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7. COSETS AND LAGRANGE’S THEOREM
Theorem (The rotation Group of the Cube). The group of rotations of
a cube is isomorphic to S4.
Proof.
Since |G| = |S4|, we need only show that G is isomorphic to a subgroup of S4.
Now a cube has 4 diagonals and any rotation induces a permutation of these
diagonals. But we cannot just assume that di↵erent rotations correspond to
di↵erent rotations.
↵ = (1 2 3 4)
= (1 4 3 2)
We need to show all 24 permutations of the diagonals come from rotations.
Numbering the diagonals as above, we see two perpendicular axes where 90
rotations give the permutations ↵ = (1 2 3 4) and = (1 4 3 2). These induce
an 8 element subgroup {", ↵, ↵2, ↵3, 2, 2↵, 2↵2, 2↵3} and the 3 element
subgroup {", ↵ , (↵ )2}. Thus the rotations induce all 24 permutations since
24 = lcm(8, 3).
⇤