutation of it's 4 main diagonals. in this case, a 90 degree rotation is a 4-cycle (d1 d2 d3 d4) for example, a 120 degree rotation about a main diagonal is a 3-cycle (d2 d3 d4) for example, and an 180 degree rotation about a midpoint diagonal is a 2-cycle (d3 d4) for example.

a main diagonal corresponds to opposite vertex pairs (one from each tetrahedron). so a transposition of diagonals, swaps the tetrahedra. even permutations consist of pairs of transpositions, each of which swap the tetrahedra, so even permutations preserve the pair of tetrahedra.