Greg Egan on Nostr: I think I’ve found a class-preserving outer automorphism of Aff(ℤ/8)! You map: x ...

I think I’ve found a class-preserving outer automorphism of Aff(ℤ/8)!

You map:

x ↦ a x + b

to:

x ↦ a x + (b + D(a))

where

D(1) = D(3) = 0
D(5) = D(7) = 4

I don’t think this is unique. But I’m still searching through the (4!)^8 choices of permutations of the odd and even parts of ℤ/8, for each value of a.

Conjugation preserves the parity of b, so rather than there being 8! choices of permutations of ℤ/8 for each value of a, there are (4!)^2.