A Cayley Graph of PSL(2,7)

The smallest non-Abelian simple group is A5 (it also has other names), with 60 elements. Its Cayley graph can be drawn, without the arcs crossing, on a sphere: see four Cayley graphs of A5.

The second smallest non-Abelian simple group is PSL(2,7) (also known as, or isomorphic to, A₁(7), A₂(2), and GL(3,2) ). It has 168 elements. It is a (simple, normal) subgroup of the automorphism group of C7×C7. The isomorphic group GL(3,2) is the automorphism group of C2×C2×C2.

The Cayley graph of PSL(2,7) cannot be drawn without the arcs crossing on a sphere, nor on a torus, nor on a genus-2 surface (like a torus but with two holes instead of one). It can be drawn on a surface of genus 3 (three holes). This page presents such a Cayley graph:

Explanation of pink parts of the diagram

To represent a genus 3 surface on your screen, the diagram uses "sewing instructions", in pink. You could in principle make a genus 3 surface, with this Cayley graph on it, as follows:

• Print out the image above on a thin sheet of rubber.
• Cut out and discard the pale pink regions.
• Glue or sew together the cut edges in accordance with the pink letters a-h, i-p, q-x.
• To prevent the result from looking like a crumpled mess, pump it up with air, or stuff it with newspaper.

A more practicable way of making a version of this Cayley graph, nicely embedded in 3-space, is:

• Take a cube of wood. Paint it white.
• Drill three holes through it, each running between two opposite faces. Offset the holes so that they do not meet in the middle.
• Copy the above diagram onto it, with the pink regions corresponding to the ends of the holes. It has been drawn so that it is clear what goes on each face. Some of the black and green lines will run right along the edges of your cube, which is not ideal. Eight parallel green lines will run, twisting slightly, through each hole.

Explanation of the Cayley graph itself

A Cayley graph uses a colour for each generator of the group, and its appearance depends fundamentally on the choice of generators. This Cayley graph uses two generators: one of order 2, shown by green arcs, and one of order 3, shown by black arcs. The black arcs need arrows on them to show which way they go; the green arcs go both ways and have no arrows.

The graph has obvious cubic symmetry. It has been drawn so as to emphasise one of the S4 subgroups of PSL(2,7). But there isn't really anything special about this subgroup: it is only one of seven conjugate S4s within PSL(2,7).

It is based on S³:{7,3}, the tiling of the genus 3 surface with 24 heptagons. Each of these heptagons is identical, and has the same relationship to each of its seven neighbouring heptagons (though this symmetry is obscured in the diagram above, by the need to portray the structure on a flat screen).

Just as you can create a Cayley graph for A5 by tiling the sphere with 12 pentagons and then "truncating" the triangular vertices, you can do the same for the tiling of the genus-3 surface by 24 heptagons. You will have to do the tiling more carefully; you can't really go wrong when tiling a sphere, but with a higher-genus surface, if you get the wrong amount of twist into one of the "tunnels", you are likely to end up with an irregular figure. Once you have achieved a regular arrangement of the 24 heptagons, you can again truncate the triangular vertices to give the cayley diagram.

Regular maps drawn on the genus-3 surface.
Some more Cayley diagrams drawn on surfaces appropriate to their genus.
Some more Cayley diagrams
and other pages on groups
Copyright N.S.Wedd 2009