Let G be the group of symmetries of a cube. Describe (geometrically) an element of G of order 3 and an element of order 4.

Question

Unsure of how the symmetries of a cube work in group theory. How would you describe an element of order 3 and an element of order 4?

Answer 1

Consider a $120^\circ$ rotation around the line passing through the center of the cube and one of its vertices. It's a symmetry of order $3$.

And if you consider a rotation of $90^\circ$ around the line passing through the center of the cube and the center of one of the faces, you'll have a symmetry of order $4$.

Answer 2

A symmetry of the cube is some sort of repositioning of the cube -- perhaps a rotation -- such that the vertices, edges, and faces line up with where the vertices, edges, and faces used to be. "Order $n$" means if you repeat the repositioning maneuver $n$ times (group-theoretically, this is raising the element to the power of $n$), you get back the original position (i.e., the identity symmetry).

Can you think of a rotation of the cube such that when you repeat the rotation $4$ times, you get back the original position? What about $3$?

Answer 3

[Hint: Draw a cube, together with all possible rotational axes!]

For an element of order 3, consider the rotation by $\frac{2}{3}\pi$ about the diagonal of the cube; for an element of order 4, take the rotation by $\frac{1}{2}\pi$ about the axes through the centers of a pair of opposite faces.

You may refer to the pictures here.