Identify both the orientation-preserving rotational symmetry group and the full Euclidean symmetry group of a regular tetrahedron, cube, and icosahedron.
Solution
For a regular tetrahedron, the rotational symmetry group acts faithfully on the four vertices. The orientation-preserving permutations are exactly the even permutations, so \[ \operatorname{Rot}(\text{tetrahedron})\cong A_4. \] Every permutation of the four vertices is realized by a Euclidean symmetry, hence \[ \operatorname{Sym}(\text{tetrahedron})\cong S_4. \]
For a cube, rotations act faithfully on the four body diagonals. Every permutation of these four diagonals is realized by a rotation, so \[ \operatorname{Rot}(\text{cube})\cong S_4. \] The central inversion \(x\mapsto -x\) is an orientation-reversing symmetry commuting with every rotation. Every full symmetry is either a rotation or central inversion followed by a rotation. Therefore \[ \operatorname{Sym}(\text{cube})\cong S_4\times C_2. \]
For an icosahedron, the rotational symmetry group has order \(60\) and is isomorphic to \[ A_5. \] One realization is its faithful action on the five inscribed cubes in the icosahedron. The icosahedron is centrally symmetric, and central inversion again commutes with all rotations. Hence \[ \operatorname{Sym}(\text{icosahedron})\cong A_5\times C_2. \]
Thus \[ \begin{array}{c|cc} & \text{rotations} & \text{full symmetries}\\ \hline \text{tetrahedron} & A_4 & S_4\\ \text{cube} & S_4 & S_4\times C_2\\ \text{icosahedron} & A_5 & A_5\times C_2 \end{array} \]