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Tetrahedral and Octahedral Groups

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Let G=O be the group of rotations of a cube. Two regular tetrahedra can be inscribed in this cube, each using half of the vertices. Let H be the subgroup carrying one of the two inscribed tetrahedra to itself. If T is the tetrahedral group, show that H=T.

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We show that a particular subgroup of the octahedral group fixes an inscribed tetrahedron.

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By definition, T is the group of rotations of a regular tetrahedron and H is the subgroup of O ...

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