[Math] Which elements of $S_8$ are in the subgroup of rigid motions of a cube

abstract-algebrafinite-groupsgroup-theorypermutationssymmetric-groups

Let the set $S\colon= \{ 1, 2, 3, 4, 5, 6, 7, 8 \}$. Then which permutations of $S$ will appear in the group of rigid motions of a cube, which is a subgroup of $S_8$, the symmetric group on 8 letters? Here the elements of $S$ are to represent the eight different vertices of the cube when in its original position.

In the original position of the cube, we assume that, going counter-clockwise, the vertices $1$, $2$, $3$, and $4$ form the bottom face; the vertices $5$, $6$, $7$, and $8$ the top face; and the vertices $1$, $2$, $7$, and $6$ the front face. This way, $1$, $8$; $2$, $5$; $3$, $6$; and $4$, $7$ are the four pairs of opposite vertices.

Is it a correct way of beginning? And if so, then what are the elements alongwith their respective geometric interpretations?

Best Answer

Yes, you've got a correct beginning. The group of permutations on $8$ letters is indeed $S_8$. The elements of $S_8$ are the $8!$ possible permutations of these letters. And you've got a good start on identifying these $8$ elements with the eight vertices of a cube. Now, only some of the permutations of $S_8$ belong to the group of rigid motions of a cube: these can be thought of the possible ways in which a cube can be rotated so that it starts snug in a box, is rotated about any number of axes of rotation, and placed back in the snug box, with vertices permuted (or not, if it's placed back in the box with vertices in the exactly same position as they were to begin with.).

Hints:

  • The group of rigid motions of the cube is isomorphic to $S_4$. And $|S_4| = 4! = 24$, hence the subgroup of elements of $S_8$ that is isomorphic to $S_4$ (the group of rigid motions of a cube) has 24 elements.

  • Consider the permutations of your named vertices when the cube is rotated about each of three axes which pierce opposite faces. There are 9 such permutations.

  • Consider the permutations of your named vertices when the cube is rotated about each of six axes which penetrate the midpoints of opposite edges. There are 6 of such permutations.

  • Consider the permutations of your named vertices when the cube is rotated about each of four long diagonals connecting opposite vertices of the cube. There are 8 such permutations.

  • Don't leave out the identity: the "do nothing" action in which the cubes vertices remain, or return to, their original locations.

Related Question