# A4 Subset of S4: Artin 6.2 Question

• Juneu436
In summary, the question asks to relate the fact that two tetrahedra can be inscribed into a cube C, each using half the vertices, to the inclusion A4 being a subset of S4. The solution involves showing that the group of rotational symmetries of a tetrahedron is isomorphic to A4 and that the full group of symmetries of a tetrahedron is isomorphic to S4, and using this to conclude that A4 is a subset of S4. This can be done by considering the symmetries of the cube that map the two tetrahedra back into themselves and identifying A4 with these symmetries. The example of different rotations and their corresponding cycles is also

## 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

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?

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

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

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.