A Cayley Graph of PSL(2,7)

The smallest non-Abelian simple group is A5 or A1(5) or A1(4) (these are all isomorphic). It has 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 A1(7)) or GL(3,2) (also known as A2(2)) these being isomorphic. It has 168 elements. Its Cayley graph 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:

Cayley graph of PSL(2,7)

Explanation of pink parts of the diagram

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

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

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 A1(7). But there isn't really anything special about this subgroup: it is only one of seven conjugate S4s within A1(7).

It is based on 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.

Some more Cayley diagrams
Some more pages on groups

Copyright N.S.Wedd 2009