Octahedral group is isomorphic to S4

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This article gives a proof/explanation of the equivalence of multiple definitions for the term symmetric group:S4
View a complete list of pages giving proofs of equivalence of definitions

Statement

Consider the octahedral group, also known as a cube group. This is the group of all self-isometries of \R^3 that send a particular regular octahedron to itself (different choices of such regular octahedra yield isomorphic groups). It can also be defined as the group of all self-isometries of \R^3 that send a particular cube to itself.

This group is isomorphic to symmetric group:S4.

Proof

Why the octahedral group and the cube group are the same

The regular octahedron and the cube are dual Platonic solids. Explcitly, if we take the centers of the faces of a regular octahedron, we get a cube, and if we take the centers of the faces of a cube, we get a regular octahedron. Composing these processes (going from a regular octahedron to a cube and then again to a regular octahedron) gives a scaled version of the original. The process preserves isometry groups, i.e., the group for the original regular octahedron is the same as the group for the corresponding cube, and also the same as the group for the smaller regular octahedron.

Construction of the homomorphism

The cube has eight vertices (these correspond to the eight faces of the regular octahedron). The vertices come in pairs of opposite vertices, and there are four such pairs. The midpoint of each such pair is the center of the cube.

Any element of the cube group must send a pair of opposite vertices to a pair of opposite vertices. Thus, we obtain a well-defined action of the cube group on the set of four pairs of opposite vertices. This group action defines a homomorphism from the cube group to symmetric group:S4. We will show that this homomorphism is an isomorphism.

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