## Cayley graphs

Posted by ubpdqn on February 7, 2011

I was inspired by the comment of Yaroslav on the post regarding rotational symmetries of the cube.

The group multiplication table of the SymmetricGroup [4] allows one to better understand the 2-,3- and 4- cycles.

This combined with the following graphic helped me understand the Cayley graphs using different generators. The associated animated gifs are examples of the invariance of composition of a two of the generator elements , e.g. 90 degree rotation about face-face axis then 180 degree rotation about opposite edge axis.

Thank you Yaroslav for this education.

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## Yaroslav Bulatov said

No problem, I’m learning about groups myself. Apparently choosing set of generators for a group is known as endowing group with a “geometry”, Terry Tao talks a bit about it in his blog — http://terrytao.wordpress.com/2010/07/10/cayley-graphs-and-the-geometry-of-groups/

## Yaroslav Bulatov said

An interesting “coincidence” — Cayley graphs of S4 can be plotted in 3d as truncated cubes or octahedrons like here http://mathematica-bits.blogspot.com/2010/12/group-theory-bits.html Cube and octahedron have group of symmetries isomorphic to S4. Cayley diagram for AlternatingGroup[4] (A4) can be arranged as truncated tetrahedron. Tetrahedron symmetries are isomorphic to A4 . Cayley graph for A5 can be arranged as truncated icosahedron or dodecahedron. Symmetries of icosahedron/dodecahedron is A5 group. Nathan Carter’s book and group explorer http://groupexplorer.sourceforge.net/ has examples of this