Unformatted text preview: The telling observation now is that vertices (as well as edges and faces) are really permuted in pairs. So we consider the action of the rotation group on pairs of opposite vertices, or, what amounts to the same thing, on the four diagonals of the cube. See Figure 4.1.8 . This gives a homomorphism of the rotation group of the cube into S 4 . Since both the rotation group and S 4 have 24 elements, to show that this is an isomorphism, it sufﬁces to show that it is injective, that is, that no rotation leaves all four diagonals ﬁxed. This is easy to check. Proposition 4.1.4. The rotation group of the cube is isomorphic to the permutation group S 4 . The close relationship between the rotation groups of the tetrahedron and the cube suggests that there should be tetrahedra related geometrically...
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 Fall '08
 EVERAGE
 Algebra, Rotation group

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