Exercise 1.7.22 (Group of rigid motions of a regular octahedron)

Question

Show that the group of rigid motions of an octahedron is isomorphic to a subgroup (cf. Exercise 26 of Section1) of $S_4$. [This group acts on the set of four pairs of opposite faces.] Deduce that the groups of rigid motions of a cube and an octahedron are isomorphic. (These groups are isomorphic because these solids are "dual" - see Introduction to Geometry by H. Coxeter, Wiley, 1961. We shall see later that the groups of rigid motions of the dodecahedron and icosahedron are isomorphic as well - these solids are also dual.)

Richard Ganaye · Accepted Answer

\begin{proof} Consider the octahedron $O$ obtained from the cube $C$ by taking the center of each face of the cube (See the figure in the solution of Exercise 7.5.10 of ``Cox, Galois Theory'' on this site).

Then every rigid motion that maps $C$ on itself maps $O$ on itself, and conversely. So the group of rigid motions of the octahedron $O$ is the same than the group of rigid motions of the cube $C$, and is isomorphic to $S_4$.
\end{proof}

Exercise 1.7.22 (Group of rigid motions of a regular octahedron)

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Exercise 1.7.22 (Group of rigid motions of a regular octahedron)

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