John Carlos Baez on Nostr: HARDCORE MATH POST The McGee graph, animated here by Mamuka Jibladze, is the unique ...

HARDCORE MATH POST

The McGee graph, animated here by Mamuka Jibladze, is the unique graph where each vertex has 3 neighbors and the shortest cycles have length 7. As it moves, each vertex comes back to its original position after two turns, so this illustrates an 8-fold symmetry of the McGee graph.

The symmetry group of the McGee graph has order 32, and Greg Egan and I figured out a bunch of what's going on here:

https://blogs.ams.org/visualinsight/2015/09/15/mcgee-graph/

thought I'd still like a simpler construction of the McGee graph based on this group.

This group, let's call it the 'McGree group', is the semidirect product of ℤ/8 with its automorphism group. Alternatively, it's the group of affine transformations of ℤ/8, i.e. transformations

x ↦ a x + b

where a,b ∈ ℤ/8 and a is invertible, so a = 1,3,5,7. That gives 32 elements.

I gave a talk about this today and someone instantly pointed out that the McGee group is the smallest group G with an outer automorphism f: G → G that maps each element of G to an element in the same conjugacy class! So for each g∈G we have

f(g) = hgh⁻¹

for some h∈G, but we can't use the same h for all g! You could call this a 'pseudo-inner automorphism'.

𝐏𝐮𝐳𝐳𝐥𝐞: Explicitly describe a pseudo-inner automorphism of the McGee group.

There should be a nice description in terms of ℤ/8, and there should also be a visually nice description of how to turn any symmetry of the McGee graph into a new symmetry that is conjugate to g, but not always via the same element h!