Consider the complete graph on 6 vertices. There are, up to color permutation, 6 ways of coloring its edges in 5 colors such that each vertex has exactly one edge of each color, one of which is shown below:
It turns out that every permutation of the 6 vertices permutes the 6 edge-colorings in a unique way, leading to an isomorphism between the group of permutations of the 6 vertices and the group of permutations of the edge-colorings.
These groups are, of course, both S6. Rotating the above image by 60 degrees maps every vertex to a different vertex but it maps the shown coloring to itself, meaning that this isomorphism from S6 to S6 is not just a relabeling of the six objects. This means that it is an outer automorphism.
A much more elegant construction of the exceptional outer automorphism of S6 involves this arrangement of the numbers 1 to 6:
The single swap of 6 and another element X is mapped to a triple swap that includes the swap of 6 and X, as well as two other swaps between the pairs of elements equally far away from X around the circle. For example, the swap (16) maps to the triple-swap (16)(25)(34). All permutations of six objects can be generated by swaps of the form (X6), and it can be confirmed that all equations involving said swaps are preserved.
Leave a Reply